diff --git "a/4dE4T4oBgHgl3EQfbgxF/content/tmp_files/load_file.txt" "b/4dE4T4oBgHgl3EQfbgxF/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/4dE4T4oBgHgl3EQfbgxF/content/tmp_files/load_file.txt" @@ -0,0 +1,1448 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf,len=1447 +page_content='Gradient TRIX CHRISTOPH LENZEN, CISPA Helmholtz Center for Information Security, Germany SHREYAS SRINIVAS, CISPA Helmholtz Center for Information Security, Germany and Saarbrucken Graduate School for Computer Science, Saarland University, Germany Gradient clock synchronization (GCS) algorithms minimize the worst-case clock offset between the nodes in a distributed network of diameter 𝐷 and size 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' They achieve optimal offsets of Θ(log 𝐷) locally, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' between adjacent nodes [18], and Θ(𝐷) globally [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As demonstrated in [3], this is a highly promising approach for improved clocking schemes for large-scale synchronous Systems-on-Chip (SoC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, in large systems, faults hinder their practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' State of the art fault-tolerant GCS [4] has a drawback that is fatal in this setting: It relies on node and edge replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑓 = 1, this translates to at least 16-fold edge replication and high degree nodes, far from the optimum of 2𝑓 + 1 = 3 for tolerating up to 𝑓 faulty neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this work, we present a self-stabilizing GCS algorithm for a grid-like directed graph with optimal node in- and out-degrees of 3 that tolerates 1 faulty in-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If nodes fail with independent probability 𝑝 ∈ π‘œ(π‘›βˆ’1/2), it achieves asymptotically optimal local skew of Θ(log 𝐷) with probability 1 βˆ’ π‘œ(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' this holds under general worst-case assumptions on link delay and clock speed variations, provided they change slowly relative to the speed of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The failure probability is the largest possible ensuring that with probabity 1 βˆ’ π‘œ(1) for each node at most one in-neighbor fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As modern hardware is clocked at gigahertz speeds and the algorithm can simultaneously sustain a constant number of arbitrary changes due to faults in each clock cycle, this results in sufficient robustness to dramatically increase the size of reliable synchronously clocked SoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' CCS Concepts: β€’ Hardware β†’ Very large scale integration design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Very large scale integration design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Computing methodologies β†’ Distributed algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Additional Key Words and Phrases: Clock Synchronisation, Fault Tolerance, VLSI, Self-Stabilisation 1 INTRODUCTION In their seminal work from 2004 [7], Fan and Lynch introduced the task of Gradient Clock Synchro- nization (GCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In a network of nodes that synchronize their clocks, it requires to minimize the worst-case clock offset between neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Two key insights motivate minimizing this local skew: In many applications the skew between adjacent nodes is the appropriate measure of quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The global skew, the maximum clock offset between any pair of nodes in the network, grows linearly with the diameter 𝐷 of the network [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Defying the intuition of many, Fan and Lynch proved a lower bound of Ξ©(log 𝐷/log log 𝐷) on the local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Follow-up work then established that this bound was very close to the mark: the best local skew that can be achieved is Θ(log 𝐷) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This exponential gap between global and local skew strongly suggests better scalability of systems employing this approach to synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Yet, more than a decade after these results have been published, we know of no efforts to apply these techniques in products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is not for want of demand!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To drive this point home, consider the case of clocking synchronous hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Conceptually speaking, state of the art hardware that operates synchronously distributes a clock signal from a single source using a tree network, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [9, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, for any tree spanning a square grid, there will be adjacent grid points whose distance in the tree is proportional to the side length of the grid [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, the worst-case local skew on a computer chip clocked by a clock tree must grow linearly with the side length of the chip [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Indeed, these theoretical results are reflected in the reality of hardware suppliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Modern systems gave up on maintaining globally synchronous operation, instead communicating Authors’ addresses: Christoph Lenzen, lenzen@cispa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='de, CISPA Helmholtz Center for Information Security, SaarbrΓΌcken, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Shreyas Srinivas, shreyas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='srinivas@cispa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='de, CISPA Helmholtz Center for Information Security, SaarbrΓΌcken, Germany and Saarbrucken Graduate School for Computer Science, Saarland University, SaarbrΓΌcken, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='05073v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='DC] 12 Jan 2023 2 Christoph Lenzen and Shreyas Srinivas asynchronously between multiple clock islands [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This comes at a steep cost, both in terms of communication latency [12] and ease of design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' So, which obstacle prevents application?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' At least in the above setting, neither large hidden constants nor an overly complex algorithm get in the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On the contrary, recent work demon- strates that implementation effort is easily managable and pays off already for moderately-sized systems [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Instead, the main obstacle are faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To see that this is the key issue, recall that today’s hardware comprises an enourmous number of individual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Recent off-the-shelf hardware has transistor counts beyond the 10 billion mark [22], requiring either incredibly low fault rates or some degree of fault-tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In a system composed of multiple clock islands that interact asynchronously, these islands are canonical choices for fault-containment regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, one can get away with using a clocking scheme in each island that cannot sustain faults, interpreting a fault of the clocking subsystem as a fault of the respective island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In contrast, when the clocking subsystems of the islands interact with each other via a clock synchronization algorithm, we must ensure that a clock fault in a single island does not bring down the entire system!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Fault-Tolerant Clocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' When clocking hardware, high connectivity networks are not scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This limits the number of concurrent faults that can be sustained, as tolerating up to 𝑓 faults requires a node connectivity of 2𝑓 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In [4], this bound is matched asymptotically by augmenting an arbitrary network such that the GCS algorithm from [18] is simulated in a fault-tolerant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Here, augmentation means to replace each node in the original network by a clique of size 3𝑓 + 1 and each edge by a biclique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The clique then synchronizes internally using the classic Lynch-Welch algorithm [10], and the resulting local outputs are interpreted as (an approximation of) a joint cluster clock on which the (non-tolerant) GCS algorithm from [18] is simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, this approach is impractical due to the large overhead in terms of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Leaving asymptotics aside – the edge overhead compared to the original graph is Θ(𝑓 2) rather than 𝑂(𝑓 ) – even for the important special case of 𝑓 = 1 node degrees will be at least 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is a far cry from the simplicity of current distribution techniques, and factor 5 beyond the minimum node degree of 2𝑓 + 1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' What might look like a β€œmoderate constant” to a theoretician will not only cause a headache to the engineer trying to route all of these edges with few layers and precise timing, it will also substantially increase communication delay uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This, in turn, directly translates into an increased skew, placing the break-even point with prior art beyond relevant limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In summary, it is essential to get as close as possible to the minimum required connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This train of thought led to the study of fault-tolerant clock distribution in low-degree networks [6, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Both of these works have in common that they assume that the clock signal is generated at a central location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This enables these approaches to achieve self-stabilization and tolerance to isolated faults with very simple pulse forwarding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The basic idea is to propagate the signal from layer to layer, having each node wait for two nodes signaling a clock pulse before locally generating and forwarding their own pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, it is assumed that in absence of faults delays are changing only slowly over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, matching the input frequency to the expected delay between grid layers results in clock pulses that are well-synchronized between adjacent layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The above works differ in the grid structure they use (Figure 1) and the skew bounds they provide: Denoting by 𝑑 βˆ’ 𝑒 and 𝑑 the minimum and maximum end-to-end communication delay, in a grid of width 𝐷 [6] bounds the local skew by 𝑑 + 𝑂(𝑒2𝐷/𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since in practice 𝑑 ≫ 𝑒, this is a non-trivial bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, the fact that 𝑑 ≫ 𝑒 also means that this bound is too large for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Even worse, for each fault this bound increases by 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In [20], each fault adds at most 𝑒 to the local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observe that the used grid also has the minimum required connectivity, as each node has only 3 incoming and outgoing edges each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 3 0 𝑒 2𝑒 𝑑 𝑑 𝑑 π‘‘βˆ’π‘’ π‘‘βˆ’π‘’ π‘‘βˆ’π‘’ 𝑑 𝑑 𝑑 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' TRIX [20] (top) and HEX [6] (bottom) grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' TRIX uses the naive pulse forwarding scheme of waiting for the second copy of each pulse before forwarding it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We see how the TRIX grid can accumulate a skew of Θ(𝑒𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In the HEX grid, each node waits for two copies of a pulse from in-neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, 2 of the 4 in-neighbors are on the same layer, causing a skew of 𝑑 if a neighbor on the preceding layer crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Alas, these advantages come at the expense of poor scaling of worst-case skews with the number of layers: on layer β„“, adjacent nodes may pulse up to 𝑒ℓ time apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that in order to tolerate failure of an arbitrary component, also the clock source has to be replicated and the replicas to be synchronized in a fault-tolerant and self-stabilizing manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, here one can employ techniques for fully connected networks [11, 19];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' using them in a single location for 𝑓 = 1 does not constitute a scalability issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In light of the above, in this work we ask the question β€œCan a small local skew be achieved in a fault-tolerant way at minimal connectivity?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Our Contribution We provide a positive answer to the above question for the special case of 𝑓 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is achieved by using the same grid as in [20], but with a different rule for forwarding pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Our novel algorithm is designed as a discrete and fault-tolerant counterpart to the GCS algorithm from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Making this work requires substantial conceptual innovation and technical novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On the conceptual level, our algorithm simulates a discretized variant of the (non-fault-tolerant!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=') GCS algorithm from [18] on an arbitrary base graph of minimum degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In more detail, each copy of the graph, referred to as layer, represents a β€œtime step” of the GCS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For each node, there is an edge from its copy on a given layer to the copies of itself and its neighbors on the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The forwarded pulses along these edges serve two very different functions: The pulse messages sent to copies of neigbhors correspond to the GCS algorithm’s messages for estimating clock offsets to neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The pulse messages sent between copies of the same node convey its local time from one of its copies to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that this turns a permanently faulty node in the grid into a simulated node being faulty in a single time step only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is of vital importance, because it enables us to rely on the self-stabilization properties of the GCS algorithm from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' These are implicitly shown in [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' we prove them explicitly in the different setting of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, by itself this does not guarantee bounded skew between correct nodes, since we also need to contain the effect of such a β€œtransient” fault on the state of the simulated algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 4 Christoph Lenzen and Shreyas Srinivas Otherwise, a fault would increase skews arbitrarily, effectively corrupting downstream nodes: at any given node, the smallest or largest time at which a pulse from neighbors on the preceding layer is received could be determined by a faulty node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We can overcome this issue if there is at most one faulty in-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The key observation to controlling the impact of a faulty node on the pulse time lies in that it can indeed affect only one of three times: the smallest or largest time at which a pulse from copies of neighbors on the previous layer is received, or the time at which the pulse from the copy of the node itself is received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, the median of these three times lies within the interval spanned by the correct in-neighbors’ pulse times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By imposing the additional constraint to always tie the time at which a pulse is generated closely to this median, we can limit the local impact of a fault on skews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In summary, we seek to simultaneously simulate a time-discrete variant of the GCS algorithm from [18], while also guaranteeing that pulse forwarding times are, up to a sufficiently small deviation, identical to median reception times plus a fixed offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, no existing GCS algorithm that achieves a small local skew [14, 15, 17, 18] can be used for this purpose as-is, since their decision rules are in conflict with the above β€œstick to the median” requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As our main technical contribution, we resolve this conflict, simultaneously adapting the resulting algorithm to the discrete setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To do so, we determine suitably weakened discrete variants of the slow and fast conditions introduced in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In essence, we allow that a simulated node whose pulse time is ahead all of its neighbors’ pulse times to delay its next pulse by the difference to the fastest neighbor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' an analogous rule applies to nodes pulsing later than all of their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From the perspective of the GCS algorithm in [18], this constitutes a potentially arbitrarily large clock β€œjump,” which we leverage to implement the stick-to-the-median requirement despite the arbitrary changes in timing faulty nodes may apply to their pulse messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To prevent uncontrolled oscillatory behavior arising from adjacent nodes β€œjumping” in opposite directions, we introduce an additional condition, which we refer to as jump condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Essentially, it slightly reduces how large jumps are to avoid that uncertainty in message delays and local clock speeds cause nodes to β€œoverswing,” potentially resulting in arbitrarily large skews, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Turning so many knobs at once meant that it was not clear that such a scheme would work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Indeed, bounding the skew of this novel algorithm turned out to be highly challenging, as jumps that delay pulses rather than speeding them up invalidate the fundamental assumption that clocks progress at rate at least 1 present in all prior work [14, 15, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As a result, the main technical hurdle and contribution turned out to be proving a bound on the local skew Lβ„“ between neighbors in the same layer β„“ for the fault-free case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If there are no faults, then Lβ„“ ≀ 4πœ…(2 + log 𝐷) for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Here, choosing the input clock frequency to be 1/(2𝑑) results in πœ… ∈ Θ(𝑒 + (πœ— βˆ’ 1)𝑑), where it is assumed that local clocks run at rates between 1 and πœ— > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' All of our results require that 𝑑 ≫ 𝑒 + (πœ— βˆ’ 1)𝑑, or equivalently, that the local skew remains small compared to 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that if this condition does not hold, we are outside the parameter range of interest: then skews become large compared to the length of a clock cycle under ideal conditions and clock frequency has to be reduced substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To address faults, we bound by how much faults can affect timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Due to the aforementioned stick to the median rule, we can bound the local impact of a fault on timing in terms of the local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, applying this argument repeatedly, skews would grow exponentially in the number of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' While tolerating a constant number of faults is certainly better than tolerating none, this is unsatisfactory, since the requirement of one faulty in-neighbor holds with probability 1 βˆ’ π‘œ(1) for a fairly high independent probability of 𝑝 ∈ π‘œ(1/βˆšπ‘›).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Given that the topology we are most Gradient TRIX 5 interested in is roughly a square grid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', there are roughly βˆšπ‘› layers, the naive approach outlined above does not result in a non-trivial bound on the skew unless 𝑝 is very close to 1/𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We provide an improved analysis exploiting that our base graph is almost a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, the 𝑑-hop neighborhood grows linearly with 𝑑 and hence the number of nodes in layers β„“β€² ∈ [β„“ βˆ’ 𝑛1/12, β„“] that affect the pulse time of a node in layer β„“ is in Θ(𝑛1/6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, if nodes fail with probability 𝑝 ∈ π‘œ(1/βˆšπ‘›), the probability that there are more than 2 faulty nodes within distance 𝑛1/12 that affect a given node is π‘œ(1/𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Intuitively, this buys enough time for the self-stabilization properties of the simulated algorithm to reduce its local skew again before it spirals out of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' With probability 1 βˆ’ π‘œ(1), Lβ„“ ∈ 𝑂(πœ… log 𝐷) for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The final step is to extend this bound on the local skew within a layer to one that includes adjacent nodes in different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As we propagate pulses layer by layer, we cannot hope to match pulse times of the π‘˜-th pulse between different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Instead, we match the input period to the nominal time a pulse spends on each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This works neatly so long as there are no changes in message delay, clock speed, and behavior of faulty nodes between consecutive pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If faulty nodes do not change the timing of their output pulses, then L ∈ 𝑂(πœ… log 𝐷) with probability 1 βˆ’ π‘œ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To a large extent, this strong assumption is justified in our specific context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Clock speeds of modern systems are in the gigahertz range, and the amount of change in timing that occurs within a single clock cycle is much smaller than over the lifetime of a system [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Similarly, the by far most common timing faults are stuck-at faults, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the signal observed by downstream nodes remains constant logical 0 or 1, and broken connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From the point of view of the receiving node, this is equivalent to an early or late pulse, respectively, without any change between pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Of course, timing will still change slowly, the above benign faults will occur at some point, before which the nodes worked correctly, and some faults may be more severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using once more that faulty nodes’ impact on timing is bounded by the local skew, the bound from Theorem 7 extends to a constant number of arbitrary faults in each pulse alongside small changes in delays and hardware clock speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' With probability 1βˆ’π‘œ(1), L ∈ 𝑂(πœ… log 𝐷) even when in each pulse (i) a constant number of faulty nodes change their output behavior and timing, (ii) link delays vary by up to π‘›βˆ’1/2𝑒 log 𝐷, and (iii) hardware clock speeds vary by up to π‘›βˆ’1/2(πœ— βˆ’ 1) log 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Finally, if all else fails, we can fall back on the ability of the pulse progation algorithm to recover from arbitrary transient faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In constrast to the simulated GCS algorithm, achieving self-stabilization of the pulse propagation scheme itself is straightforward due to the directionality of the propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The pulse propagation algorithm can be implemented in a self-stabilizing way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It stabilizes within 𝑂(βˆšπ‘›) pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In light of these results, we view this work as a major step towards simultaneously achieving high performance and strong robustness in the practical setting of clock distribution in hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In alignment with the theoretical question motivating this work, we achieve an asymptotically optimal local skew at the minimum possible node degree under the assumption of node failures with probability π‘œ(π‘›βˆ’1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Organization of this Article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Section 2, we discuss the system model, introduce the graph on which we run our synchronization algorithm, and motivate our modeling choices, including its non- standard aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We then present a simplified version of the algorithm that better highlights the 6 Christoph Lenzen and Shreyas Srinivas conceptual approach in Section 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' the full algorithm and its equivalence without faulty predecessors is shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We follow with the formal derivation of the skew bounds in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 2 MODELING The model we use is non-standard, as it is tailored to the specific setting outlined in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Accordingly, we will emphasize and discuss model choices where this seems prudent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Recall that our goal is to provide a synchronized clock signal to a large System-on-Chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Physically, this means that we need to provide the clock signal to a rectangular area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' for simplicity, we will assume it to be square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We want to supply a uniform grid of nodes in the square area with this signal, which then will serve as roots of relatively small local clock trees supplying the low-level components with the clock signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If these trees contribute a maximum clock skew of Ξ” and the skew between adjacent grid points is at most L, the triangle inequality guarantees a worst-case skew of L + 2Ξ” between adjacent components of the System-on-Chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The local clock trees can be designed using standard methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, in the following we will focus exclusively on the grid of their roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A key assumption we make is that communication delay between correct adjacent nodes changes only slowly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This enables us to generate synchronized pulses at all grid nodes by matching the input frequency with the (inverse) propagation time between consecutive layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is justified for two reasons: The dominant sources of uncertainty in propagation delay are inaccuracies in component fabrication, aging, and temperature and frequency variations that are slow relative to the time it takes to propagate an input clock pulse across even a large System-on-Chip [23] Changing delays of all links between a pair of adjacent layers by up to 𝛿 increases skew bounds by at most 𝛿, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In order to generate sufficiently synchronized pulses at the nodes of layer 0, a straightforward solution is to use a redundant path, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', a path of 3-cliques in which adjacent cliques are fully bipartitely connected, to propagate pulses from the clock reference along an edge of the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As we show in Corollary 6, this results in input pulses of small enough local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For each clique, one of the nodes will be the layer-0 node providing its output pulse to close-by nodes of layer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In a perfect grid, all layers would consist of a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, this results in the issue that the endpoints of the path, lacking one neighbor, would have only two adjacent nodes in the preceding and subsequent layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A naive solution is to insert a additional edges between the boundary nodes, turning the layer into a cycle and the entire graph into a cylinder (with some special treatment of layer 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, realizing such a solution on the square would result in far too long edges between boundary nodes or require to, essentially, replicate each layer, effectively doubling the number of nodes and edges in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Instead, we choose to replicate the boundary nodes only, which then provides the β€œmissing” input to the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that this increases the degree of the nodes next to the boundary nodes by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We cope with this by a general analysis allowing for the layers to be copies of an arbitrary base graph of minimum degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Figure 2 and Figure 3, we show the base graph and the connectivity of nodes between adjacent layers of our synchronization network in our assumed setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Network Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We are given a simple connected base graph 𝐻 = (𝑉, 𝐸) of minimum degree 2 and diameter 𝐷 ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑣,𝑀 ∈ 𝑉 , denote by 𝑑(𝑣,𝑀) ≀ 𝐷 the distance from 𝑣 to 𝑀 in 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To derive the graph 𝐺 = (𝑉𝐺, 𝐸𝐺) we use for synchronization, for each β„“ ∈ N we create a copy 𝑉ℓ of 𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denoting by (𝑣, β„“) the copy of 𝑣 ∈ 𝑉 in 𝑉ℓ, we define 𝐸ℓ := {((𝑣, β„“), (𝑀, β„“ + 1)) | {𝑣,𝑀} ∈ 𝐸 ∨ 𝑣 = 𝑀}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We now obtain 𝐺 by setting 𝑉𝐺 := οΏ½ β„“ ∈N 𝑉ℓ and 𝐸𝐺 := οΏ½ β„“ ∈N 𝐸ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' That is, for each layer β„“ ∈ N we have a copy Gradient TRIX 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Base graph 𝐻 used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Rather than using a cycle, which would result in a TRIX grid, we replicate the end nodes of a line to ensure a minimum degree of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Alternatively, one could use a line and exploit that the probability that one of the 𝑂(βˆšπ‘›) boundary nodes fails is π‘œ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Layer structure of 𝐺 resulting from our choice of 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Most nodes have in- and out-degree 3, some 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' of 𝑣 ∈ 𝑉 , which has outgoing edges to the copies of itself and all its neighbors on layer β„“ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='1 Sine 𝑉𝐺 is a DAG, we refer to out-neighbors as successors and in-neighbors as predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Fault Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' An unknown subset 𝐹 βŠ‚ 𝑉𝐺 is Byzantine faulty, meaning that these nodes may violate the protocol arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Edge faults are mapped to node faults, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', if edge ((𝑣, β„“), (𝑀, β„“ +1)) is faulty, we instead consider (𝑣, β„“) faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We impose the constraint that no node has two faulty predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Formally, for all β„“ ∈ N and 𝑣 ∈ 𝑉 , |({(𝑣, β„“)} βˆͺ οΏ½ {𝑣,𝑀}∈𝐸{(𝑀, β„“)}) ∩ 𝐹 | ≀ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' When analyzing the system under random faults, we will assume that each node fails independently with probability 𝑝 ∈ π‘œ(1/βˆšπ‘›), which ensures that the above constraint is met with probability 1 βˆ’ π‘œ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In addition, we impose the restriction that at most a constant number of such faulty nodes change their timing behavior between consecutive pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Each node has the ability to broadcast pulse messages on its outgoing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If node 𝑣ℓ ∈ 𝑉ℓ broadcasts at time 𝑑𝑣,β„“, its successors receive its message at a (potentially different) time from [𝑑𝑣,β„“ + 𝑑 βˆ’ 𝑒,𝑑𝑣,β„“ + 𝑑].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The maximum end-to-end delay 𝑑 includes any delay caused by computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Typically, the delay uncertainty 𝑒 is much smaller than 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As discussed above, we assume delays to be static, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', each edge 𝑒 = ((𝑣, β„“), (𝑀, β„“ +1)) has an unknown, but fixed associated delay 𝛿𝑒 ∈ [𝑑 βˆ’ 𝑒,𝑑] applied to each pulse sent from (𝑣, β„“) to (𝑀, β„“ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that faulty nodes can send pulses at arbitrary times, without being required to broadcast;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' even if physical node implementations disallow point-to-point communication, edge faults could still result in this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Local Clocks and Computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Each node is able to approximately measure the progress of time by means of a local time reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We model this by node (𝑣, β„“) having query access to a hardware clock 𝐻𝑣,β„“ : Rβ‰₯0 β†’ Rβ‰₯0 satisfying βˆ€π‘‘ < 𝑑 β€² ∈ Rβ‰₯0, 𝑑 β€² βˆ’ 𝑑 ≀ 𝐻𝑣,β„“ (𝑑 β€²) βˆ’ 𝐻𝑣,β„“ (𝑑) ≀ πœ—(𝑑 β€² βˆ’ 𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' for some πœ— > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' No known phase relation is assumed between the hardware clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The algorithm will use them exclusively to measure how much time passes between local events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Analogous to delays, we assume that hardware clock speeds are static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is justified in the same way as for delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 1This is an abuse of notation, since in a (roughly) square grid of 𝑛 := |𝑉𝐺 | nodes, we have Θ(βˆšπ‘›) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝑛, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the size of the grid, will only play a role when making probabilistic statements, we opted for this more convenient notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 8 Christoph Lenzen and Shreyas Srinivas Computations are deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, in addition to receiving a message, the hardware clock reaching a time value previously determined by the algorithm can also trigger computations and possibly broadcasting a pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Output and Skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The goal of the algorithm is to synchronize the pulses generated by correct nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We assume that correct nodes on layer 0 generate well-synchronized pulses at times π‘‘π‘˜ 𝑣,0 for π‘˜ ∈ N>0 at a frequency we control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Appendix A, we discuss how to realize this assumption in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' All other correct nodes generate pulses π‘‘π‘˜ 𝑣,β„“, π‘˜ ∈ N>0, based on the pulse messages received from their predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Our measure of quality is the worst-case local skew the algorithm guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We define the local skew as the largest offset between the π‘˜-th pulses of adjacent nodes on the same layer or pulses π‘˜ and π‘˜ + 1 of adjacent nodes on layers β„“ and β„“ + 1, whichever is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Formally, for β„“ ∈ N we define Lβ„“ := sup π‘˜ ∈N max {𝑣,𝑀}∈𝐸 (𝑣,β„“),(𝑀,β„“)βˆ‰πΉ {|π‘‘π‘˜ 𝑣,β„“ βˆ’ π‘‘π‘˜ 𝑀,β„“|}, Lβ„“,β„“+1 := sup π‘˜ ∈N max ((𝑣,β„“),(𝑀,β„“+1)) βˆˆπΈβ„“ (𝑣,β„“),(𝑀,β„“+1)βˆ‰πΉ {|π‘‘π‘˜ 𝑣,β„“ βˆ’ π‘‘π‘˜+1 𝑀,β„“+1|}, and L := supβ„“ ∈N max{Lβ„“, Lβ„“,β„“+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This deviates from the standard definition of the local skew: The definition is adjusted to pulse synchronization, which can be viewed as an essentially equivalent time-discrete variant of clock synchronization [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Between consecutive layers, we synchronize consecutive pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' After initialization, which is complete once the first pulse propagated through the (in practice finite) grid, this is equivalent to a layer-dependent index shift of pulse numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 3 ALGORITHM In this section, we discuss the pulse forwarding algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We provide a simplified version of the algorithm that behaves identical so long as the predecessors of the executing node are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The full algorithm needs to handle the possibility that faulty nodes send multiple messages or none at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This complicates bookkeeping and loop control, distracting from the principles underlying the algorithm’s operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Accordingly, we defer the full algorithm to Appendix B, where we show the equivalence to the simplified variant when there are no faulty predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='1 Simplified Pulse Forwarding Algorithm The algorithm proceeds in iterations corresponding to pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In each iteration, node (𝑣, β„“) (1) timestamps the arrival times of the pulses of its predecessors using its hardware clock, (2) determines a correction value C𝑣,β„“ based on these timestamps, and (3) forwards the pulse Ξ› βˆ’ 𝑑 βˆ’ C𝑣,β„“ time after receiving the pulse from π‘£β„“βˆ’1, measured by its hardware clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If all reception times are close to each other, then C𝑣,β„“ will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Recalling that messages are in transit for roughly 𝑑 time, this translates to Ξ› being the nominal time for a pulse to propagate from layer β„“ βˆ’ 1 to layer β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We need to choose Ξ› large enough such that the above sequence can be always realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' That is, we need to consider how far apart the reception times of messages from the previous layer can be, and ensure that Ξ› βˆ’ 𝑑 exceeds this value plus the resulting C𝑣,β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assuming that this precondition holds, Algorithm 1 implements the above approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In each loop iteration, it initializes three reception times to ∞: 𝐻own, which stores the arrival time of the pulse from (𝑣, β„“ βˆ’ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From the perspective of the simulated GCS algorithm, this reflects the state of the node 𝑣 ∈ 𝑉 simulated by (𝑣, β„“), β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻min, which stores the minimum arrival time of a pulse from a neighbor π‘€β„“βˆ’1, 𝑀 β‰  𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This corresponds to the first pulse received from a neighbor 𝑀 of 𝑣 in 𝐺 in this iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 9 Algorithm 1 Simplified pseudocode for discrete GCS at node (𝑣, β„“), β„“ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As shown in Lemma 29, this code is equivalent to Algorithm 3 in the absence of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The parameters Ξ› and πœ… will be determined later, based on the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' loop 𝐻own,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻max := ∞ do if received pulse from (𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β„“ βˆ’ 1) then 𝐻own := 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ (𝑑) if received pulse from first (𝑀,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β„“ βˆ’ 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' {𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝑀} ∈ 𝐸 then 𝐻min := 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ (𝑑) if received pulse from last (𝑀,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β„“ βˆ’ 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' {𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝑀} ∈ 𝐸 then 𝐻max := 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ (𝑑) until 𝐻own,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻max < ∞ C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ := min𝑠 ∈N{max{𝐻own βˆ’ 𝐻max + 4π‘ πœ…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ…}} βˆ’ πœ…/2 if C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ < 0 then C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ := min{𝐻own βˆ’ 𝐻min βˆ’ πœ…/2 + 2πœ…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 0} else if C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ > πœ—πœ… then C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ := max{𝐻own βˆ’ 𝐻max βˆ’ πœ…/2 βˆ’ πœ…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='πœ—πœ…} wait until 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ (𝑑) = 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ broadcast pulse 𝐻max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' which stores the maximum arrival time of a pulse from a neighbor π‘€β„“βˆ’1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑀 β‰  𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This corresponds to the last pulse received from a neighbor 𝑀 of 𝑣 in 𝐺 in this iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The do-until loop fills these variables with the correct values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' At the heart of the algorithm lies the computation of 𝐢𝑣,β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If there were no faults, one could always compute Ξ” := min 𝑠 ∈N {max{𝐻own βˆ’ 𝐻max + 4π‘ πœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ…}} βˆ’ πœ… 2 and then choose the closest value from the range [0,πœ—πœ…], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', set C𝑣,β„“ := \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 Ξ” if Ξ” ∈ [0,πœ—πœ…], 0 if Ξ” < 0, and πœ—πœ… if Ξ” > πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To get intuition on this choice, observe that minπ‘₯ ∈R{max{𝐻own βˆ’ 𝐻min βˆ’ π‘₯, 𝐻own βˆ’ 𝐻max + π‘₯}} is attained when 𝐻own βˆ’ 𝐻max + π‘₯ = 𝐻own βˆ’ 𝐻min βˆ’ π‘₯, which is equivalent to π‘₯ = (𝐻max βˆ’ 𝐻min)/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', 𝐻own βˆ’ Ξ” = (𝐻max βˆ’ 𝐻min)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If timing was perfectly accurate, the reception times of the pulse messages could serve as exact proxies for the actual pulse forwarding times of the nodes on layer β„“ βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In iteration π‘˜, this would mean to generate the pulse at (𝑣, β„“) faster if (𝑣, β„“ βˆ’ 1) generated its pulse later than the average of min{𝑣,𝑀}∈𝐸{π‘‘π‘˜ 𝑀,β„“βˆ’1} and max{𝑣,𝑀}∈𝐸{π‘‘π‘˜ 𝑀,β„“βˆ’1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, any (𝑣, β„“) for which π‘‘π‘˜ 𝑣,β„“βˆ’1 βˆ’ min{𝑣,𝑀}∈𝐸{π‘‘π‘˜ 𝑀,β„“βˆ’1} > max{𝑣,𝑀}∈𝐸{π‘‘π‘˜ 𝑀,β„“βˆ’1} βˆ’ π‘‘π‘˜ 𝑣,β„“βˆ’1 would choose 𝐢𝑣,β„“ > 0, attempting to reduce max{𝑣,𝑀}∈𝐸{|π‘‘π‘˜ 𝑣,β„“ βˆ’ π‘‘π‘˜ 𝑀,β„“|} compared to max{𝑣,𝑀}∈𝐸{|π‘‘π‘˜ 𝑣,β„“βˆ’1 βˆ’ π‘‘π‘˜ 𝑀,β„“βˆ’1|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This can be viewed as trying to reduce the local skew by a greedy strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, this naive strategy fails to account for inaccuracies due to message delay uncer- tainty and drifting hardware clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Nonetheless, we follow this strategy up to deviations of 𝑂(πœ…).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The additional terms serve the following purposes: 10 Christoph Lenzen and Shreyas Srinivas Considering only discrete choices for π‘₯ ∈ 4πœ…N rather than arbitrary π‘₯ ∈ R is the key ingredient that makes the algorithmic approach succeed, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Essentially, this is necessary because there is no way to determine π‘‘π‘£β„“βˆ’1,π‘˜ βˆ’ π‘‘π‘€β„“βˆ’1,π‘˜ precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Discretizing observed skews in units of πœ… ∈ Θ(𝑒 + (πœ— βˆ’ 1)(Ξ› βˆ’ 𝑑)) enables a delicate strategy that alternates between overestimating skews to locally generate the next pulse earlier for the sake of β€œcatching up” with others and underestimating skews to β€œwait” for others catch up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Substracting πœ…/2 accounts for errors in measuring skews, which are caused by uncertainty in message delay and hardware clock speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To limit the damage that a faulty predecessor of (𝑣, β„“) can do, we ensure that (𝑣, β„“) generates its pulse without too large of a deviation from the median of 𝑑𝑣,β„“βˆ’1, min{𝑣,𝑀}∈𝐸{π‘‘π‘˜ 𝑀,β„“βˆ’1}, and max{𝑣,𝑀}∈𝐸{π‘‘π‘˜ 𝑀,β„“βˆ’1} (plus the nominal offset of Ξ›).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is achieved by permitting corrections 𝐢𝑣,β„“ < 0 if (𝑣, β„“ βˆ’ 1) clearly generated its pulse earlier than min{𝑣,𝑀}∈𝐸{π‘‘π‘˜ 𝑀,β„“βˆ’1} and 𝐢𝑣,β„“ > πœ—πœ… if it clearly generated its pulse later than max{𝑣,𝑀}∈𝐸{π‘‘π‘˜ 𝑀,β„“βˆ’1}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To further motivate the last point, recall that there can be at most one fault among the predecessors of (𝑣, β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A single faulty predecessor can only affect only one of the three values 𝐻own, 𝐻min, and 𝐻max: control 𝐻own arbitrarily, 𝐻min to be smaller than the minimum reception time from a correct node (𝑀, β„“ βˆ’ 1), {𝑣,𝑀} ∈ 𝐸, or 𝐻max to exceed the maximum reception time from correct nodes (𝑀, β„“ βˆ’ 1), {𝑣,𝑀} ∈ 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, ensuring that pulses are generated with only a small offset relative to median {𝐻own, 𝐻min, 𝐻max} + Ξ› βˆ’ 𝑑 indeed limits the damage that a fault can do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Achieving all of the desired properties is non-trivial, leading to the fairly involved choice of C𝑣,β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It can be viewed as simultaneously implementing relaxed fast and slow conditions (as introduced in [15]), an additional jump condition required to make the GCS algorithm work under these relaxed fast and slow conditions, and the requirement to stick close to the median of predecessors’ pulse times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='2, we specify the (relaxed) slow and fast condition, as well as the jump condition, and show that the algorithm implements them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemmas 19 and 20 show that the algorithm also enforces deviates little from the time interval spanned by correct predecessors (offset by Ξ›).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' There is some freedom in the choice of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For simplicity, we fix a good choice of πœ… and note that 𝑑 must satisfy a lower bound 𝐡 ∈ 𝑂(supβ„“ ∈N{Lβ„“} +πœ…).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observe that this constraint simply means that the skew bounds are useful, as a skew that is of similar size as the maximum end-to-end delay requires to slow the system down substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Finally, Ξ› must be at least 𝑑 +𝑂(supβ„“ ∈N{Lβ„“}), which due to the previous constraint holds e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' for the choice Ξ› = 2𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Formally, for a sufficiently large constant 𝐢,2 πœ… := 2 οΏ½ 𝑒 + οΏ½ 1 βˆ’ 1 πœ— οΏ½ (Ξ› βˆ’ 𝑑) οΏ½ , (1) Ξ› β‰₯ πΆπœ—(sup β„“ ∈N {Lβ„“} + 𝑒) + 𝑑, and (2) 𝑑 β‰₯ 𝐢(πœ—(sup β„“ ∈N {Lβ„“} + 𝑒) + πœ…).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (3) Complete Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The complete algorithm cannot wait for messages from all predecessors to determine when to send its pulse, as a faulty node not sending its pulse then would deadlock all its descendants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As discussed above, the hardware clock time of the next pulse time does not deviate much from median {𝐻own, 𝐻min, 𝐻max} + Ξ› βˆ’π‘‘, but does depend on max{𝐻min, 𝐻own, 𝐻max} in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we will prove that Lβ„“βˆ’1 is small enough such that all pulse messages from correct nodes will be received in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, it is sufficient to wait until median {𝐻own, 𝐻min, 𝐻max}+πœ—Lβ„“βˆ’1 (or later) according to 𝐻𝑣,β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Provided that Ξ› βˆ’ 𝑑 is large enough, this implies that any message for 2We do not attempt to optimize constants in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 11 computing C𝑣,β„“ missing is due to a fault;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' in fact, at the point in time when this becomes clear, C𝑣,β„“ is already determined, regardless of how late the message would arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The complete algorithm differs from Algorithm 1 by covering the case that a signal does not arrive in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Intuitively, one can treat the respective message arrival time (𝐻own or 𝐻max, 𝐻min is not possible) as ∞, while allowing such an ∞ to cancel out in substraction: If 𝐻own = ∞, then 𝐢𝑣,β„“ ∈ 𝐻own βˆ’ 𝐻max βˆ’ 𝑂(πœ…), and (𝑣, β„“) will generate its pulse at local time 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ 𝐢𝑣,β„“ ∈ 𝐻max + Ξ› βˆ’ 𝑑 + 𝑂(πœ…).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝐻max = ∞ and 𝐻own β‰₯ 𝐻min, then 𝐢𝑣,β„“ ∈ 𝐻own βˆ’ 𝐻min Β± Θ(πœ…) and (𝑣, β„“) will generate its pulse at local time 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ 𝐢𝑣,β„“ ∈ 𝐻min + Ξ› βˆ’ 𝑑 Β± 𝑂(πœ…).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝐻max = ∞ and 𝐻own < 𝐻min, then 𝐢𝑣,β„“ ∈ [0, 2πœ…] and (𝑣, β„“) will generate its pulse at local time 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ 𝐢𝑣,β„“ ∈ 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ 𝑂(πœ…).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that in all cases, the pulse is generated with an offset of Ξ›βˆ’π‘‘ βˆ’Ξ˜(πœ…) from the median reception time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The complete algorithm follows the above intuition, leveraging the fact that there is no need to wait indefinitely to determine that the third signal is late, and is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Last, but not least, it is of interest to make the pulse forwarding algorithm self-stabilizing [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Due to the design choice of propagating the clock signal from a single source along a DAG, this will immediately translate to the overall scheme being self-stabilizing, so long as the clock generation is self-stabilizing, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is straightforward, because one can assume that the signals from the previous layer are already well-synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, all that nodes need to do is to detect when all but possibly one (faulty) pulse signal arrive in close temporal proximity to determine when to clear their memory and start a new iteration of the main loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='1, we sketch how this can be achieved using standard techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 4 ANALYSIS We now analyze the pulse progagation scheme under the assumption that layer 0 generates well- synchronized pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We discuss a suitable method for achieving this in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Our analysis proceeds along the following lines: (1) We show that, if the local skew is small enough compared to Ξ›, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', Equation (2) holds, all correct nodes execute their iterations as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' That is, each correct node on layer β„“ > 0 receives the π‘˜-th pulses of its correct predecessors in its π‘˜-th loop iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is deferred to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We then proceed under the assumption that this holds true, which will be justified retroactively once we establish that the local skew is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (2) Since delays and hardware clock speeds are (approximated as being) static, any (substantial) change in relative timing of consecutive pulses is due to faulty nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, the task of bounding the local skew reduces to bounding the intra-layer skew Lβ„“ for a single pulse, since such a bound must take into account the full variability introduced by faulty nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This reasoning is deferred to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (3) Based on potential functions, we analyze Lβ„“ in the absence of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The results entail not only bounded skew, but also that the potentials recover if they become unexpectedly large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (4) We show that faulty nodes have limited impact on the potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From this and the above recovery property, we conclude that skews behave favorably also when there are faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As stated above, the first two steps of our line of reasoning are deferred to the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The main challenge is to bound Lβ„“ for a single pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Due to the first step, we know that the π‘˜-th pulse at correct nodes depends only on the π‘˜-th pulses of their predecessors (Lemma 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, in the following fix π‘˜ and denote the π‘˜-th pulse time of correct (𝑣, β„“) ∈ 𝑉𝐺 by 𝑑𝑣,β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Recall that for 𝑣,𝑀 ∈ 𝑉 , we denote by 𝑑(𝑣,𝑀) their distance in the base graph 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Our analysis is built around the following potential functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 12 Christoph Lenzen and Shreyas Srinivas Definition 1 (Potential Functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let 𝑣,𝑀 ∈ 𝑉 and 𝑠, β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We define πœ“π‘  𝑣,𝑀(β„“) := 𝑑𝑣,β„“ βˆ’ 𝑑𝑀,β„“ βˆ’ 4π‘ πœ…π‘‘(𝑣,𝑀), Ψ𝑠 (β„“) := max 𝑣,π‘€βˆˆπ‘‰{πœ“π‘  𝑣,𝑀(β„“)}, πœ‰π‘  𝑣,𝑀(β„“) := 𝑑𝑣,β„“ βˆ’ 𝑑𝑀,β„“ βˆ’ (4𝑠 βˆ’ 2)πœ…π‘‘(𝑣,𝑀), and Ξžπ‘  (β„“) := max 𝑣,π‘€βˆˆπ‘‰{Ξžπ‘  𝑣,𝑀(β„“)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Bounding Ψ𝑠 (β„“) readily translates to bounding Lβ„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If for 𝑠, β„“ ∈ N and some Ψ𝑠 ∈ Rβ‰₯0 it holds that Ψ𝑠 (β„“) ≀ Ψ𝑠, then Lβ„“ ≀ Ψ𝑠 + 4π‘ πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Fix π‘˜ ∈ N and suppose that {𝑣,𝑀} ∈ 𝐸 maximizes |𝑑𝑣,β„“ βˆ’ 𝑑𝑀,β„“|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', assume that 𝑑𝑣,β„“ β‰₯ 𝑑𝑀,β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since {𝑣,𝑀} ∈ 𝐸, we have that 𝑑(𝑣,𝑀) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, |𝑑𝑣,β„“ βˆ’ 𝑑𝑀,β„“| = 𝑑𝑣,β„“ βˆ’ 𝑑𝑀,β„“ = πœ“π‘  𝑣,𝑀(β„“) + 4π‘ πœ… ≀ Ψ𝑠 (β„“) + 4π‘ πœ… ≀ Ψ𝑠 + 4π‘ πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since π‘˜ ∈ N is arbitrary, it follows that Lβ„“ ≀ Ψ𝑠 + 4π‘ πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ In summary, the goal of our analysis will be to bound Ψ𝑠 (β„“) by a small value for some 𝑠 satisfying 4π‘ πœ… ∈ 𝑂(𝑒 log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We first study the behavior of the algorithm if there are no faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Accordingly, this will be tacitly assumed in all statements of this section, with the expection of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that by Lemma 29, this means that we may also tacitly assume that Algorithm 1 is run by all nodes in layers β„“ ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='4, we will then bound the impact of faulty layers on the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='1 Basic Statements We first show three basic lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The first relates the local reception times of pulses to the actual sending times, bounding the error by πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For (𝑣, β„“) ∈ 𝑉ℓ, β„“ ∈ N>0, set 𝑑min := min{𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} and 𝑑max := min{𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max βˆ’ πœ… ≀ 𝐻own βˆ’ 𝐻max βˆ’ πœ… 2 ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑min βˆ’ πœ… ≀ 𝐻own βˆ’ 𝐻min βˆ’ πœ… 2 ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We prove the first inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' the second is shown analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let 𝑑 β€² 𝑣,β„“βˆ’1 and 𝑑 β€² max denote the times when the pulse messages sent at time 𝑑𝑣,β„“βˆ’1 and 𝑑max are received at 𝑣ℓ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From the bounds on message delays, it follows that 𝑑𝑣,β„“βˆ’1 + 𝑑 βˆ’ 𝑒 ≀ 𝑑 β€² 𝑣,β„“βˆ’1 ≀ 𝑑𝑣,β„“βˆ’1 + 𝑑 and 𝑑max + 𝑑 βˆ’ 𝑒 ≀ 𝑑 β€² max ≀ 𝑑max + 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max βˆ’ 𝑒 ≀ 𝑑 β€² 𝑣,β„“βˆ’1 βˆ’ 𝑑 β€² max ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max + 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using the bounds on hardware clock rates, we get that |𝑑 β€² 𝑣,β„“βˆ’1 βˆ’ 𝑑 β€² max βˆ’ (𝐻own βˆ’ 𝐻max)| ≀ (πœ— βˆ’ 1)|𝑑 β€² 𝑣,β„“βˆ’1 βˆ’ 𝑑 β€² max| ≀ (πœ— βˆ’ 1)(|𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max| + 𝑒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 13 Applying Equation (2), we infer that |𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max βˆ’ (𝐻own βˆ’ 𝐻max)| ≀ |𝑑 β€² 𝑣,β„“βˆ’1 βˆ’ 𝑑 β€² max βˆ’ (𝐻own βˆ’ 𝐻max)| + 𝑒 ≀ (πœ— βˆ’ 1)|𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max| + πœ—π‘’ ≀ (πœ— βˆ’ 1)Lβ„“βˆ’1 + πœ—π‘’ ≀ (πœ— βˆ’ 1) οΏ½Ξ› βˆ’ 𝑑 πœ— βˆ’ 𝑒 οΏ½ + πœ—π‘’ = οΏ½ 1 βˆ’ 1 πœ— οΏ½ (Ξ› βˆ’ 𝑑) + 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Finally, using Equation (1), we conclude that 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max βˆ’ πœ… ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max βˆ’ 2 οΏ½οΏ½ 1 βˆ’ 1 πœ— οΏ½ (Ξ› βˆ’ 𝑑) + 𝑒 οΏ½ ≀ 𝐻own βˆ’ 𝐻max βˆ’ πœ… 2 ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ The second lemma shows that corrections are not too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all 𝑣 ∈ 𝑉 and β„“ ∈ N>0, C𝑣,β„“ ≀ Ξ› βˆ’ 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Abbreviate Ξ” = min 𝑠 ∈N {max{𝐻own βˆ’ 𝐻max + 4π‘ πœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ…}} βˆ’ πœ… 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Ξ” < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Algorithm 1 sets C𝑣,β„“ ≀ min οΏ½ 𝐻own βˆ’ 𝐻min βˆ’ πœ… 2 + 2πœ…, 0 οΏ½ ≀ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As Ξ› β‰₯ 𝑑 by Equation (2), the claim of the lemma holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 0 ≀ Ξ” ≀ πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then, using the notation of Lemma 1, C𝑣,β„“ = Ξ” < min 𝑠 ∈N {max{𝐻own βˆ’ 𝐻max + 4π‘ πœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ…}} βˆ’ πœ… 2 ≀ min 𝑠 ∈N οΏ½ max{𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max + 4π‘ πœ…,𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑min βˆ’ 4π‘ πœ…} οΏ½ ≀ min 𝑠 ∈N {max{Lβ„“βˆ’1 + 4π‘ πœ…, Lβ„“βˆ’1 βˆ’ 4π‘ πœ…}} = Lβ„“βˆ’1, which is smaller than Ξ› βˆ’ 𝑑 by Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Ξ” > πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that then πœ—πœ… < Ξ” ≀ max{𝐻own βˆ’ 𝐻max, 𝐻own βˆ’ 𝐻min} βˆ’ πœ… 2 = 𝐻own βˆ’ 𝐻max βˆ’ πœ… 2, 14 Christoph Lenzen and Shreyas Srinivas as 𝐻max β‰₯ 𝐻min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, applying Lemma 2, C𝑣,β„“ = max οΏ½ 𝐻own βˆ’ 𝐻max βˆ’ πœ… 2 βˆ’ πœ…,πœ—πœ… οΏ½ ≀ 𝐻own βˆ’ 𝐻max βˆ’ πœ… 2 ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max ≀ Lβ„“βˆ’1 < Ξ› βˆ’ 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ The third lemma bounds the time difference between the pulses of (𝑣, β„“ βˆ’ 1) and (𝑣, β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all 𝑣 ∈ 𝑉 and β„“ ∈ N>0 it holds that 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 βˆ’ C𝑣,β„“ πœ— ≀ 𝑑𝑣,β„“ βˆ’ 𝑑𝑣,β„“βˆ’1 ≀ Ξ› βˆ’ C𝑣,β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let 𝑑 β€² 𝑣,β„“βˆ’1 denote the time at which (𝑣, β„“) receives the pulse sent by (𝑣, β„“ βˆ’ 1) at time 𝑑𝑣,β„“βˆ’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Inspecting the code of Algorithm 1, we see that 𝐻𝑣,β„“ (𝑑𝑣,β„“) = 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ C𝑣,β„“ = 𝐻𝑣,β„“ (𝑑 β€² 𝑣,β„“βˆ’1) + Ξ› βˆ’ 𝑑 βˆ’ C𝑣,β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since C𝑣,β„“ ≀ Ξ› βˆ’π‘‘ by Lemma 2, it follows that 𝐻𝑣,β„“ (𝑑 β€² 𝑣,β„“βˆ’1) β‰₯ 𝐻𝑣,β„“ (𝑑𝑣,β„“) and hence 𝑑𝑣,β„“ β‰₯ 𝑑 β€² 𝑣,β„“βˆ’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using the bounds on message delays and hardware clock speeds, we get that 𝑑𝑣,β„“ βˆ’ 𝑑𝑣,β„“βˆ’1 = 𝑑𝑣,β„“ βˆ’ 𝑑 β€² 𝑣,β„“βˆ’1 + 𝑑 β€² 𝑣,β„“βˆ’1 βˆ’ 𝑑𝑣,β„“βˆ’1 ≀ 𝐻𝑣,β„“ (𝑑𝑣,β„“) βˆ’ 𝐻𝑣,β„“ (𝑑 β€² 𝑣,β„“βˆ’1) + 𝑑 = Ξ› βˆ’ C𝑣,β„“ and 𝑑𝑣,β„“ βˆ’ 𝑑𝑣,β„“βˆ’1 = 𝑑𝑣,β„“ βˆ’ 𝑑 β€² 𝑣,β„“βˆ’1 + 𝑑 β€² 𝑣,β„“βˆ’1 βˆ’ 𝑑𝑣,β„“βˆ’1 β‰₯ 𝐻𝑣,β„“ (𝑑𝑣,β„“) βˆ’ 𝐻𝑣,β„“ (𝑑 β€² 𝑣,β„“βˆ’1) πœ— + 𝑑 βˆ’ 𝑒 = Ξ› βˆ’ 𝑑 βˆ’ C𝑣,β„“ πœ— + 𝑑 βˆ’ 𝑒, which can be rearranged into the claimed inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='2 The Slow, Fast, and Jump Conditions The key to bounding the local skew without faults is to find the right balance between two conflicting goals: choosing C𝑣,β„“ large enough to β€œcatch up” to predecessors π‘€β„“βˆ’1 β‰  π‘£β„“βˆ’1 that generated their pulse earlier than π‘£β„“βˆ’1, but small enough to β€œwait” for predecessors π‘€β„“βˆ’1 β‰  π‘£β„“βˆ’1 that generated their pulse later than π‘£β„“βˆ’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The following condition captures what we need regarding the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Definition 2 (Slow Condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all 𝑠 ∈ N, correct layers β„“ βˆ’ 1 ∈ N, and 𝑣ℓ ∈ 𝑉ℓ \\ 𝐹, we require the slow condition SC(𝑠) := SC-1(𝑠) ∨ SC-2(𝑠) ∨ SC-3 to hold, where SC-1(𝑠) : C𝑣,β„“ πœ— ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ max {𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} + 4π‘ πœ… SC-2(𝑠) : C𝑣,β„“ πœ— ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ min {𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} βˆ’ 4π‘ πœ… SC-3: C𝑣,β„“ ≀ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 15 This can be viewed as a variant of the slow condition from [15], adjusted to our setting by quantifying by how much 𝑣ℓ may safely shift the timing of its pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The main conceptual difference to [15] is that we relax the slow condition by adding SC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In what follows, we drop 𝑠 from the notation when it is clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all 𝑠 ∈ N and (𝑣, β„“) ∈ 𝑉ℓ, β„“ ∈ N>0, SC(𝑠) holds at (𝑣, β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using Lemma 29, we prove the claim for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Set 𝑑min := min{𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} and 𝑑max := min{𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If C𝑣,β„“ ≀ 0, SC-3 is trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, assume that C𝑣,β„“ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Abbreviate Ξ” = min 𝑠 ∈N {max{𝐻own βˆ’ 𝐻max + 4π‘ πœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ…}} βˆ’ πœ… 2 = max{𝐻own βˆ’ 𝐻max + 4𝑠minπœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4𝑠minπœ…} βˆ’ πœ… 2, where 𝑠min ∈ N is an index for which the minimum is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If Ξ” ≀ πœ—πœ…, then C𝑣,β„“ = Ξ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Otherwise, C𝑣,β„“ = max οΏ½ 𝐻own βˆ’ 𝐻max βˆ’ πœ… 2 βˆ’ πœ…,πœ—πœ… οΏ½ ≀ max{Ξ”,πœ—πœ…} = Ξ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Either way, we get that C𝑣,β„“/πœ— < C𝑣,β„“ ≀ Ξ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻own βˆ’ 𝐻max + 4𝑠minπœ… β‰₯ 𝐻own βˆ’ 𝐻min βˆ’ 4𝑠minπœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then for 𝑠 ∈ N, 𝑠 β‰₯ 𝑠min, by Lemma 1 we have that Ξ” ≀ 𝐻own βˆ’ 𝐻max + 4𝑠minπœ… βˆ’ πœ… 2 ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max + 4π‘ πœ…, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', SC-1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Now consider 𝑠 ∈ N, 𝑠 < 𝑠min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝐻own βˆ’π»max βˆ’πœ—π‘’ + 4π‘ πœ… < 𝐻own βˆ’π»max βˆ’ πœ—π‘’ + 4𝑠minπœ… ≀ Ξ”, but the minimum is attained at index 𝑠min, we must have that Ξ” ≀ 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ… βˆ’ πœ… 2 ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑min βˆ’ 4π‘ πœ…, where the second step again applies Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, SC-2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻own βˆ’ 𝐻max + 4𝑠minπœ… < 𝐻own βˆ’ 𝐻min βˆ’ 4𝑠minπœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, we analogously infer that SC-1 holds for 𝑠 > 𝑠min and SC-2 holds for 𝑠 ≀ 𝑠min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ The fast condition is the counterpart to Definition 2 addressing the need to β€œcatch up” to neighbors that are ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Definition 3 (Fast Condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all 𝑠 ∈ N>0, correct layers β„“ βˆ’ 1 ∈ N>0, and 𝑣ℓ ∈ 𝑉ℓ \\ 𝐹, we require the fast condition FC(𝑠) := FC-1(𝑠) ∨ FC-2(𝑠) ∨ FC-3 to hold, where FC-1(𝑠) : C𝑣,β„“ β‰₯ 𝑑𝑣,β„“βˆ’1 βˆ’ max {𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} + (4𝑠 βˆ’ 2)πœ… + πœ… FC-2(𝑠) : C𝑣,β„“ β‰₯ 𝑑𝑣,β„“βˆ’1 βˆ’ min {𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} βˆ’ (4𝑠 βˆ’ 2)πœ… + πœ… FC-3: C𝑣,β„“ β‰₯ πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This can be viewed as a variant of the fast condition from [15], adjusted to our setting by quantifying by how much 𝑣ℓ may safely shift the timing of its pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The main conceptual difference to [15] is that we relax the fast condition by adding FC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In addition, note that there is an additive term of πœ… that does not change sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Its purpose is to account for the fact that our simulation of the GCS algorithm from [18] operates in discrete time steps corresponding to the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The continuous versions of the GCS algorithm in [14, 15, 18] can choose this term arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In contrast, we need it to exceed the maximum error in time measurement accumulated in a step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We remark that, in principle, one could choose this term 16 Christoph Lenzen and Shreyas Srinivas 𝑣 𝑑𝑣 βˆ’ 4π‘ πœ…π‘‘(𝑣,𝑀) 𝑀 𝑑𝑀 βˆ’ (4𝑠 βˆ’ 2)πœ…π‘‘(𝑣,𝑀) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Slow condition (left) and fast condition (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' SC(𝑠) is tailored to ensuring that maxπ‘€βˆˆπ‘‰ {πœ“π‘ π‘£,𝑀(β„“)} (the length of the green arrow) cannot grow quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Nodes 𝑀 with C𝑀,β„“ ≀ 0 (SC-3 holds) cannot apply a correction pushing them below the red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If C𝑀,β„“ > 0, then both SC-1 and SC-2 will ensure that there is a neighbor π‘₯ of 𝑀 such that the offset of 𝑑𝑀,β„“βˆ’1 βˆ’ C𝑀,β„“/πœ— to the black line does not exceed the one of 𝑑π‘₯,β„“βˆ’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In other words, SC ensures that the blue arrows indicating C𝑀,β„“/πœ— do not reach below the red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This means that any increase of maxπ‘€βˆˆπ‘‰ {πœ“π‘ π‘£,𝑀(β„“)} is caused by delay and clock speed variation, which in turn is bounded by πœ…/2 per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Similarly, FC(𝑠) is tailored to ensuring that maxπ‘£βˆˆπ‘‰ {πœ‰π‘ π‘£,𝑀(β„“)} (the length of the green arrow), if positive, decreases by at least πœ…/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To ensure this, C𝑀,β„“ (indicated by blue arrows) must be large enough to reach below the red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is achieved by FC(𝑠) having an additional β€œslack” term of πœ…, which overcomes the β€œloss” of πœ…/2 due to uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' different from πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, since both need to meet the same lower bound of 𝑒 + (1 βˆ’ 1/πœ—)(Ξ› βˆ’ 𝑑), there is no asymptotic gain in introducing a separate parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all 𝑠 ∈ N and (𝑣, β„“) ∈ 𝑉ℓ, β„“ ∈ N>0, FC(𝑠) holds at (𝑣, β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using Lemma 29, we prove the claim for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Set 𝑑min := min{𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} and 𝑑max := min{𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If C𝑣,β„“ β‰₯ πœ—πœ…, trivially FC-3 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, assume that C𝑣,β„“ < πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Abbreviate Ξ” = min 𝑠 ∈N {max{𝐻own βˆ’ 𝐻max + 4π‘ πœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ…}} βˆ’ πœ… 2 = max{𝐻own βˆ’ 𝐻max + 4𝑠minπœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4𝑠minπœ…} βˆ’ πœ… 2, where 𝑠min ∈ N is an index for which the minimum is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If Ξ” β‰₯ 0, then C𝑣,β„“ = Ξ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Otherwise, C𝑣,β„“ = min οΏ½ 𝐻own βˆ’ 𝐻min βˆ’ πœ… 2 + 2πœ…, 0 οΏ½ β‰₯ Ξ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Either way, we get that C𝑣,β„“ β‰₯ Ξ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑠 ∈ N, 𝑠 ≀ 𝑠min, by Lemma 1 and Equation (1) it holds that Ξ” β‰₯ 𝐻own βˆ’ 𝐻max + 4π‘ πœ… βˆ’ πœ… 2 β‰₯ 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max + (4𝑠 βˆ’ 2)πœ… + πœ…, proving that FC-1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑠 ∈ N, 𝑠 > 𝑠min, by Lemma 1 and Equation (1) we get that Ξ” β‰₯ 𝐻own βˆ’ 𝐻min βˆ’ 4(𝑠 βˆ’ 1)πœ… βˆ’ πœ… 2 β‰₯ 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑min βˆ’ (4𝑠 βˆ’ 2)πœ… + πœ…, showing that FC-2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Our relaxation of the slow and fast conditions adds a substantial complication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From the per- spective of the time-continuous variant of the algorithm in [15], we now allow for arbitrarily large clock β€œjumps,” rather than bounded clock rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In our discrete version, the rate bound from [15] corresponds to C𝑣,β„“ ∈ [0,πœ—πœ…].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Without this additional constraint, the slow and fast conditions are insufficient to bound skews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 17 𝑣ℓ+2 𝑣ℓ+1 𝑣ℓ 𝑣ℓ+2 𝑣ℓ+1 𝑣ℓ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On the left, it is shown how skews increase without JC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' While SC(0) disallows that (𝑣, β„“) speeds up its pulse by more than the equivalent of (𝑣, β„“ βˆ’ 1) matching the earliest pulse of any (𝑀, β„“ βˆ’ 1), {𝑣,𝑀} ∈ 𝐸, FC permits that a node (𝑣, β„“) with slow (𝑣, β„“ βˆ’ 1) to β€œovershoot,” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', C𝑣,β„“ (shown as blue arrow) gets large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This results in an amplifying oscillatory behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On the right, the same scenario is shown with JC in effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' JC forces the corrections to stop πœ… before the earliest or latest neighbor, respectively, resulting in a dampened oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is illustrated in Figure 5, showing an execution that satisfies SC and FC, but suffers from skews that grow without bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The key issue is that adjacent nodes could β€œjump” in opposite directions, resulting in an oscillatory behavior in which measurement errors accumulate indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To avoid this kind of behavior, we add an additional condition that β€œdampens” such oscillations, yet limits by how much a faulty predecessor can cause an increase in skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Definition 4 (Jump Condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all correct layers β„“ βˆ’ 1 ∈ N>0 and 𝑣ℓ ∈ 𝑉ℓ \\ 𝐹, we require the jump condition JC := JC-1 ∨ JC-2 ∨ JC-3 to hold, where JC-1: πœ… < C𝑣,β„“ πœ— ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ max {𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} βˆ’ πœ… JC-2: 0 > C𝑣,β„“ β‰₯ 𝑑𝑣,β„“βˆ’1 βˆ’ min {𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} + πœ… JC-3: 0 ≀ C𝑣,β„“ πœ— ≀ πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that layer β„“ βˆ’ 1 ∈ N and 𝑣ℓ ∈ 𝑉ℓ are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then JC holds at 𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using Lemma 29, we prove the claim for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Set 𝑑min := min{𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} and 𝑑max := min{𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 0 ≀ C𝑣,β„“ ≀ πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then JC-3 is satisfied trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 18 Christoph Lenzen and Shreyas Srinivas C𝑣,β„“ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 1 and Equation (1), then C𝑣,β„“ = 𝐻own βˆ’ 𝐻min βˆ’ πœ… 2 + 2πœ… β‰₯ 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑min + πœ…, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', JC-2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' C𝑣,β„“ > πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 1, then C𝑣,β„“ = 𝐻own βˆ’ 𝐻max βˆ’ πœ… 2 βˆ’ πœ… ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑max βˆ’ πœ…, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', JC-3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='3 Bounding Ψ𝑠 in the Absence of Faults With the conditions established, we are ready to study how Ψ𝑠 (β„“) evolves in the fault-free setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The main technical challenge in bounding Ψ𝑠 lies in performing the induction step from 𝑠 βˆ’ 1 ∈ N to 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We will argue that for Ψ𝑠 ( Β―β„“ ) to be large for some Β―β„“, Ξžπ‘  (β„“ ) must have been large for some β„“ < Β―β„“, with an additive term growing with Β―β„“ βˆ’ β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑠 ∈ N>0 and layers β„“ ≀ Β―β„“, it holds that Ψ𝑠 ( Β―β„“ ) ≀ max οΏ½ 0, Ξžπ‘  (β„“ ) βˆ’ ( Β―β„“ βˆ’ β„“ + 1)πœ… οΏ½ + ( Β―β„“ βˆ’ β„“ ) Β· πœ… 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Intuitively, we intend to argue that if Ψ𝑠 ( Β―β„“ ) is large, so must be Ξžπ‘  (β„“ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Tracing back the cause for this, we show that in every step, we have that Ξžπ‘  (β„“ βˆ’ 1) is larger than Ξžπ‘  (β„“) by at least πœ…/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since Ξžπ‘  ( Β―β„“ ) β‰₯ Ψ𝑠 ( Β―β„“ ), as πœ“π‘  𝑣,𝑀(β„“) β‰₯ πœ‰π‘  𝑣,𝑀(β„“) for all 𝑣, 𝑀, 𝑠, and β„“, this yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To formalize that Ξžπ‘  (β„“) must have been decreasing steadily, we seek to show that the minimal layer β„“ for which there are nodes 𝑣ℓ,𝑀ℓ ∈ 𝑉 satisfying that πœ‰π‘  𝑣ℓ,𝑀ℓ (β„“) is large enough is β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To this end, we identify nodes 𝑀 and 𝑣 – either 𝑀ℓ and 𝑣ℓ themselves or neighbors of them – which cause the large skew on layer β„“ by a having a large skew on layer β„“ βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is done based on SC(𝑠) and FC(𝑠), with JC kicking in for the special case that 𝑀 = 𝑣ℓ and 𝑣 = 𝑀ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A key obstacle is that if 𝑀 is a neighbor of 𝑀ℓ, this results in a larger difference in skew than if 𝑣 is a neighbor of 𝑣ℓ, namely 4π‘ πœ… versus (4𝑠 βˆ’ 2)πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, when 𝑀 is closer to 𝑣ℓ than 𝑀ℓ, we β€œlose” 2πœ… relative to the skew bound on layer β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑑(𝑣 Β―β„“,𝑀 Β―β„“) many steps, we can compensate for this based on the initial skew between 𝑣 Β―β„“ and 𝑀 Β―β„“, but not more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To address this, essentially we need to show that for any additional steps β€œtowards” 𝑣ℓ there will be a corresponding step β€œaway” from 𝑣ℓ, on which we β€œgain” additional 2πœ… relative to the skew bound on the layer β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If corrections were always positive, this would be straightforward: Steps towards 𝑣ℓ would also be steps towards 𝑣 Β―β„“, and upon 𝑀ℓ = 𝑣 Β―β„“ we would reach a contradiction to the skew bounds shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, negative corrections foreclose this simple argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To address this, we introduce a third β€œprover” node 𝑝ℓ, where 𝑝 Β―β„“ = 𝑣 Β―β„“, which never increases its distance to 𝑀ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' if 𝑝ℓ performs a negative correction, then 𝑝 is a neighbor of 𝑝ℓ that is closer to 𝑀ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We then can infer that 𝑝 β‰  𝑀 from the skew bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A major complication this approach faces is the special case 𝑝 = 𝑀ℓ and 𝑀 = 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Again, JC kicks in to show that we have sufficiently large skew between 𝑝 and 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, now 𝑝 lies β€œbehind” 𝑀 from the perspective of 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A later reversal of this situation by repeating the case that 𝑝 = 𝑀ℓ and 𝑀 = 𝑝ℓ results in 𝑀 being farther away from 𝑣ℓ, yet 𝑑(𝑝,𝑀) = 𝑑(𝑝ℓ,𝑀ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The proof covers this case by adding an additional (4𝑠 βˆ’ 2)πœ… to the skew bound if the above situation occured an odd number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Finally, we seek to avoid the case that 𝑣 = 𝑝ℓ and 𝑝 = 𝑣ℓ for analogous reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Fortunately, here we can exploit that the skew bound between 𝑣ℓ and 𝑀ℓ is stronger than the one between 𝑝ℓ and 𝑀ℓ, meaning that we can simply choose 𝑝 = 𝑣 instead in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In the proof, we do so Gradient TRIX 19 whenever 𝑣 lies on the path connecting 𝑝ℓ and 𝑀ℓ that we maintain to keep track of hop counts in the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume towards a contradiction that the statement of Theorem 1 is false for minimal Β―β„“, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', there are 𝑣 Β―β„“ and 𝑀 Β―β„“ such that πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ > ( Β―β„“ βˆ’ β„“ ) Β· πœ… 2 (4) and πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ > Ξžπ‘  (β„“ ) βˆ’ ( Β―β„“ βˆ’ β„“ ) Β· πœ… 2 βˆ’ πœ… (5) and there is no smaller Β―β„“β€² for which this applies for some pair of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let β„“ ∈ [β„“, Β―β„“] be minimal such that are 𝑣ℓ, 𝑝ℓ,𝑀ℓ ∈ 𝑉 , a path 𝑄ℓ in 𝐻 from 𝑝ℓ to 𝑣ℓ, and a path 𝑃ℓ in 𝐻 from 𝑝ℓ to 𝑀ℓ with the following properties: (P1) 𝑀ℓ β‰  𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P2) 𝑀ℓ β‰  𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P3) 𝑑𝑝ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ βˆ’ 4π‘ πœ…|𝑃ℓ| β‰₯ πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) βˆ’ ( Β―β„“ βˆ’ β„“) Β· πœ… 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P4) Denote by |𝑃ℓ| and |𝑄ℓ| the length of 𝑃ℓ and 𝑄ℓ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' With the shorthand Ξ”β„“ := οΏ½ |𝑃ℓ| + |𝑄ℓ| βˆ’ 1 if 𝑃ℓ and 𝑄ℓ have the same first edge |𝑃ℓ| + |𝑄ℓ| else, it holds that 𝑑𝑣ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ βˆ’ (4𝑠 βˆ’ 2)πœ…Ξ”β„“ β‰₯ πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) + ( Β―β„“ βˆ’ β„“) Β· πœ… 2 + 2πœ…|𝑃ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P5) If 𝑣ℓ ∈ 𝑃ℓ, then 𝑝ℓ = 𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To see that such an index must indeed exist, let 𝑝 Β―β„“ := 𝑣 Β―β„“, 𝑃 Β―β„“ be a shortest path in 𝐻 from 𝑝 Β―β„“ to 𝑀 Β―β„“, and 𝑄 Β―β„“ := (𝑝 Β―β„“) = (𝑣 Β―β„“), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the 0-length path from 𝑝 Β―β„“ to 𝑣 Β―β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This choice satisfies (P1) and (P2), because Ψ𝑠 𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) β‰  0 implies that 𝑣 Β―β„“ β‰  𝑀 Β―β„“;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P4), because 𝑑𝑣 Β―β„“,Β―β„“ βˆ’ 𝑑𝑀 Β―β„“,Β―β„“ βˆ’ (4𝑠 βˆ’ 2)πœ…Ξ”β„“ = 𝑑𝑣 Β―β„“,Β―β„“ βˆ’ 𝑑𝑀 Β―β„“,Β―β„“ βˆ’ (4𝑠 βˆ’ 2)πœ…|𝑃 Β―β„“| = πœ“π‘£ Β―β„“,𝑀 Β―β„“ + 2πœ…|𝑃 Β―β„“|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' and (P3) and (P5), because 𝑝 Β―β„“ = 𝑣 Β―β„“ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', 𝑑𝑝 Β―β„“,Β―β„“ = 𝑑𝑣 Β―β„“,Β―β„“ and Δ¯ℓ = |𝑃 Β―β„“|) and (P4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Corollary 2 proves that in fact β„“ = β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that 𝑑(𝑣ℓ,𝑀ℓ) ≀ οΏ½ |𝑃ℓ| + |𝑄ℓ| βˆ’ 2 if 𝑃ℓ and 𝑄ℓ share the first edge |𝑃ℓ| + |𝑄ℓ| else ≀ Ξ”β„“ 20 Christoph Lenzen and Shreyas Srinivas and that |𝑃ℓ| β‰₯ 1 due to (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, (P4) yields that Ξžπ‘  (β„“ ) β‰₯ 𝑑𝑣ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ βˆ’ (4𝑠 βˆ’ 2)πœ…π‘‘(𝑣ℓ,𝑀ℓ) β‰₯ 𝑑𝑣ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ βˆ’ (4𝑠 βˆ’ 2)πœ…Ξ”β„“ β‰₯ πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) + ( Β―β„“ βˆ’ β„“) Β· πœ… 2 + 2πœ…|𝑃ℓ| β‰₯ πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) + ( Β―β„“ βˆ’ β„“) Β· πœ… 2 + 2πœ…, contradicting Equation (5) and completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ The remainder of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='3 is dedicated to proving Corollary 2, which is the missing step in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To this end, until the end of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='3 we consider the setting of the proof of Theorem 1 and assume for contradiction that β„“ > β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We take note of some straightforward implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For any fixed index β„“, we have the following implications: (P3) β‡’ (P1) (P4) β‡’ (P2) (𝑣ℓ = π‘β„“βˆ§ (P4)) β‡’ (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, πœ“π‘  𝑣ℓ,𝑀ℓ (β„“) βˆ’ ( Β―β„“ βˆ’ β„“) Β· πœ… 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We prove each implication separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From (P3), 𝑑𝑝ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ > 4π‘ πœ…|𝑃ℓ| β‰₯ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This implies 𝑑𝑝ℓ,β„“ > 𝑑𝑀ℓ,β„“ and hence 𝑀ℓ β‰  𝑝ℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that Ξ”β„“ β‰₯ 0, |𝑃ℓ| β‰₯ 0, and 4𝑠 βˆ’ 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, (P4) and Equation (4) imply that 𝑑𝑣ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ β‰₯ πœ“π‘  𝑣ℓ,𝑀ℓ ( Β―β„“ ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It follows that 𝑀ℓ β‰  𝑣ℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑣ℓ = 𝑝ℓ, then 𝑑𝑣ℓ,β„“ = 𝑑𝑝ℓ,β„“, |𝑄ℓ| = 0, and Ξ”β„“ = |𝑃ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, (P4) implies that 𝑑𝑝ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ βˆ’ (4𝑠 βˆ’ 2)πœ…|𝑃ℓ| β‰₯ πœ“π‘  𝑣ℓ,𝑀ℓ (¯𝑙) + ( Β―β„“ βˆ’ β„“) Β· πœ… 2 + 2πœ…|𝑃ℓ| β‰₯ πœ“π‘  𝑣ℓ,𝑀ℓ (¯𝑙) βˆ’ ( Β―β„“ βˆ’ β„“) Β· πœ… 2 + 2πœ…|𝑃ℓ|, which can be rearranged to yield (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ A Step in the Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We now identify nodes that are suitable for taking the role of 𝑣ℓ, 𝑝ℓ, and 𝑀ℓ on layer β„“ βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' These are either the nodes themselves or neighbors of them in 𝐻, where FC(𝑠), SC(𝑠), and JC serve to relate respective pulse times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' There is a node 𝑣 ∈ 𝑉 such that 𝑑𝑣ℓ,β„“βˆ’1 βˆ’ C𝑣ℓ,β„“ ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ (4𝑠 βˆ’ 2)πœ…Ξ”π‘£ βˆ’ πœ…, where Δ𝑣 = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 0 and 𝑣 = 𝑣ℓ, βˆ’1 and {𝑣, 𝑣ℓ} is the last edge of 𝑄ℓ or the first edge of 𝑃ℓ, or 1 and {𝑣ℓ, 𝑣} ∈ 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 5, 𝑣ℓ obeys the fast condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus one of three things is true for 𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' FC-1(𝑠) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, let 𝑣 = arg max{π‘₯,𝑣ℓ }∈𝐸{𝑑π‘₯,β„“βˆ’1} and bound 𝑑𝑣ℓ,β„“βˆ’1 βˆ’ C𝑣ℓ,β„“ ≀ max {π‘₯,𝑣ℓ }∈𝐸 οΏ½ 𝑑π‘₯,β„“βˆ’1 οΏ½ βˆ’ (4𝑠 βˆ’ 2)πœ… βˆ’ πœ… = 𝑑𝑣,β„“βˆ’1 βˆ’ (4𝑠 βˆ’ 2)πœ… βˆ’ πœ…, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the claim of the lemma holds with Δ𝑣 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 21 FC-2(𝑠) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, let {𝑣, 𝑣ℓ} be the last edge of 𝑄ℓ if |𝑄ℓ| β‰  0 or the first edge of 𝑃ℓ otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' the latter is feasible, because then 𝑣ℓ = 𝑝ℓ, and |𝑃ℓ| β‰  0 due to (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We get that 𝑑𝑣ℓ,β„“βˆ’1 βˆ’ C𝑣ℓ,β„“ ≀ min {π‘₯,𝑣ℓ }∈𝐸 οΏ½ 𝑑π‘₯,β„“βˆ’1 οΏ½ + (4𝑠 βˆ’ 2)πœ… βˆ’ πœ… ≀ 𝑑𝑣,β„“βˆ’1 + (4𝑠 βˆ’ 2)πœ… βˆ’ πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, the claim of the lemma holds with Δ𝑣 = βˆ’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' FC-3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, 𝑑𝑣ℓ,β„“βˆ’1 βˆ’ C𝑣ℓ,β„“ ≀ 𝑑𝑣ℓ,β„“βˆ’1 βˆ’ πœ…, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the claim of the lemma holds with Δ𝑣 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' There is a node 𝑀 ∈ 𝑉 such that 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀ℓ,β„“ πœ— β‰₯ 𝑑𝑀,β„“βˆ’1 + 4π‘ πœ…Ξ”π‘€, where Δ𝑀 = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 0 and 𝑀 = 𝑀ℓ, βˆ’1 and {𝑀,𝑀ℓ} is the last edge of 𝑃ℓ, or 1 and {𝑀ℓ,𝑀} ∈ 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 4, 𝑀ℓ satisfies SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We make a case distinction based on which one of SC-1, SC-2, and SC-3 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' SC-1(𝑠) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let {𝑀,𝑀ℓ} be the last edge of 𝑃ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' by (P1), |𝑃ℓ| β‰  0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', this edge exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀ℓ,𝑙 πœ— β‰₯ max {π‘₯,𝑀ℓ }∈𝐸{𝑑π‘₯,β„“βˆ’1} βˆ’ 4π‘ πœ… β‰₯ 𝑑𝑀,β„“βˆ’1 βˆ’ 4π‘ πœ…, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the claim of the lemma holds with Δ𝑀 = βˆ’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' SC-2(𝑠) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, let 𝑀 = arg min{π‘₯,𝑣ℓ }∈𝐸{𝑑π‘₯,β„“βˆ’1} and bound 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀ℓ, β„“ πœ— β‰₯ min {π‘₯,𝑀ℓ }∈𝐸{𝑑π‘₯,β„“βˆ’1} + 4π‘ πœ… = 𝑑𝑀,β„“βˆ’1 + 4π‘ πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, the lemma holds with Δ𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' SC-3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀,β„“ β‰₯ 𝑑𝑀ℓ,β„“βˆ’1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the claim of the lemma holds with Δ𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' There is a node 𝑝 ∈ 𝑉 such that 𝑑𝑝ℓ,β„“βˆ’1 βˆ’ C𝑝ℓ,β„“ ≀ οΏ½ 𝑑𝑝,β„“βˆ’1 and 𝑝 = 𝑝ℓ, or 𝑑𝑝,β„“βˆ’1 βˆ’ πœ… and {𝑝ℓ, 𝑝} is the first edge of 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If C𝑝ℓ,β„“ β‰₯ 0, the claim holds with 𝑝 = 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, suppose that C𝑝ℓ,β„“ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let {𝑝ℓ, 𝑝} be the first edge of 𝑃ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' such an edge exists, as by (P1) we have that 𝑝ℓ β‰  𝑀ℓ and hence |𝑃ℓ| β‰  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 6, 𝑝ℓ satisfies JC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As C𝑝ℓ,β„“ < 0, JC-2 must apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We conclude that C𝑝ℓ,β„“ β‰₯ 𝑑𝑝ℓ,β„“βˆ’1 βˆ’ min {π‘₯,𝑝ℓ }∈𝐸 οΏ½ 𝑑π‘₯,β„“βˆ’1 οΏ½ + πœ… β‰₯ 𝑑𝑝ℓ,β„“βˆ’1 βˆ’ 𝑑𝑝,β„“βˆ’1 + πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Rearranging terms, the desired inequality follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ 22 Christoph Lenzen and Shreyas Srinivas In the following, let (𝑣, 𝑝,𝑀) be the triple of nodes guaranteed by Lemmas 7 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote by β—¦ concatenation of paths, by prefix(𝑅,π‘₯) the prefix of path 𝑅 ending at node π‘₯ ∈ 𝑅, and by suffix(𝑅,π‘₯) the suffix of path 𝑅 starting at node π‘₯ ∈ 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let 𝑝′ = οΏ½ 𝑣 if 𝑣 lies on suffix(𝑃ℓ, 𝑝), 𝑝 else, 𝑃 := οΏ½ prefix(𝑃ℓ,𝑀) if 𝑀 lies on 𝑃ℓ, 𝑃ℓ β—¦ (𝑀ℓ,𝑀) else, 𝑃 β€² := οΏ½ suffix(𝑃, 𝑝′) if 𝑝′ lies on 𝑃, (𝑝′,𝑀) else, 𝑄 := οΏ½ prefix(𝑄ℓ, 𝑣) if 𝑣 lies on 𝑄ℓ, 𝑄ℓ β—¦ {𝑣ℓ, 𝑣} else, 𝑄 β€² := οΏ½ suffix(𝑄, 𝑝′) if 𝑝′ lies on 𝑄, (𝑝′, 𝑝ℓ) β—¦ 𝑄 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For notational convenience, in analogy to Ξ”β„“ we also define Ξ” := οΏ½ |𝑃 β€²| + |𝑄 β€²| βˆ’ 1 if 𝑃 β€² and 𝑄 β€² have the same first edge |𝑃 β€²| + |𝑄 β€²| else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We will show that this construction satisfies properties (P1) to (P5) for layer β„“ βˆ’ 1 with π‘£β„“βˆ’1 = 𝑣, π‘β„“βˆ’1 = 𝑝′, π‘€β„“βˆ’1 = 𝑀, π‘ƒβ„“βˆ’1 = 𝑃 β€², and π‘„β„“βˆ’1 = 𝑄 β€²;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' this will constitute the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we first point out that indeed 𝑃 β€² and 𝑄 β€² are paths in 𝐻 from 𝑝′ to 𝑀 and 𝑣, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To this end, we first cover the special case that 𝑝′ does not lie on 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑝′ does not lie on 𝑃, then 𝑝′ = 𝑀ℓ and either 𝑀 = 𝑝ℓ or 𝑝′ = 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 9, 𝑝 lies on the first edge of 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, if 𝑝′ = 𝑝, 𝑝′ lies on 𝑃 unless prefix(𝑃ℓ,𝑀) does not contain this edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 8, this can only happen if the first edge of 𝑃ℓ is also the last edge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', 𝑃ℓ = (𝑝ℓ,𝑀ℓ) = (𝑀, 𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It remains to consider the case that 𝑝′ β‰  𝑝, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', 𝑝′ = 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Again, we use that all edges but the last of 𝑃ℓ are also contained in 𝑃 by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝑝′ = 𝑣 = 𝑀ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑃 β€² is a path in 𝐻 from 𝑝′ to 𝑀 and 𝑄 β€² is a path in 𝐻 from 𝑝′ to 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To show that 𝑃 β€² is a path from 𝑝′ to 𝑀, note that by Lemma 8, 𝑃 is a path in 𝐻, which by definition ends at 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, if 𝑃 β€² = suffix(𝑃, 𝑝′), 𝑃 β€² is a path from 𝑝′ to 𝑀 in 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Otherwise, by Observation 3, 𝑝′ = 𝑀ℓ, and {𝑝′,𝑀} = {𝑀ℓ,𝑀} ∈ 𝐸 by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To show that 𝑄 β€² is a path from 𝑝′ to 𝑣, note that by Lemma 7, 𝑄 is a path in 𝐻, which by definition ends at 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑝′ = 𝑝, by Lemma 9 𝑄 β€² is also a path in 𝐻, which by definition begins at 𝑝′ and has the same endpoint as 𝑄, which is 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On the other hand, if 𝑝′ = 𝑣, suffix(𝑄, 𝑝′) = suffix(𝑄, 𝑣) = (𝑣), which is the 0-length path from 𝑝′ = 𝑣 to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Proving the Properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To prove Corollary 2, we establish that the tuple (𝑣, 𝑝′,𝑀, 𝑃 β€²,𝑄′) satisfies properties (P1) to (P5) for layer β„“ βˆ’ 1, contradicting the minimality of β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Observation 4, indeed 𝑃 β€² and 𝑄 β€² are paths from 𝑣 to 𝑀 and 𝑝′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In the following, we will repeatedly use this fact and the property that {π‘₯β„“,π‘₯} ∈ 𝐸 for π‘₯ ∈ {𝑣,𝑀, 𝑝} whenever π‘₯ β‰  π‘₯β„“, without explicitly invoking Observation 4 and Lemmas 7 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We first rule out the special case that 𝑣 = 𝑀ℓ and 𝑀 = 𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 23 Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The case that 𝑣 = 𝑀ℓ and 𝑀 = 𝑣ℓ is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume towards a contradiction that 𝑣 = 𝑀ℓ and 𝑀 = 𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We use (P4), Lemma 3, and Lemma 8 to bound βˆ’C𝑀,β„“ β‰₯ 𝑑𝑀,β„“ βˆ’ 𝑑𝑀,β„“βˆ’1 βˆ’ Ξ› = 𝑑𝑣ℓ,β„“ βˆ’ (𝑑𝑀,β„“βˆ’1 βˆ’ 4π‘ πœ…) βˆ’ Ξ› βˆ’ 4π‘ πœ… β‰₯ 𝑑𝑣ℓ,β„“ βˆ’ οΏ½ 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀ℓ,β„“ πœ— οΏ½ βˆ’ Ξ› βˆ’ 4π‘ πœ… = 𝑑𝑣ℓ,β„“ βˆ’ οΏ½ 𝑑𝑀ℓ,β„“βˆ’1 + 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 βˆ’ C𝑀ℓ,β„“ πœ— οΏ½ βˆ’ πœ… 2 βˆ’ 4π‘ πœ… β‰₯ 𝑑𝑣ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ βˆ’ 4π‘ πœ… βˆ’ πœ… 2 β‰₯ πœ“π‘£ Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) βˆ’ ( Β―β„“ βˆ’ β„“ + 1)πœ… 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, by JC, it holds that 𝑑𝑀,β„“βˆ’1 ≀ 𝑑𝑀ℓ,β„“βˆ’1 + C𝑀,β„“ βˆ’ πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that by (P1), |𝑃ℓ| β‰  0 and hence |𝑃ℓ|, Ξ”β„“ β‰₯ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, by (P4) and Equation (4) 𝑑𝑣ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ βˆ’ 4π‘ πœ… β‰₯ πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) + ( Β―β„“ βˆ’ β„“)πœ… 2 β‰₯ πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' C𝑀ℓ,β„“ ≀ πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then by Lemma 3 4π‘ πœ… < 𝑑𝑣ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ = 𝑑𝑀,β„“ βˆ’ 𝑑𝑀ℓ,β„“ ≀ 𝑑𝑀,β„“βˆ’1 βˆ’ C𝑀,β„“ βˆ’ οΏ½ 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀ℓ,β„“ πœ— οΏ½ + 𝑒 + οΏ½ 1 βˆ’ 1 πœ— οΏ½ (Ξ› βˆ’ 𝑑) ≀ 𝑒 + οΏ½ 1 βˆ’ 1 πœ— οΏ½ (Ξ› βˆ’ 𝑑) < πœ…, which is a contradiction, because 𝑠 β‰₯ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' C𝑀ℓ,β„“ > πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By JC, it follows that 𝑑𝑀ℓ,β„“βˆ’1 β‰₯ 𝑑𝑀,β„“βˆ’1 + C𝑀ℓ,β„“ πœ— + πœ…, yielding by Lemma 3 that 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑𝑀,β„“βˆ’1 = 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ 𝑑𝑀,β„“βˆ’1 β‰₯ 𝑑𝑀,β„“βˆ’1 + C𝑀ℓ,β„“ πœ— + πœ… βˆ’ (𝑑𝑀ℓ,β„“βˆ’1 + C𝑀,β„“ βˆ’ πœ…) = 𝑑𝑣ℓ,β„“βˆ’1 βˆ’ C𝑣ℓ,β„“ βˆ’ οΏ½ 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀ℓ,β„“ πœ— οΏ½ + 2πœ… β‰₯ 𝑑𝑣ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ + 2πœ… βˆ’ πœ… 2 > 𝑑𝑣ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ + πœ… 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 24 Christoph Lenzen and Shreyas Srinivas Recall that by (P1), |𝑃ℓ| β‰  0 and hence |𝑃ℓ|, Ξ”β„“ β‰₯ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, 𝑑(𝑣,𝑀) = 𝑑(𝑀ℓ,𝑀) ≀ 1, since by Lemma 8 𝑀 is either 𝑀ℓ or a neighbor of 𝑀ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, (P4) implies that πœ“π‘  𝑣,𝑀(β„“ βˆ’ 1) = 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑𝑀,β„“βˆ’1 βˆ’ 4π‘ πœ…π‘‘(𝑣,𝑀) > 𝑑𝑣ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ βˆ’ 4π‘ πœ…|𝑃ℓ| + πœ… 2 β‰₯ πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) + ( Β―β„“ βˆ’ (β„“ βˆ’ 1))πœ… 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝑣 and 𝑀 satisfy Equation (4) and Equation (5) with index Β―β„“ replaced by index β„“ βˆ’ 1 < Β―β„“, contradicting the minimality of Β―β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Next, we prove a helper lemma relating 𝑑𝑀ℓ,β„“ and 𝑑��,β„“βˆ’1 by a stronger bound than Lemma 8 for the special case that 𝑝′ = 𝑀ℓ and 𝑀 = 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This follows similar reasoning as the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, it does not yield an immediate contradiction, as we need to rely on the weaker bound provided by (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑝′ = 𝑀ℓ and 𝑀 = 𝑝ℓ, then 𝑑𝑀ℓ,β„“ βˆ’ 𝑑𝑀,β„“βˆ’1 > 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 πœ— .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We use (P3) and Lemma 3 to bound βˆ’C𝑀,β„“ β‰₯ 𝑑𝑀,β„“ βˆ’ 𝑑𝑀,β„“βˆ’1 βˆ’ Ξ› = 𝑑𝑝ℓ,β„“ βˆ’ (𝑑𝑀,β„“βˆ’1 βˆ’ 4π‘ πœ…) βˆ’ Ξ› βˆ’ 4π‘ πœ… β‰₯ 𝑑𝑝ℓ,β„“ βˆ’ οΏ½ 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀ℓ,β„“ πœ— οΏ½ βˆ’ Ξ› βˆ’ 4π‘ πœ… = 𝑑𝑝ℓ,β„“ βˆ’ οΏ½ 𝑑𝑀ℓ,β„“βˆ’1 + 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 βˆ’ C𝑀ℓ,β„“ πœ— οΏ½ βˆ’ πœ… 2 βˆ’ 4π‘ πœ… β‰₯ 𝑑𝑝ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ βˆ’ 4π‘ πœ… βˆ’ πœ… 2 β‰₯ πœ“π‘£ Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) βˆ’ ( Β―β„“ βˆ’ β„“ + 1)πœ… 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, by JC, it holds that 𝑑𝑀,β„“βˆ’1 ≀ 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' C𝑀ℓ,β„“ ≀ πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑑𝑀ℓ,β„“ βˆ’ 𝑑𝑀,β„“βˆ’1 β‰₯ 𝑑𝑀ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“βˆ’1 + πœ… β‰₯ 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 βˆ’ C𝑀ℓ,β„“ πœ— + πœ… β‰₯ 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 πœ— .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' C𝑀ℓ,β„“ > πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By JC, it follows that 𝑑𝑀ℓ,β„“βˆ’1 β‰₯ 𝑑𝑀,β„“βˆ’1 + C𝑀ℓ,β„“ πœ— + πœ…, Gradient TRIX 25 yielding that 𝑑𝑀ℓ,β„“ βˆ’ 𝑑𝑀,β„“βˆ’1 β‰₯ 𝑑𝑀ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“βˆ’1 + C𝑀ℓ,β„“ πœ— + πœ… > 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 πœ— .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Using Lemma 11, we establish (P4) for the special case of 𝑝′ = 𝑀ℓ and 𝑀 = 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that this entails that 𝑀 is closer to 𝑝ℓ, yet 𝑃 is not shorter than 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is accounted for by the case distinction in the definition of Ξ”β„“, which covers the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑝′ = 𝑀ℓ and 𝑀 = 𝑝ℓ, then (P4) holds for 𝑣, 𝑝′, 𝑀, |𝑃 β€²|, |𝑄 β€²|, and layer β„“ βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote by Δ𝑣 ∈ {βˆ’1, 0, 1} the value such that 𝑑𝑣ℓ,β„“βˆ’1 βˆ’ C𝑣ℓ,β„“ ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ (4𝑠 βˆ’ 2)πœ…Ξ”π‘£ βˆ’ πœ… according to Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemmas 3 and 11, 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑𝑀,β„“βˆ’1 > 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑𝑀ℓ,β„“ + 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 πœ— β‰₯ 𝑑𝑣ℓ,β„“βˆ’1 βˆ’ C𝑣ℓ,β„“ + (4𝑠 βˆ’ 2)πœ…Ξ”π‘£ + πœ… οΏ½οΏ½ 𝑑𝑀ℓ,β„“ + 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 πœ— β‰₯ 𝑑𝑣ℓ,β„“ βˆ’ Ξ› + (4𝑠 βˆ’ 2)πœ…Ξ”π‘£ + πœ… βˆ’ 𝑑𝑀ℓ,β„“ + 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 πœ— =𝑑𝑣ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ + (4𝑠 βˆ’ 2)πœ…Ξ”π‘£ + πœ… 2 β‰₯ (4𝑠 βˆ’ 2)πœ…(Ξ”β„“ + Δ𝑣) +πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) + ( Β―β„“ βˆ’ (β„“ βˆ’ 1))πœ… 2 + πœ…π‘ |𝑃ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We claim that Ξ” ≀ Ξ”β„“ + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that plugging this into the above inequality yields 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑𝑀,β„“βˆ’1 β‰₯ (4𝑠 βˆ’ 2)πœ…Ξ” +πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) + ( Β―β„“ βˆ’ (β„“ βˆ’ 1))πœ… 2 + πœ…π‘ |𝑃 β€²|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', (P4) for 𝑣, 𝑝′, 𝑀, |𝑃 β€²|, |𝑄 β€²|, and layer β„“ βˆ’ 1, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, proving the above claim will complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To show the claim, we first note that 𝑃 β€² = (𝑝′,𝑀) = (𝑀ℓ, 𝑝ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝑀 = 𝑝ℓ, by Lemma 8 we also have that 𝑃ℓ = (𝑝ℓ,𝑀ℓ) = (𝑀, 𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, |𝑃ℓ| = |𝑃 β€²|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑃ℓ and 𝑄ℓ share the first edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It follows that |𝑄ℓ| β‰₯ 2, as otherwise 𝑣ℓ = 𝑀ℓ, contradicting (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑣 = 𝑝′, then |𝑄 β€²| = |𝑄| = |(𝑣)| = 0 ≀ |𝑄ℓ| βˆ’ 2 ≀ |𝑄ℓ| + Δ𝑣 βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Otherwise, the first edge of𝑄 is the first edge of𝑄ℓ and thus 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This edge is {𝑝ℓ,𝑀ℓ} = {𝑝ℓ, 𝑝′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, |𝑄 β€²| = | suffix(𝑄, 𝑝′)| ≀ |𝑄| βˆ’ 1 = |𝑄ℓ| + Δ𝑣 βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Either way, we get that Ξ” ≀ |𝑃 β€²| + |𝑄 β€²| ≀ |𝑃ℓ| + |𝑄ℓ| + Δ𝑣 βˆ’ 1 = Ξ”β„“ + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑃ℓ and 𝑄ℓ do not share the first edge, but 𝑃 β€² and 𝑄 β€² do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Ξ” = |𝑃 β€²| + |𝑄 β€²| βˆ’ 1 ≀ |𝑃ℓ| + |𝑄ℓ| + Δ𝑣 βˆ’ 1 = Ξ”β„“ + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑃ℓ and 𝑄ℓ do not share the first edge and neither do 𝑃 β€² and 𝑄 β€².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As the first (and only) edge of 𝑃 β€² is {𝑝′,𝑀} = {𝑝′, 𝑝ℓ}, this entails that 𝑄 β€² = suffix(𝑄, 𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – | suffix(𝑄, 𝑝′)| ≀ |𝑄| βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Ξ” = |𝑃 β€²| + |𝑄 β€²| ≀ |𝑃ℓ| + |𝑄| βˆ’ 1 ≀ |𝑃ℓ| + |𝑄ℓ| + Δ𝑣 = Ξ”β„“ + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 26 Christoph Lenzen and Shreyas Srinivas – | suffix(𝑄, 𝑝′)| = |𝑄| and 𝑣ℓ β‰  𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑝′ is the last node on 𝑄, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', 𝑣 = 𝑝′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As by Observa- tion 4 𝑄 β€² is a path from 𝑝′ to 𝑣, it follows that |𝑄 β€²| = 0 < |𝑄ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We conclude that Ξ” = |𝑃 β€²| + |𝑄 β€²| ≀ |𝑃ℓ| + |𝑄ℓ| + Δ𝑣 = Ξ”β„“ + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – | suffix(𝑄, 𝑝′)| = |𝑄| and 𝑣ℓ = 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝑀 = 𝑝ℓ and 𝑝′ = 𝑣 = 𝑀ℓ as in the previous subcase, this contradicts Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Before proceeding to the case that 𝑣 β‰  𝑀ℓ or 𝑀 β‰  𝑣ℓ, we prove another helper statement ruling out the specific case that 𝑣 β‰  𝑝′ = 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It is not possible that 𝑣 β‰  𝑝′ = 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume towards a contradiction that 𝑣 β‰  𝑝′ = 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝑝′ = 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemmas 8 and 9 yield that 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀ℓ,β„“ πœ— β‰₯ 𝑑𝑀,β„“βˆ’1 βˆ’ 4π‘ πœ… and 𝑑𝑝ℓ,β„“βˆ’1 βˆ’ C𝑝ℓ,β„“ ≀ 𝑑𝑝′,β„“βˆ’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using (P3) and Lemma 3, it follows that 0 = 𝑑𝑝′,β„“βˆ’1 βˆ’ 𝑑𝑀,β„“βˆ’1 β‰₯ 𝑑𝑝ℓ,β„“βˆ’1 βˆ’ C𝑝ℓ,β„“ βˆ’ οΏ½ 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀ℓ,β„“ πœ— οΏ½ βˆ’ 4π‘ πœ… β‰₯ 𝑑𝑝ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ βˆ’ 4π‘ πœ… βˆ’ πœ… 2 β‰₯ πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) βˆ’ ( Β―β„“ βˆ’ (β„“ βˆ’ 1))πœ… 2 > 0, arriving at the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ We now establish (P4) for the case that 𝑣 β‰  𝑀ℓ or 𝑀 β‰  𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑝′ β‰  𝑀ℓ or 𝑀 β‰  𝑝ℓ, then (P4) holds for 𝑣, 𝑝′, 𝑀, |𝑃 β€²|, |𝑄 β€²|, and layer β„“ βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote by Δ𝑀, Δ𝑣 ∈ {βˆ’1, 0, 1} the values such that 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀ℓ,β„“ β‰₯ 𝑑𝑀,β„“βˆ’1 + 4π‘ πœ…Ξ”π‘€ 𝑑𝑣ℓ,β„“βˆ’1 βˆ’ C𝑣ℓ,β„“ ≀ 𝑑𝑣,β„“βˆ’1 βˆ’ (4𝑠 βˆ’ 2)πœ…Ξ”π‘£ βˆ’ πœ… according to Lemmas 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using (P4) and Lemma 3, we bound 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑��,β„“βˆ’1 β‰₯ 𝑑𝑣ℓ,β„“βˆ’1 βˆ’ C𝑣,β„“ πœ— + (4𝑠 βˆ’ 2)πœ…Ξ”π‘£ + πœ… βˆ’ (𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀,β„“ βˆ’ 4π‘ πœ…Ξ”π‘€) β‰₯ 𝑑𝑣ℓ,β„“ + (4𝑠 βˆ’ 2)πœ…Ξ”π‘£ + πœ… βˆ’ 𝑑𝑀ℓ,β„“ + 4π‘ πœ…Ξ”π‘€ βˆ’ πœ… 2 β‰₯ (4𝑠 βˆ’ 2)πœ…(Ξ”β„“ + Δ𝑣 + Δ𝑀) + ( Β―β„“ βˆ’ (β„“ βˆ’ 1))πœ… 2 + πœ…π‘  (|𝑃ℓ| + Δ𝑀) β‰₯ (4𝑠 βˆ’ 2)πœ…(Ξ”β„“ + Δ𝑣 + Δ𝑀) + ( Β―β„“ βˆ’ (β„“ βˆ’ 1))πœ… 2 + πœ…π‘ |𝑃 β€²|, where the last step exploits that |𝑃 β€²| = | suffix(𝑃, 𝑝′)| ≀ |𝑃| ≀ |𝑃ℓ| + Δ𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We claim that Ξ” ≀ Ξ”β„“ + Δ𝑣 + Δ𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proving this claim will complete the proof, as by the above inequality then 𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑𝑀,β„“βˆ’1 β‰₯ (4𝑠 βˆ’ 2)πœ…Ξ” + ( Β―β„“ βˆ’ (β„“ βˆ’ 1))πœ… 2 + πœ…π‘ |𝑃 β€²|, Gradient TRIX 27 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', (P4) for 𝑣, 𝑝′, 𝑀, |𝑃 β€²|, |𝑄 β€²|, and layer β„“ βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Observation 3 and the prerequisites of the lemma, 𝑃 β€² = suffix(𝑃, 𝑝′) or 𝑝′ = 𝑣 = 𝑀ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To cover the possibility that 𝑃 β€² = suffix(𝑃, 𝑝′), we distinguish several cases: 𝑝′ = 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑃 β€² = 𝑃 and 𝑄 β€² = 𝑄, as 𝑝′ is the first node of both 𝑃 and 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, |𝑃 β€²| + |𝑄 β€²| = |𝑃| + |𝑄| ≀ |𝑃ℓ| + |𝑄ℓ| + Δ𝑀 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish three subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – 𝑃ℓ and 𝑄ℓ do not share their first edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Ξ” ≀ |𝑃 β€²| + |𝑄 β€²| = |𝑃ℓ| + |𝑄ℓ| + Δ𝑀 + Δ𝑣 = Ξ”β„“ + Δ𝑀 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – 𝑃ℓ, 𝑄ℓ, and 𝑄 β€² share the same first edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 13, 𝑀 β‰  𝑝′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, 𝑃 β€² = 𝑃 β‰  (𝑝′), which means that 𝑃ℓ and 𝑃 β€² have the same first edge, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝑄 β€² and 𝑃 β€² have the same first edge as well, and Ξ” = |𝑃 β€²| + |𝑄 β€²| βˆ’ 1 = |𝑃ℓ| + |𝑄ℓ| βˆ’ 1 + Δ𝑀 + Δ𝑣 = Ξ”β„“ + Δ𝑀 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – 𝑃ℓ and 𝑄ℓ have the same first edge, but 𝑄 β€² does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝑄ℓ β‰  (𝑝ℓ), we have that 𝑣ℓ β‰  𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By (P5), this implies that 𝑣ℓ βˆ‰ 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, 𝑣ℓ cannot be part of the first edge of 𝑄ℓ and |𝑄ℓ| β‰₯ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝑝′ = 𝑝ℓ, 𝑄 and 𝑄 β€² both start with 𝑝′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, 𝑄 β€² is a prefix of 𝑄ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, 𝑄ℓ has the same first edge as 𝑃ℓ, while 𝑄 β€² does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, |𝑄 β€²| = 0 ≀ |𝑄ℓ| + Δ𝑣 βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We conclude that Ξ” = |𝑃 β€²| + |𝑄 β€²| ≀ |𝑃ℓ| + |𝑄ℓ| βˆ’ 1 + Δ𝑀 + Δ𝑣 = Ξ”β„“ + Δ𝑀 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑣 = 𝑝′ β‰  𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑄 β€² = (𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, by the prerequisites of the lemma, 𝑃 β€² = suffix(𝑃, 𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝑝′ β‰  𝑝ℓ, we have that | suffix(𝑃, 𝑝′)| ≀ |𝑃| βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By construction, |𝑄 β€²| ≀ |𝑄| + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Overall, Ξ” ≀ |𝑃 β€²| + |𝑄 β€²| ≀ |𝑃| βˆ’ 1 + |𝑄| + 1 = |𝑃ℓ| + Δ𝑀 + |𝑄ℓ| + Δ𝑣 = Ξ”β„“ + Δ𝑀 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑣 β‰  𝑝′ β‰  𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝑝′ = 𝑝 and by Lemma 9 {𝑝ℓ, 𝑝′} is the first edge of 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, |𝑃 β€²| = | suffix(𝑃, 𝑝′)| ≀ |𝑃| βˆ’ 1 ≀ |𝑃ℓ| + Δ𝑀 βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – 𝑃ℓ and 𝑄ℓ do not share their first edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Ξ” ≀ |𝑃 β€²| + |𝑄 β€²| ≀ |𝑃ℓ| + Δ𝑀 + |𝑄ℓ| + Δ𝑣 = Ξ”β„“ + Δ𝑀 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – 𝑃ℓ and 𝑄ℓ share their first edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝑣 β‰  𝑝′, 𝑄 has the same first edge as 𝑄ℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', {𝑝ℓ, 𝑝′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, |𝑄 β€²| = | suffix(𝑄, 𝑝′| = |𝑄| βˆ’ 1 ≀ |𝑄ℓ| + Δ𝑣 βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We conclude that Ξ” ≀ |𝑃 β€²| + |𝑄 β€²| ≀ |𝑃ℓ| + Δ𝑀 + |𝑄ℓ| + Δ𝑣 βˆ’ 2 < Ξ”β„“ + Δ𝑀 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It remains to consider the case that 𝑃 β€² β‰  suffix(𝑃, 𝑝′) and 𝑝′ = 𝑣 = 𝑀ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then |𝑄 β€²| = (𝑣) and |𝑃 β€²| = |(𝑝′,𝑀)| = 1, implying that Ξ” = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By (P2), 𝑣ℓ β‰  𝑀ℓ = 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If Δ𝑣 = 1, then Ξ” = 1 ≀ Ξ”β„“ ≀ Ξ”β„“ + Δ𝑀 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 7, the remaining case is that Δ𝑣 = βˆ’1 and {𝑣, 𝑣ℓ} is the last edge of 𝑄ℓ or the first edge of 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 10, it is impossible that 𝑣 = 𝑀ℓ, so this edge must be the last one of 𝑄ℓ and distinct from the first one of 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, by the prerequisites of the lemma, 𝑝ℓ β‰  𝑀, so it must hold that |𝑃ℓ| β‰₯ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Overall, either |𝑄ℓ| β‰₯ 2 and Ξ” = 1 ≀ |𝑃ℓ| + |𝑄ℓ| βˆ’ 3 ≀ Ξ”β„“ βˆ’ 2 = Ξ”β„“ + Δ𝑀 + Δ𝑣, or |𝑄ℓ| = 1 and 𝑄ℓ and 𝑃ℓ do not share the first edge, yielding Ξ” = 1 ≀ |𝑃ℓ| + |𝑄ℓ| βˆ’ 2 = Ξ”β„“ βˆ’ 2 = Ξ”β„“ + Δ𝑀 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P4) and (P2) hold for 𝑣, 𝑝′, 𝑀, |𝑃 β€²|, |𝑄 β€²|, and layer β„“ βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 28 Christoph Lenzen and Shreyas Srinivas Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Follows from Lemma 12, Lemma 14, and Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ It remains to prove (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P3) holds for 𝑣, 𝑝′, |𝑃 β€²|, and layer β„“ βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑣 = 𝑝′, the statement readily follows from Corollary 1 and Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, assume that 𝑣 β‰  𝑝′ and hence 𝑝′ = 𝑝 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote by Δ𝑀 ∈ {βˆ’1, 0, 1} the value such that 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀ℓ,β„“ β‰₯ 𝑑𝑀,β„“βˆ’1 + 4π‘ πœ…Ξ”π‘€ 𝑑𝑝ℓ,β„“βˆ’1 βˆ’ C𝑝ℓ,β„“ ≀ 𝑑𝑝′,β„“βˆ’1 according to Lemmas 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using (P3) and Lemma 3, it follows that 𝑑𝑝′,β„“βˆ’1 βˆ’ 𝑑𝑀,β„“βˆ’1 β‰₯ 𝑑𝑝ℓ,β„“βˆ’1 βˆ’ C𝑝ℓ,β„“ βˆ’ οΏ½ 𝑑𝑀ℓ,β„“βˆ’1 βˆ’ C𝑀ℓ,β„“ πœ— οΏ½ + Δ𝑀4π‘ πœ… β‰₯ 𝑑𝑝ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ + 4π‘ πœ…Ξ”π‘€ βˆ’ πœ… 2 β‰₯ 4π‘ πœ…(|𝑃ℓ| + Δ𝑀) +πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) βˆ’ ( Β―β„“ βˆ’ (β„“ βˆ’ 1))πœ… 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑃 β€² = suffix(𝑃, 𝑝′), then |𝑃 β€²| ≀ |𝑃| ≀ |𝑃ℓ| + Δ𝑀 and (P3) for 𝑣, 𝑝′, |𝑃 β€²|, and layer β„“ βˆ’ 1 readily follows from the above inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Otherwise, by the assumption that 𝑣 β‰  𝑝′ and Observation 3, it holds that 𝑝′ = 𝑀ℓ and 𝑀 = 𝑝ℓ, and |𝑃 β€²| = |𝑃ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using Lemmas 3 and 11 together with (P3), we arrive at 𝑑𝑝′,β„“βˆ’1 βˆ’ 𝑑𝑀,β„“βˆ’1 β‰₯ 𝑑𝑝′,β„“βˆ’1 βˆ’ 𝑑𝑀ℓ,β„“ + οΏ½ 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 πœ— οΏ½ β‰₯ 𝑑𝑝ℓ,β„“βˆ’1 βˆ’ C𝑝ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ + οΏ½ 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 πœ— οΏ½ β‰₯ 𝑑𝑝ℓ,β„“ βˆ’ 𝑑𝑀ℓ,β„“ βˆ’ πœ… 2 β‰₯ 4π‘ πœ…|𝑃ℓ| +πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) βˆ’ ( Β―β„“ βˆ’ (β„“ βˆ’ 1))πœ… 2 β‰₯ 4π‘ πœ…|𝑃 β€²| +πœ“π‘  𝑣 Β―β„“,𝑀 Β―β„“ ( Β―β„“ ) βˆ’ ( Β―β„“ βˆ’ (β„“ βˆ’ 1))πœ… 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', (P3) for 𝑣, 𝑝′, |𝑃 β€²|, and layer β„“ βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Finally, using these results it is not hard to show that (P5) is satisfied as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P5) holds for 𝑣, 𝑝′, |𝑃 β€²|, and layer β„“ βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that 𝑣 lies on 𝑃 β€².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Corollary 1, 𝑣 β‰  𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, if 𝑃 β€² = (𝑝′,𝑀), 𝑣 = 𝑝′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', (P5) holds for 𝑣, 𝑝′, |𝑃 β€²|, and layer β„“ βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Otherwise, 𝑃 β€² = suffix(𝑃, 𝑝′), implying that 𝑣 lies on suffix(𝑃, 𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝑣 β‰  𝑀, this implies that 𝑣 lies on 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assuming for contradiction that 𝑣 β‰  𝑝′ = 𝑝, by Lemma 9 we have that prefix(𝑃, 𝑝′) = prefix(𝑃ℓ, 𝑝′), which equals either (𝑝ℓ) = (𝑝′) or (𝑝ℓ, 𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, the above entails that 𝑣 actually lies on suffix(𝑃ℓ, 𝑝′) = suffix(𝑃ℓ, 𝑝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As then 𝑝′ = 𝑣, this is a contradiction and we must indeed have that 𝑝′ = 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In the proof of Theorem 1, it must hold that β„“ = β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assuming for contradiction that β„“ > β„“, Corollary 1, Lemmas 15 and 16, and Observation 2 show that layer β„“ βˆ’ 1 also satisfies the properties (P1) to (P5) for some π‘£β„“βˆ’1, π‘β„“βˆ’1,π‘€β„“βˆ’1, and paths π‘ƒβ„“βˆ’1, π‘„β„“βˆ’1, contradicting the minimality of β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Bounding Skews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' With our machinery for bounding Ψ𝑠 in place, it remains to perform the induction on 𝑠 ∈ N>0 to wrap things up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To anchor the induction at 𝑠 = 1, we exploit that Ξ¨1(β„“) ≀ Ξ1(β„“)+2πœ…π·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Ξ¨1(β„“) ≀ οΏ½ Ξ1(0) if β„“ < 4Ξ1(0)/πœ… 4πœ…π· else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Recall that πœ… = 2(𝑒 + (1 βˆ’ 1/πœ—)(Ξ› βˆ’ 𝑑)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that Ξ1(β„“) ≀ Ξ¨1(β„“) + 2πœ…π· for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Theorem 1, we thus have for any β„“ ≀ Β―β„“ that Ξ¨1( Β―β„“ ) ≀ max οΏ½ 0, Ξ1(β„“ ) βˆ’ ( Β―β„“ βˆ’ β„“ + 1)πœ… οΏ½ + ( Β―β„“ βˆ’ β„“ )πœ… 2 ≀ max οΏ½ 0, Ξ¨1(β„“ ) + 2πœ…π· βˆ’ ( Β―β„“ βˆ’ β„“ + 1)πœ… οΏ½ + ( Β―β„“ βˆ’ β„“ )πœ… 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, we have that Ξ¨1(β„“) ≀ οΏ½ max οΏ½ 4πœ…π·, Ξ1(0) οΏ½ if β„“ < 8𝐷 max οΏ½ 4πœ…π·, Ξ¨1(β„“ βˆ’ 8𝐷) βˆ’ 2πœ…π· οΏ½ else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By induction on π‘˜ ∈ N, we thus have that Ξ¨1(β„“) ≀ max{4πœ…π·, Ξ1(0) βˆ’ 2π‘˜πœ…π·} for all β„“ ∈ [8π‘˜π·, 8(π‘˜ + 1)𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The claim of the lemma follows by noting that β„“ β‰₯ 4Ξ1(0)/πœ… results in π‘˜ β‰₯ Ξ1(0)/(2πœ…π·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Note that this lemma shows that Ξ¨1 self-stabilizes [5] within 𝑂(Ξ1(0)/πœ…) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We remark that a more careful analysis reveals a bound on Ξ¨1(β„“) that converges to 2πœ…π·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We confine ourselves to stating this result for the small input skew that we guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If L0 ≀ 4πœ…, then Ξ¨1(β„“) ≀ 2πœ…π· for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that Ξ1(0) = max 𝑣,π‘€βˆˆπ‘‰{𝑑𝑣,0 βˆ’ 𝑑𝑀,0 βˆ’ 2πœ…π‘‘(𝑣,𝑀)} ≀ max 𝑣,π‘€βˆˆπ‘‰{(L0 βˆ’ 2πœ…)𝑑(𝑣,𝑀)} ≀ (L0 βˆ’ 2πœ…)𝐷 ≀ 2πœ…π·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By replacing 8𝐷 with 4𝐷 in the induction from the proof of Lemma 17, we get that Ξ¨1(β„“) ≀ οΏ½ max οΏ½ 2πœ…π·, Ξ1(0) οΏ½ if β„“ < 4𝐷 max οΏ½ 2πœ…π·, Ξ¨1(β„“ βˆ’ 4𝐷) οΏ½ else, implying a uniform bound of Ξ¨1(β„“) ≀ 2πœ…π· for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ For the sake of completeness, we also infer that supβ„“ ∈N{Ξ¨0(β„“)}, also referred to as the global skew in the literature, is in 𝑂(𝑒 + (1 βˆ’ 1/πœ—)(Ξ› βˆ’π‘‘)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Provided that Ξ› ∈ 𝑂(𝑑 +𝑒/(πœ— βˆ’ 1)), this bound is asymptotically optimal [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If L0 ≀ 4πœ…, then Ξ¨0(β„“) ≀ 6πœ…π· ∈ 𝑂(𝑒 + (1 βˆ’ 1/πœ—)(Ξ› βˆ’ 𝑑)) for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Follows from Corollary 3, the fact that Ξ¨0(β„“) ≀ Ξ¨1(β„“) + 4πœ…π·, and the choice of πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ In order to bound the local skew, we now turn to attention to Ψ𝑠 (β„“) for 𝑠 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 30 Christoph Lenzen and Shreyas Srinivas Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For some 𝑠 ∈ N, 𝑠 > 0, suppose that Ξ¨π‘ βˆ’1(β„“) ≀ Ξ¨π‘ βˆ’1 for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Ψ𝑠 (β„“) ≀ οΏ½ Ξžπ‘  (0) + Ξ¨π‘ βˆ’1 2 if β„“ < Ξ¨π‘ βˆ’1/πœ… Ξ¨π‘ βˆ’1 2 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Recall that πœ… = 2(𝑒 + (1 βˆ’ 1/πœ—)(Ξ› βˆ’ 𝑑)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For β„“ < Ξ¨π‘ βˆ’1/πœ…, by Theorem 1 with Β―β„“ = β„“ and β„“ = 0 we have that Ψ𝑠 (β„“) ≀ Ξžπ‘  (0) + πœ…β„“ 2 ≀ Ξžπ‘  (0) + Ξ¨π‘ βˆ’1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that Ξžπ‘  (β„“) ≀ Ξ¨π‘ βˆ’1(β„“) ≀ Ξ¨π‘ βˆ’1 for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, for β„“ β‰₯ Ξ¨π‘ βˆ’1/πœ… by Theorem 1 with Β―β„“ = β„“ and β„“ = β„“ βˆ’ βŒŠΞ¨π‘ βˆ’1/πœ…βŒ‹ we have that Ψ𝑠 (β„“) ≀ max οΏ½ 0, Ξžπ‘  οΏ½ β„“ βˆ’ οΏ½ Ξ¨π‘ βˆ’1 πœ… οΏ½οΏ½ βˆ’ οΏ½οΏ½ Ξ¨π‘ βˆ’1 πœ… οΏ½ + 1 οΏ½ πœ… οΏ½ + οΏ½ Ξ¨π‘ βˆ’1 πœ… οΏ½ πœ… 2 ≀ Ξ¨π‘ βˆ’1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Using this lemma, we can bound the local skew by 𝑂(πœ…(1 + log 𝐷)) = 𝑂((𝑒 + (1 βˆ’ 1/πœ—)(Ξ› βˆ’ 𝑑))(1 + log 𝐷)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If there are no faults, then Lβ„“ ≀ 4πœ…(2 + log 𝐷) for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 27, L0 ≀ 4πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Corollary 3, Ξ¨1(β„“) ≀ 2πœ…π· for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By the assumption that L0 ≀ 4πœ…, for all 𝑠 > 1 we have that Ξžπ‘  (0) = max 𝑣,π‘€βˆˆπ‘‰{𝑑𝑣,0 βˆ’ 𝑑𝑀,0 βˆ’ (4𝑠 βˆ’ 2)πœ…π‘‘(𝑣,𝑀)} ≀ max 𝑣,π‘€βˆˆπ‘‰{(L0 βˆ’ 6πœ…)𝑑(𝑣,𝑀)} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, inductive use of Lemma 18 yields that Ψ𝑠 (β„“) ≀ 22βˆ’π‘ πœ…π·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, Ψ⌊log π·βŒ‹ ≀ 8πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The claim now follows by Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Moreover, in addition we obtain the following self-stabilization property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If for 𝑠,𝑠′ ∈ N, 𝑠 ≀ 𝑠′, we have that Ψ𝑠 (β„“) ≀ Ψ𝑠 for all β„“ β‰₯ β„“ ∈ N, then for β„“ β‰₯ β„“ Lβ„“ ≀ οΏ½ 4π‘ πœ… + Ψ𝑠 if β„“ ≀ β„“ < β„“ + 2Ψ𝑠/πœ… and 4π‘ β€²πœ… + Ψ𝑠 2π‘ β€²βˆ’π‘  if β„“ β‰₯ β„“ + 2Ψ𝑠/πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Inductive use3 of Lemma 18 yields for 𝑠′ β‰₯ 𝑠 and β„“ β‰₯ οΏ½οΏ½οΏ½ + �𝑠′ 𝜎=𝑠+1 Ψ𝑠/(2πœŽβˆ’π‘ πœ…) that Ψ𝑠′ ≀ Ψ𝑠 2π‘ β€²βˆ’π‘  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since the sum forms a geometric series, this in particular applies to all β„“ β‰₯ β„“ + 2Ψ𝑠/πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The claim now follows by applying Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='4 Bounding Skews in the Presence of Faults To analyze how skews evolve with faults, we relate the setting with faults to the bounds we have for a fault-free system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The key property the algorithm guarantees is that, up to an additive 2πœ…, the pulse time is within the interval spanned by the correct predecessors’ pulse times plus Ξ›.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We first show this for the case that for some node (𝑣, β„“), (𝑣, β„“ βˆ’ 1) is faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 3As is, the lemma applies only if β„“ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, the algorithm and hence all statements are invariant under shifting indices by β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 31 Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that the only faulty predecessor of (𝑣, β„“) ∈ 𝑉ℓ, β„“ > 0, is (𝑣, β„“ βˆ’ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote 𝑑min := min {𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} and 𝑑max := max {𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑑min + Ξ› βˆ’ 2πœ… ≀ 𝑑𝑣,β„“ ≀ 𝑑max + Ξ› + 2πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By the assumption of the lemma, for all {𝑣,𝑀} ∈ 𝐸, (𝑀, β„“ βˆ’ 1) βˆ‰ 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We have that 𝐻own βˆ’ 𝐻max = min 𝑠 ∈N {𝐻own βˆ’ 𝐻max + 4π‘ πœ…} ≀ min 𝑠 ∈N {max{𝐻own βˆ’ 𝐻max + 4π‘ πœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ…}} ≀ max{𝐻own βˆ’ 𝐻max, 𝐻own βˆ’ 𝐻min} = 𝐻own βˆ’ 𝐻min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, abbreviating Ξ” = min 𝑠 ∈N {max{𝐻own βˆ’ 𝐻max + 4π‘ πœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ…}} βˆ’ πœ… 2, it holds that 𝐻own βˆ’ 𝐻max βˆ’ πœ… 2 ≀ Ξ” ≀ 𝐻own βˆ’ 𝐻min βˆ’ πœ… 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Taking into account the adjustments in case Ξ” βˆ‰ [0,πœ—πœ…] and using that 𝐻min ≀ 𝐻max we get that 𝐻own βˆ’ 𝐻max βˆ’ 3πœ… 2 ≀ C𝑣,β„“ ≀ 𝐻own βˆ’ 𝐻min + 3πœ… 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, the local time 𝐻𝑣,β„“ (𝑑𝑣,β„“) = 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ C𝑣,β„“ at which (𝑣, β„“) generates its pulse satisfies 𝐻min + Ξ› βˆ’ 𝑑 βˆ’ 3πœ… 2 ≀ 𝐻𝑣,β„“ (𝑑𝑣,β„“) ≀ 𝐻max + Ξ› βˆ’ 𝑑 + 3πœ… 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝐻min > 𝐻𝑣,β„“ (𝑑𝑣,β„“), we have that 𝑑min βˆ’ 𝑑𝑣,β„“ ≀ 𝐻min βˆ’ 𝐻𝑣,β„“ (𝑑𝑣,β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Applying the lower bound of 𝑑 βˆ’ 𝑒 on message delay and Equation (1), we get that 𝑑𝑣,β„“ β‰₯ 𝑑min + 𝑑 βˆ’ οΏ½οΏ½οΏ½ + Ξ› βˆ’ 𝑑 βˆ’ 3πœ… 2 > 𝑑min + Ξ› βˆ’ 2πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝐻min ≀ 𝐻𝑣,β„“ (𝑑𝑣,β„“), the bounds on message delays and hardware clock drift together with Equa- tion (1) yield that 𝑑𝑣,β„“ β‰₯ 𝑑min + 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 βˆ’ 3πœ…/2 πœ— > 𝑑min + Ξ› βˆ’ 3πœ… 2 βˆ’ 𝑒 βˆ’ οΏ½ 1 βˆ’ 1 πœ— οΏ½ (Ξ› βˆ’ 𝑑) = 𝑑min + Ξ› βˆ’ 2πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Concerning the upper bound on 𝑑𝑣,β„“, note that because 𝑑𝑣,β„“ is increasing in 𝐻𝑣,β„“ (𝑑𝑣,β„“), to bound 𝑑𝑣,β„“ from above we may assume that 𝐻𝑣,β„“ (𝑑𝑣,β„“) = 𝐻max + Ξ› βˆ’ 𝑑 + 3πœ… 2 > 𝐻max, where the last step uses Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, 𝑑𝑣,β„“ βˆ’ 𝑑max ≀ 𝐻𝑣,β„“ (𝑑𝑣,β„“) βˆ’ 𝐻max + 𝑑 ≀ Ξ› + 3πœ… 2 < Ξ› + 2πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ 32 Christoph Lenzen and Shreyas Srinivas Similar reasoning covers the case that for some (𝑣, β„“) ∈ 𝑉ℓ and {𝑣,𝑀} ∈ 𝐸, (𝑀, β„“ βˆ’ 1) is faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that for (𝑣, β„“) ∈ 𝑉ℓ, β„“ > 0, (𝑣, β„“ βˆ’ 1) is not faulty, and at most one predecessor is faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denoting 𝑑min := min ((𝑀,β„“βˆ’1),(𝑣,β„“)) βˆˆπΈβ„“βˆ’1 (𝑀,β„“βˆ’1)βˆ‰πΉ {𝑑𝑀,β„“βˆ’1} and 𝑑max := max ((𝑀,β„“βˆ’1),(𝑣,β„“)) βˆˆπΈβ„“βˆ’1 (𝑀,β„“βˆ’1)βˆ‰πΉ {𝑑𝑀,β„“βˆ’1}, then 𝑑min + Ξ› βˆ’ 2πœ… ≀ 𝑑𝑣,β„“ ≀ 𝑑max + Ξ›.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 3, C𝑣,β„“ β‰₯ 0 implies that 𝑑𝑣,β„“ βˆ’ 𝑑min ≀ 𝑑𝑣,β„“ βˆ’ 𝑑𝑣,β„“βˆ’1 ≀ Ξ›, while C𝑣,β„“ ≀ πœ—πœ… yields that 𝑑𝑣,β„“ βˆ’ 𝑑max β‰₯ 𝑑𝑣,β„“ βˆ’ 𝑑𝑣,β„“βˆ’1 β‰₯ 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 πœ— βˆ’ πœ… β‰₯ Ξ› βˆ’ 2πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It remains to show the upper bound on 𝑑𝑣,β„“ if C𝑣,β„“ < 0 and the lower bound if C𝑣,β„“ > πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consider first the case that C𝑣,β„“ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Accordingly, C𝑣,β„“ = 𝐻own βˆ’ 𝐻min βˆ’ πœ… 2 + 2πœ… > 𝐻own βˆ’ 𝐻min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It follows that 𝐻𝑣,β„“ (𝑑𝑣,β„“) = 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ C𝑣,β„“ ≀ 𝐻min + Ξ› βˆ’ 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Noting that the reception time of the first message from a predecessor is bounded from above by the reception time of the message from a correct predecessor, we conclude that 𝑑𝑣,β„“ ≀ 𝑑min + Ξ›.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Now consider the case that C𝑣,β„“ > πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consequently, C𝑣,β„“ = 𝐻own βˆ’ 𝐻max βˆ’ πœ… 2 βˆ’ πœ… > 𝐻own βˆ’ 𝐻max βˆ’ πœ—π‘’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It follows that the local time 𝐻 at which (𝑣, β„“) generates its pulse satisfies that 𝐻 = 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ C𝑣,β„“ β‰₯ 𝐻max + Ξ› βˆ’ 𝑑 + πœ—π‘’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Noting that the reception time of the latest message from a predecessor is bounded from below by the reception time of the latest message from a correct predecessor, by Equation (1) we conclude that 𝑑𝑣,β„“ β‰₯ 𝑑max + 𝑑 + Ξ› βˆ’ 𝑑 πœ— > 𝑑𝑣,β„“βˆ’1 + Ξ› βˆ’ πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote 𝑑min := min ((𝑀,β„“βˆ’1),(𝑣,β„“)) βˆˆπΈβ„“βˆ’1 (𝑀,β„“βˆ’1)βˆ‰πΉ {𝑑𝑀,β„“βˆ’1} and 𝑑max := max ((𝑀,β„“βˆ’1),(𝑣,β„“)) βˆˆπΈβ„“βˆ’1 (𝑀,β„“βˆ’1)βˆ‰πΉ {𝑑𝑀,β„“βˆ’1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑑min + Ξ› βˆ’ 2πœ… ≀ 𝑑𝑣,β„“ ≀ 𝑑max + Ξ› + 2πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 33 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Immediate from Lemmas 19 and 20 and the assumption that no node has more than one faulty predecessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Using this result, we can bound the impact of a fault in layer β„“ βˆ’ 1 on successors via the skew bounds of close-by nodes on layer β„“ βˆ’ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' we exploit that all bounds we show would in fact also apply to the faulty node if it was correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose for a node (𝑣, β„“) ∈ 𝑉ℓ, β„“ > 0, that one of its predecessors is faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, assume that in an execution that differs only in that the faulty predecessor of (𝑣, β„“) is correct, it holds that max{𝑣,𝑀}∈𝐸{|𝑑𝑣,β„“βˆ’1 βˆ’ 𝑑𝑀,β„“βˆ’1|} ≀ 𝐡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then in the execution with the predecessor being faulty, the pulse time of (𝑣, β„“) differs by at most 2𝐡 + 4πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote by standard variables values in the execution without the predecessor being faulty and by primed variables values in the one where it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, for node (𝑣, β„“) ∈ 𝑉ℓ \\ 𝐹 𝑑min := min ((𝑀,β„“βˆ’1),(𝑣,β„“)) βˆˆπΈβ„“βˆ’1{𝑑𝑀,β„“βˆ’1}, 𝑑max := max ((𝑀,β„“βˆ’1),(𝑣,β„“)) βˆˆπΈβ„“βˆ’1 {𝑑𝑀,β„“βˆ’1}, 𝑑 β€² min := min ((𝑀,β„“βˆ’1),(𝑣,β„“)) βˆˆπΈβ„“βˆ’1 (𝑀,β„“βˆ’1)βˆ‰πΉ {𝑑𝑀,β„“βˆ’1}, and 𝑑 β€² max := max ((𝑀,β„“βˆ’1),(𝑣,β„“)) βˆˆπΈβ„“βˆ’1 (𝑀,β„“βˆ’1)βˆ‰πΉ {𝑑𝑀,β„“βˆ’1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' denote the earliest and latest pulsing times of (correct) predecessors without and with faults on layer β„“ βˆ’ 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observe that 𝑑𝑣,β„“βˆ’1 βˆ’ 𝐡 ≀ 𝑑min ≀ 𝑑minβ€² ≀ 𝑑maxβ€² ≀ 𝑑max ≀ 𝑑𝑣,β„“βˆ’1 + 𝐡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, Corollary 5 (applied to both executions) shows that 𝑑𝑣,β„“βˆ’1 βˆ’ 𝐡 βˆ’ 2πœ… ≀ 𝑑𝑣,β„“ ≀ 𝑑𝑣,β„“βˆ’1 + 𝐡 + 2πœ… and 𝑑𝑣,β„“βˆ’1 βˆ’ 𝐡 βˆ’ 2πœ… ≀ 𝑑 β€² 𝑣,β„“ ≀ 𝑑𝑣,β„“βˆ’1 + 𝐡 + 2πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Finally, we observe that such a β€œtime shift” propagates without further increase, so long as there are no faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, a subtlety here is that this is only true for our bounds on timing: a change in timing might leave more time for drift of the local clock to accumulate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' since our worst-case bounds include the maximum time error that can possibly be accumulated from drift (so long as local skews do not become exceedingly large), this is already accounted for in the bound provided by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, we obtain the following generalized variant of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that for 𝑣 ∈ 𝑉 and β„“ ∈ N>0 the predecessors of (𝑣, β„“) are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If we shift the pulse times of these predecessors by at most 𝛿 ∈ R, where Equation (2) still holds for the shifted times, then 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 βˆ’ C𝑣,β„“ πœ— βˆ’ 𝛿 ≀ 𝑑 β€² 𝑣,β„“ βˆ’ 𝑑𝑣,β„“βˆ’1 ≀ Ξ› βˆ’ C𝑣,β„“ + 𝛿, where 𝑑 β€² 𝑣,β„“ denotes the pulse time of (𝑣, β„“) in the execution with the shifts applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Pulse times are increasing as functions of pulse times of predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, in order to maximize or minimize 𝑑 β€² 𝑣,β„“, we need to maximize or minimize the predecessors’ pulse times, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Shifting all predecessors’ pulse times uniformly by 𝛿 also shifts 𝑑 β€² 𝑣,β„“ by 𝛿 relative to 𝑑𝑣,β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The statement now follows analogously to the proof of Lemma 3, carrying the uniform shift through all inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ 34 Christoph Lenzen and Shreyas Srinivas With these tools in place, we can conclude that skews do not grow arbitrarily in the face of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If there are at most 𝑓 faulty nodes in the system and none in layer 0, then Lβ„“ ∈ 𝑂(5𝑓 πœ… log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We prove by induction on the number 𝑖 ≀ 𝑓 of layers β„“ > 0 with faults that the skew is bounded by 𝐡𝑖 := 4πœ…(2 + log 𝐷)5𝑖 �𝑖 𝑗=0 5βˆ’π‘— ∈ 𝑂(5𝑓 πœ… log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Corollary 6, L0 ≀ πœ…/2 < 4πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, if there are no faults in layers β„“ > 0, by Theorem 2 we have that Lβ„“ ≀ 𝐡0 := 4πœ…(2 + log 𝐷) for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume that we completed step 𝑖 ∈ N and that ℓ𝑖+1 is the next layer where faults need to be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then we have that for all β„“ ≀ ℓ𝑖+1 that Lβ„“β€² ≀ 𝐡𝑖 = 4πœ…(2 + log 𝐷)5𝑓 �𝑖 𝑗=0 5βˆ’π‘— both before and after adding the faults on layer 𝑖 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 21, it follows that pulsing times on layer ℓ𝑖+1 + 1 do not change by more than 2𝐡𝑖 + 4πœ… due to the addition of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 22, this extends to all bounds4 we compute on pulse times in layers β„“ > ℓ𝑖+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝐷 β‰₯ 1 and thus log 𝐷 β‰₯ 0, we get that the local skew in step 𝑖 + 1 is bounded by 5𝐡𝑖 + 4πœ… = 4πœ…(2 + log 𝐷)5𝑖+1 π‘–βˆ‘οΈ 𝑗=0 5βˆ’π‘— + 4πœ… ≀ 4πœ…(2 + log 𝐷)5𝑖+1 𝑖+1 βˆ‘οΈ 𝑗=0 5βˆ’π‘— = 𝐡𝑖+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Bounding Skews with Uniform Fault Distribution The bound in Theorem 4, which is exponential in 𝑓 , seems to suggest that the system can only support a very small number of faults or the local skew explodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we have not yet taken into account that the starting point of our entire approach is the assumption that faults are sufficiently sparse, meaning that it is highly unlikely that many of them cluster together in a way that causes an exponential pile-up of local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This enables the self-stabilization properties of the algorithm to prevent such a build-up altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In the following, assume that each node fails uniformly and independently with probability π‘œ(π‘›βˆ’1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is the largest probability of error we can support while guaranteeing that no node has more than one faulty predecessor with probability 1 βˆ’ π‘œ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A key observation is that this entails that within a fairly large distance of 𝑛1/12, no node has more than a constant number of faulty nodes that can influence it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We now formalize and show this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Definition 5 (Distance-𝛿 Ancestors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For node (𝑣, β„“) ∈ 𝑉ℓ and 𝛿 ∈ N, its distance-𝛿 ancestors are all nodes (𝑀, β„“β€²) ∈ 𝑉𝐺 \\ {(𝑣, β„“)} such that there is a (directed) path of length at most 𝛿 from (𝑀, β„“β€²) to (𝑣, β„“) in 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Definition 6 (Distance-𝛿 π‘˜-faulty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Node (𝑣, β„“) ∈ 𝑉ℓ, β„“ ∈ N>0 is distance-𝛿 π‘˜-faulty if π‘˜ ∈ N is minimal such that there are at most π‘˜ faulty nodes among the distance-((π‘˜ + 1)𝛿) ancestors of (𝑣, β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that 𝛿 ≀ 𝑛1/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If nodes fail independently with probability 𝑝 ∈ π‘œ(1/βˆšπ‘›), then with probability 1 βˆ’ π‘œ(1) all nodes are distance-𝛿 π‘˜-faulty for π‘˜ ≀ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In order to be distance-𝛿 π‘˜-faulty for π‘˜ > 2, a node must have at least 3 faults among its distance-(3𝛿) ancestors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The number of these ancestors is bounded by (3𝛿)2 ∈ 𝑂(𝑛1/6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝑝 ∈ π‘œ(1/βˆšπ‘›), the probability for this to happen is bounded by 𝑂(𝑝3�𝑛1/6 3 οΏ½) = 𝑂(𝑝3βˆšπ‘›) βŠ‚ π‘œ(1/𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The claim follows by applying a union bound over all 𝑛 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ 4Due to drifting hardware clocks, this does not apply to the pulse times themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we rely on Lemma 3 to prove our bounds in the absence of faults, and this is covered by Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 35 We can exploit this to control how much skews grow as the result of faults much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that Ψ𝑠 (β„“) ≀ 𝐡𝑠,β„“ and Lβ„“ ≀ 𝐡 for all layers β„“ β‰₯ β„“ and 𝑠 ∈ N, where β„“, β„“ ∈ N, if there are no faults in these layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If no node in a layer β„“ β‰₯ β„“ has more than 2 faulty nodes among its distance-(β„“ βˆ’ β„“ ) ancestors, then Ψ𝑠 (β„“) ≀ 𝐡𝑠,β„“ + 12𝐡 + 24πœ… for all β„“ β‰₯ β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We examine by how much adding faults on layers β„“ β‰₯ Β―β„“ might affect pulsing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For β„“ β‰₯ Β―β„“ and (𝑣, β„“) ∈ 𝑉ℓ, denote by 𝑓𝑣,β„“ ∈ {0, 1, 2} the number of faulty distance-(β„“ βˆ’ Β―β„“) ancestors of (𝑣, β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑓𝑣,β„“ = 0, there is no change in 𝑑𝑣,β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑓𝑣,β„“ > 0, consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If (𝑣, β„“) has no faulty predecessor, then by Lemma 22, 𝑑𝑣,β„“ is changed at most by the maximum shift that any of its predecessors undergoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On the other hand, if (𝑣, β„“) does have a faulty predecessor, then 𝑓𝑣,β„“ > 𝑓𝑀,β„“βˆ’1 for all correct predecessors of (𝑣, β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, by Lemma 21 we can bound shifts by 𝐡𝑓𝑣,β„“ , where 𝐡0 := 0 and 𝐡𝑓 +1 := 2(𝐡 + 𝐡𝑓 ) + 4πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By assumption, 𝑓𝑣,β„“ ≀ 2 and hence the maximum shift is bounded by 𝐡2 = 6𝐡 + 12πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We conclude that Ψ𝑠 (β„“) ≀ 𝐡𝑠,β„“ + 2𝐡2 = 𝐡𝑠,β„“ + 12𝐡 + 24πœ…, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Together with Lemma 23, Observation 5 shows that skews do not increase by more than a constant factor within 𝑛1/12 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we need to handle a total of Θ(βˆšπ‘›) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To this end, we slice up the task into chunks of 𝑛1/12 layers and leverage the self-stabilization properties of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For simplicity, in the following we assume that 𝑛1/12 is integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As we prove asymptotic bounds, this does not affect the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Definition 7 (Slices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Slice 𝑖 ∈ N>0 consists of layers β„“ ∈ [(𝑖 βˆ’ 1)𝑛1/12,𝑖𝑛1/12 βˆ’ 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that there are no more than 𝑛5/12 slices, because the nodes are arranged in square grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Due to the duplication of nodes on layer 0 and the boundary nodes on layers β„“ > 0, the number of slices is actually 𝑛5/12 βˆ’ Θ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As our next step towards a probabilistic skew bound, we prove that if the local skew remains bounded, then for levels 𝑠 that are not too large, Ψ𝑠 remains almost as small as without faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' First, we show a loose bound that naively accumulates shifts slice by slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that L0 ≀ 4πœ…, each node is distance-𝑛1/12 π‘˜-faulty for π‘˜ ≀ 2, and Lβ„“ ≀ 𝐡 for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then for each 𝑠 ∈ N and layer β„“ in slice 𝑖 ∈ N>0, we have that Ψ𝑠 (β„“) ≀ 22βˆ’π‘ πœ…π· + 𝑖(12𝐡 + 24πœ…).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume first that there are no faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, analogously to the proof of Theorem 2, we get that Ψ𝑠 (β„“) ≀ 22βˆ’π‘ πœ…π· for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Now we β€œadd” faults inductively slice by slice, by Lemma 23 each time increasing the bound on Ψ𝑠 (β„“) by 12𝐡 + 24πœ… for all slices 𝑗 β‰₯ 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ For larger values of 𝑠, 22βˆ’π‘ πœ…π· β‰ͺ 𝑛1/12, meaning that this naive bound is insufficient to show that Ψ𝑠 (β„“) does not increase much compared to the fault-free setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we can take things much further by leveraging Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that L0 ≀ 4πœ…, each node is distance-𝑛1/12 π‘˜-faulty for π‘˜ ≀ 2, and Lβ„“ ≀ 𝐡 ∈ π‘œ(𝑛1/12πœ…/log 𝐷) for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 36 Christoph Lenzen and Shreyas Srinivas Then for5 𝑠 ∈ N>0, 𝑠 ≀ log 𝐷 βˆ’ log(𝐡/πœ…) βˆ’ 2 log log 𝐷, it holds that Ψ𝑠 (β„“) ≀ Ψ𝑠 ∈ (1 + π‘œ(1))22βˆ’π‘ πœ…π·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that 𝐷 ∈ Θ(𝑛1/2) and hence log log 𝐷 ∈ πœ”(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Accordingly, the prerequisites of the lemma ensure that 𝑛5/12(𝐡 + πœ…) ∈ π‘œ(πœ…π·/log 𝐷) and οΏ½οΏ½οΏ½ + πœ… ∈ π‘œ(Ξ¨π‘ βˆ’1/log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, we may fix a suitable πœ€ ∈ π‘œ(1) such that 𝑛5/12(12𝐡 + 24πœ…) ≀ πœ€ log 𝐷 Β· 2πœ…π· and οΏ½οΏ½ Ξ¨π‘ βˆ’1 𝑛5/12πœ… οΏ½ + 1 οΏ½ (12𝐡 + 24πœ…) ≀ πœ€ 4 log 𝐷 Β· Ξ¨π‘ βˆ’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We claim that if 𝑛 is sufficiently large such that πœ€ ≀ 1, we have that Ψ𝑠 (β„“) ≀ Ψ𝑠 := 22βˆ’π‘ πœ…π· Β· οΏ½ 1 + πœ€π‘  log 𝐷 οΏ½ , which we show by induction on 𝑠 ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For the base case of 𝑠 = 1, note that there are no more than 𝑛5/12 slices, yielding by Lemma 24 that Ξ¨1(β„“) ≀ 2πœ…π· + 𝑛5/12(12𝐡 + 24πœ…) ≀ οΏ½ 1 + πœ€ log 𝐷 οΏ½ 2πœ…π·, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', indeed Ξ¨1(β„“) ≀ Ξ¨1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Now assume that the claim holds for 𝑠 βˆ’ 1 ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then, by Lemma 24 and the induction hypothesis, for layers β„“ in slices 𝑖 ≀ ⌈(Ξ¨π‘ βˆ’1/(𝑛1/12πœ…)βŒ‰, we have that Ψ𝑠 (β„“) ≀ 22βˆ’π‘ πœ…π· + οΏ½ Ξ¨π‘ βˆ’1 𝑛1/12πœ… οΏ½ (12𝐡 + 24πœ…) < Ξ¨π‘ βˆ’1 2 + οΏ½οΏ½ Ξ¨π‘ βˆ’1 𝑛1/12πœ… οΏ½ + 1 οΏ½ (12𝐡 + 24πœ…).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For a layer β„“ in a slice 𝑖 > ⌈(Ξ¨π‘ βˆ’1/(𝑛1/12πœ…)βŒ‰, assume first that we add only faults in slices 𝑗 < 𝑖 βˆ’ ⌈(Ξ¨π‘ βˆ’1/(𝑛1/12πœ…)βŒ‰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, we can apply Lemma 18, shifting layer indices such that β€œlayer 0” is the first layer of slice 𝑖 βˆ’ ⌈(Ξ¨π‘ βˆ’1/(𝑛1/12πœ…)βŒ‰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this setting, we thus have that Ψ𝑠 (β„“) ≀ Ξ¨π‘ βˆ’1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We now apply Lemma 23 inductively to slices 𝑗 ∈ [π‘–βˆ’βŒˆ(Ξ¨π‘ βˆ’1/(𝑛1/12πœ…)βŒ‰,𝑖], adding in total (βŒˆΞ¨π‘ βˆ’1/(𝑛1/12πœ…)βŒ‰+ 1)(12𝐡 + 24πœ…) to the bound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', Ψ𝑠 (β„“) ≀ Ξ¨π‘ βˆ’1 2 + οΏ½οΏ½ Ξ¨π‘ βˆ’1 𝑛1/12πœ… οΏ½ + 1 οΏ½ (12𝐡 + 24πœ…) ≀ οΏ½1 2 + οΏ½ πœ€ 4 log 𝐷 οΏ½οΏ½ Ξ¨π‘ βˆ’1 = οΏ½1 2 + οΏ½ πœ€ 4 log 𝐷 οΏ½οΏ½ 22βˆ’(π‘ βˆ’1)πœ…π· Β· οΏ½ 1 + πœ€(𝑠 βˆ’ 1) log 𝐷 οΏ½ = 22βˆ’π‘ πœ…π· Β· οΏ½ 1 + πœ€(𝑠 βˆ’ 1/2) log 𝐷 + πœ€2 2 log2 𝐷 οΏ½ ≀ 22βˆ’π‘ πœ…π· Β· οΏ½ 1 + πœ€π‘  log 𝐷 οΏ½ , where the last step assumes that 𝑛 is large enough so that πœ€ ≀ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ 5If 𝐷 = 1, we assume the upper bound on 𝑠 to be negative and the claim is vacuously true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that we are making an asymptotic statement in 𝑛 and that 𝐷 grows with 𝑛, so this case is actually of no concern here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 37 Our goal is to bound Ψ⌊log π·βŒ‹ by 𝑂(πœ… log 𝐷), since by Observation 1 this implies a bound of 𝑂(πœ… log 𝐷) on the local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, we will use the above lemma with 𝐡 ∈ 𝑂(πœ… log 𝐷), which gets us within 𝑂(log log 𝐷) levels of our β€œtarget” level ⌊log π·βŒ‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To bridge this remaining gap, we exploit that the time required for stabilizing the remaining 𝑂(log log 𝐷) levels after a fault-induced increase of skews takes only log𝑂 (1) 𝐷 = log𝑂 (1) 𝑛 βŠ‚ π‘œ(𝑛1/12) layers, since the involved potentials are bounded by π‘œ(πœ…π‘›1/12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that L0 ≀ 4πœ… and each node is distance-𝑛1/12 π‘˜-faulty for π‘˜ ≀ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Lβ„“ ∈ 𝑂(πœ… log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume towards a contradiction that the claim is false, and let Β―β„“ ∈ N>0 be minimal such that Lβ„“ is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, for layers β„“ < Β―β„“, we may assume that Lβ„“ ≀ πΆπœ… log 𝐷 for a sufficiently large constant 𝐢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consider 𝑠 = ⌊log 𝐷 βˆ’ log(𝐡/πœ…) βˆ’ 2 log log 𝐷 βˆ’ logπΆβŒ‹ βˆ’ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 25, for all β„“ ∈ N, β„“ < Β―β„“ it holds that Ψ𝑠 (β„“) ∈ Ψ𝑠 := (1 + π‘œ(1))22βˆ’π‘ πœ…π· βŠ† οΏ½1 4 + π‘œ(1) οΏ½ log3 𝐷, which for sufficiently large 𝑛 is smaller than ⌊log3 π·βŒ‹/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In fact, this bound also applies to layer Β―β„“, since the pulsing times of nodes on layer Β―β„“ depend only on the behavior of nodes on layer Β―β„“ βˆ’ 1 and the delays of messages sent to nodes on layer Β―β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Now assume that 𝑛 is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This ensures that log3 𝐷 ≀ 𝑛1/12, implying by the prerequisites of the lemma that each node is distance-(log3 𝐷) π‘˜-faulty for π‘˜ ≀ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consider adjacent correct nodes (𝑣, β„“), (𝑀, β„“) ∈ 𝑉ℓ \\ 𝐹 for any β„“ ∈ N, β„“ ≀ Β―β„“, and {𝑣,𝑀} ∈ 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We first show that distance-(log3 𝐷) 0-faulty nodes satisfy that 𝑑𝑣,β„“ βˆ’ 𝑑𝑀,β„“ ∈ (4 + π‘œ(1))πœ…(2 + log 𝐷) βŠ‚ 𝑂(πœ… log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (6) Since faults that are not among the ancestry of a node cannot affect its pulse time, this follows by applying Theorem 3 with β„“ = β„“ βˆ’ ⌊(log3 𝐷)βŒ‹ ≀ β„“ βˆ’ 2Ψ𝑠 and 𝑠′ := ⌊log π·βŒ‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To extend this to distance-(log3 𝐷) π‘˜-faulty nodes for π‘˜ ∈ {1, 2}, we show by induction on π‘˜ ∈ {0, 1, 2} that such nodes have their pulse time shifted by no more than 𝑂(πœ… log 𝐷) relative to an execution in which they are distance-(log3 𝐷) 0-faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The base case of π‘˜ = 0 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To perform the step from π‘˜ βˆ’ 1 ∈ {0, 1} to π‘˜, assume towards a contradiction that there is a node (𝑣, β„“) with a larger shift, on some minimal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Now consider a distance-(log3 𝐷) π‘˜-faulty node (𝑣, β„“) ∈ 𝑉ℓ \\ 𝐹, β„“ ≀ Β―β„“, whose predecessors are all correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' There must be a distance-(log3 𝐷) ancestor of (𝑣, β„“) that is faulty, since otherwise (𝑣, β„“) would be distance-(log3 𝐷) 0-faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let 𝑑 be the minimal distance in which there is a faulty ancestor of (𝑣, β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then all ancestors of (𝑣, β„“) in distance 𝑑 are distance-(log3 𝐷) π‘˜β€²-faulty for π‘˜β€² < π‘˜, as otherwise (𝑣, β„“) would be π‘˜β€²-faulty for some π‘˜β€² > π‘˜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consider an ancestor of (𝑣, β„“) in distance 𝑑 βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If its predecessors are all correct, by the induction hypothesis and Lemma 22 their pulse time is shifted by 𝑂(πœ… log 𝐷) relative to an execution in which they are distance distance-(log3 𝐷) 0-faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If there is a faulty predecessor, we infer this from the induction hypothesis, Equation (6), and Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='6 If 𝑑 > 1, we now inductively apply Lemma 22 until having extended this bound to all ancestors of (𝑣, β„“) within distance 𝑑 βˆ’ 1 and finally (𝑣, β„“) itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is a contradiction to (𝑣, β„“) violating the claimed bound on the shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 6Here the constants in the 𝑂-notation change, while Lemma 22 maintains the bound used in its prerequisites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since we perform only two inductive steps, we do not need to keep track of how much the constants increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 38 Christoph Lenzen and Shreyas Srinivas We conclude that indeed shifts are bounded by 𝑂(πœ… log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From this and Equation (6), it im- mediately follows that L Β―β„“ ∈ 𝑂(πœ… log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝐢 is sufficiently large, for sufficiently large 𝑛 this is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We conclude that Lβ„“ ∈ 𝑂(πœ… log 𝐷) for all β„“ ∈ N, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Putting these results together, we arrive the desired bound on the local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' With probability 1 βˆ’ π‘œ(1), Lβ„“ ∈ 𝑂(πœ… log 𝐷) for all β„“ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Corollary 6, with probability 1 βˆ’ π‘œ(1) it holds that L0 ≀ πœ…/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Observation 5, with probability 1 βˆ’ π‘œ(1) each node is distance-𝑛1/12 π‘˜-faulty for π‘˜ ≀ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By a union bound, both events occur concurrently with probability 1 βˆ’ π‘œ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, the claim follows by applying Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Gradient TRIX 39 REFERENCES [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Bailey.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theory of Computing Systems, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [12] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Kinniment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Synchronization and Arbitration in Digital Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Wiley Publishing, 2008.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' CoRR, abs/1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='2894, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [14] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Kuhn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lenzen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Locher, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Oshman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Optimal Gradient Clock Synchronization in Dynamic Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Symposium on Principles of distributed computing (PODC), 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [15] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Kuhn and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Oshman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient Clock Synchronization Using Reference Broadcasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Conference on Principles of Distributed Systems (OPODIS), pages 204–218, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [16] Clock Synchronisation and Adversarial Fault Tolerance, 2021, retrieved on 04 Jan 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='mpi-inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='de/ fileadmin/inf/d1/teaching/summer21/csaft/reading-material-ch09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [17] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lenzen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Locher, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Wattenhofer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Clock Synchronization with Bounded Global and Local Skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Symposium on Foundations of Computer Science (FOCS), pages 509–518, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lenzen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Locher, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Wattenhofer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Tight Bounds for Clock Synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Journal of the ACM, 57(2), 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lenzen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Rybicki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Self-Stabilising Byzantine Clock Synchronisation Is Almost as Easy as Consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Journal of the ACM, 66(5), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lenzen and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Wiederhake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' TRIX: Low-Skew Pulse Propagation for Fault-Tolerant Hardware, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' org/abs/2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='01415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Shelar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Routing with Constraints for Post-Grid Clock Distribution in Microprocessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 29(2):245–249, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [22] Transistor Count, retreived Oct 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='org/wiki/Transistor_count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [23] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Xanthopoulos, editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Clocking in Modern VLSI Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Springer US, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 40 Christoph Lenzen and Shreyas Srinivas A GENERATING SYNCHRONIZED INPUTS In this appendix we describe a method for generating well synchronised pulses at layer 0, at a rate of roughly one pulse per Ξ› time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' There are several ways of approaching this task, but even when aiming for a fault-tolerant solution, this is an easy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The reason is that we merely need to maintain a small local skew on a line topology, with no alternative propagation paths to neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since our goal is to handle an independent probability of 𝑝 ∈ π‘œ(π‘›βˆ’1/2) of node failures, in fact we can simply exploit that at most βˆšπ‘› nodes are required on layer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We provide a trivial scheme that is suitable for our specific setting of the base graph 𝐺 being a line (with replicated endpoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Algorithm 2 Pulse forwarding algorithm for nodes (𝑖, 0), 𝑖 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' , 𝐷};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' node (0, 0) is the clock source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The parameter Ξ› is as described in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻 := ∞ loop do if received pulse from (𝑖 βˆ’ 1, 0) then 𝐻 := 𝐻𝑖,0(𝑑) until 𝐻𝑖,0(𝑑) = 𝐻 + Ξ› βˆ’ 𝑑 broadcast pulse to (𝑖 + 1, 0) and successors on layer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For π‘˜ ∈ N, assume that the clock source at node (0, 0) generates its π‘˜-th pulse at time (π‘˜ βˆ’ 1)Ξ›.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If all nodes on layer 0 are correct, the scheme given in the above algorithm generates pulses with local skew L0 ≀ πœ…/2 and π‘‘π‘˜ 𝑖,0 ∈ [(π‘˜ + 𝑖 βˆ’ 1)Ξ› βˆ’ π‘–πœ…/2, (π‘˜ + 𝑖 βˆ’ 1)Ξ›].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, it stabilizes after transient faults within time 𝐷Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consider first the case that there are no transient faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We prove the statement by induction on 𝑖 ∈ N, where the base case is covered by the assumptions on node 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For the step from 𝑖 βˆ’ 1 ∈ N to 𝑖, we perform an induction over the pulse number π‘˜ ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The induction hypothesis is that pulses 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' ,π‘˜ βˆ’ 1 have been generated in accordance with the claim of the lemma and the first π‘˜ βˆ’ 1 loop iterations at node 𝑖 have been completed by the time the π‘˜-th pulse message from node 𝑖 βˆ’ 1 arrives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that we can use π‘˜ = 0 as base case for this induction, for which the claim is vacuously true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For the step from π‘˜ βˆ’ 1 ∈ N to π‘˜, denote by 𝑑 β€² π‘–βˆ’1,π‘˜ ∈ [π‘‘π‘–βˆ’1,π‘˜ + 𝑑 βˆ’ 𝑒,π‘‘π‘–βˆ’1,π‘˜ + 𝑑] the reception time of the pulse message from node (0,𝑖 βˆ’ 1) at node (0,𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By the bounds on hardware clock rates, Equation (1), and the induction hypothesis of the induction on 𝑖, node (0,𝑖) generates its π‘˜-th pulse at time 𝑑𝑖,π‘˜ ∈ οΏ½ π‘‘π‘–βˆ’1,π‘˜ + 𝑑 βˆ’ 𝑒 + Ξ› βˆ’ 𝑑 πœ— ,π‘‘π‘–βˆ’1,π‘˜ + Ξ› οΏ½ βŠ† οΏ½ π‘‘π‘–βˆ’1,π‘˜ + Ξ› βˆ’ πœ… 2,π‘‘π‘–βˆ’1,π‘˜ + Ξ› οΏ½ βŠ† οΏ½ (π‘˜ + 𝑖 βˆ’ 1)Ξ› βˆ’ π‘–πœ… 2 , (π‘˜ + 𝑖 βˆ’ 1)Ξ› οΏ½ , unless it receives another pulse message from (𝑖 βˆ’ 1, 0) before doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This, however, is not the case, since we assume that message delays and hardware clock rates do not vary over time, entailing that these reception times lie Ξ› time apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='7 7Note that a separation of Ξ› βˆ’ 𝑑 time would suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The slack of 𝑑 means that small changes in timing between pulses are unproblematic, which we exploit in Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 41 It remains to show the claimed bound on stabilization time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To this end, observe that the only state information that nodes maintain is 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On reception of a pulse message, this state is overwritten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This will remove spurious state from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We would like to argue that the above induction can therefore be performed as-is, meaning that the system has stabilized by the time each node has generated its first pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, there is a subtlety: it could happen that a spurious message that is still in transit at time 0 overwrites the state of node (1, 0) after it received the first message from (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Node (1, 0) then behaves as if the first message of (0, 0) arrived later, at the exact same time as the spurious message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Because also such a spurious message is delivered within at most 𝑑 time, we can re-interpret this as a longer delay of still at most 𝑑 of the first message sent by node (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that this modification reduces the difference between the reception times of the first and second pulse from node (0, 0) at node (1, 0) by up to 𝑒, but the separation remains at least Ξ› βˆ’ 𝑒 β‰₯ Ξ› βˆ’ 𝑑, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the second message is not received before (1, 0) generates its first pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We can apply the same scheme to nodes 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' , 𝐷, resulting in the desired bound on the stabilization time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' L0 ≀ πœ…/2 with probability 1 βˆ’ π‘œ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It is self-stabilizing with stabilization time Λ𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We remark that for a general base graph 𝐺, ensuring a small local skew is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, so long as |𝑉 | is small enough such that faults on layer 0 occur with probability π‘œ(1), one is free to fall back on a non-fault-tolerant GCS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This achieves L0 ∈ 𝑂(πœ… log 𝐷), which does not increase the asymptotic local skew bound of the pulse forwarding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 42 Christoph Lenzen and Shreyas Srinivas B FULL PULSE FORWARDING ALGORITHM Algorithm 3 Discrete GCS at node (𝑣, β„“), β„“ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The parameters Ξ›, and πœ… will be determined later, based on the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' loop 𝐻min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻own,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻max := ∞ for {𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝑀} ∈ 𝐸 do π‘Ÿπ‘€ := 0 do if received pulse from π‘£β„“βˆ’1 and 𝐻own = ∞ then 𝐻own := 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ (𝑑) if for some {𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝑀} ∈ 𝐸 received pulse from (𝑀,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β„“ βˆ’ 1) and π‘Ÿπ‘€ = 0 then if π‘Ÿπ‘€β€² = 0 for all {𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝑀 β€²} ∈ 𝐸 then 𝐻min := 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ (𝑑) π‘Ÿπ‘€ := 1 if π‘Ÿπ‘€β€² = 1 for all {𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝑀 β€²} ∈ 𝐸 then 𝐻max := 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ (𝑑) until 𝐻min < ∞ and 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ (𝑑) β‰₯ min{𝐻max + πœ…/2 + πœ—πœ…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 2𝐻own βˆ’ 𝐻min + 2πœ…)} if 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ (𝑑) = 𝐻max + πœ…/2 + πœ—πœ… then wait until 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ (𝑑) = 𝐻max + 3πœ…/2 + Ξ› βˆ’ 𝑑 else C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ := min𝑠 ∈N{max{𝐻own βˆ’ 𝐻max + 4π‘ πœ…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ…}} βˆ’ πœ…/2 if C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ < 0 then C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ := min {𝐻own βˆ’ 𝐻min + 3πœ…/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 0} else if C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ > πœ—πœ… then C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ := max {𝐻own βˆ’ 𝐻max βˆ’ 3πœ…/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='πœ—πœ…} wait until 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ (𝑑) = 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='β„“ broadcast pulse A basic requirement for the algorithm to work correctly is that (𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β„“) receives the π‘˜-th pulses of all correct predecessors within its π‘˜-th iteration of the main loop of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all π‘˜ ∈ N and (𝑣, β„“) ∈ 𝑉ℓ, β„“ > 0, node (𝑣, β„“) receives the π‘˜-th pulses of all correct predecessors within its π‘˜-th iteration of the main loop of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We show by induction on β„“ ∈ N>0 and π‘˜ ∈ N>0 that (𝑣, β„“) broadcasts the π‘˜π‘‘β„Ž pulse after receiving the π‘˜-th pulse from all correct (𝑀, β„“ βˆ’ 1) satisfying that ((𝑀, β„“ βˆ’ 1), (𝑣, β„“)) ∈ 𝐸, but before receiving the (π‘˜ + 1)-th pulse from such a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, for all π‘˜ β‰₯ 2, π‘‘π‘˜ 𝑣,β„“ βˆ’ π‘‘π‘˜βˆ’1 𝑣,β„“ = Ξ›.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For the induction on β„“, we use β„“ = 0 as base case, requiring only that nodes generate pulses at frequency 1/Ξ›.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As delays and clock speeds do not change, this holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For the step from β„“ βˆ’ 1 ∈ N to β„“, we perform the induction on π‘˜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that the claim holds for all π‘˜β€² < π‘˜ ∈ N>0 and consider the π‘˜-th iteration of the outer loop at (𝑣, β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The inner loop terminated because 𝐻𝑣,β„“ (𝑑) = 𝐻max + πœ…/2 + πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then a message from each node (𝑀, β„“ βˆ’ 1), {𝑣,𝑀} ∈ 𝐸, has been received in the current loop iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By the induction hypotheses for layer β„“ βˆ’ 1 and pulse π‘˜ βˆ’ 1, respectively, for correct such nodes this is the π‘˜-th pulse message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We need to show that the π‘˜-th message from (𝑣, β„“ βˆ’ 1) is received in time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' the induction hypothesis guarantees that it is not received too early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As the minimum degree of 𝐺 is 2, at Gradient TRIX 43 least one node (𝑀, β„“ βˆ’ 1), {𝑣,𝑀} ∈ 𝐸, is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If (𝑣, β„“ βˆ’ 1) is correct, too, it sent its pulse message at the latest at time 𝑑𝑀,β„“βˆ’1 + Lβ„“βˆ’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By the bounds on message delay and clock speed, this message is received at a local time 𝐻 ≀ 𝐻max + πœ—(Lβ„“βˆ’1 + 𝑒) ≀ 𝐻max + Ξ› βˆ’ 𝑑 < 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The inner loop terminated because 𝐻𝑣,β„“ (𝑑) = 2𝐻own βˆ’π»min +2πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝐻min < ∞, also 𝐻own < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using that 𝐻min ≀ 𝐻max, we get that Ξ” := min 𝑠 ∈N {max{𝐻own βˆ’ 𝐻max + 4π‘ πœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ…}} βˆ’ πœ… 2 ≀ max{𝐻own βˆ’ 𝐻max, 𝐻own βˆ’ 𝐻min} βˆ’ πœ… 2 ≀ 𝐻own βˆ’ 𝐻min βˆ’ πœ… 2 and hence C𝑣,β„“ ≀ 𝐻own βˆ’ 𝐻min + 3πœ…/2 ≀ 3πœ…/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It follows that 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“) β‰₯ max{𝐻min, 𝐻own} + Ξ› βˆ’ 𝑑 βˆ’ 3πœ… 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – (𝑣, β„“ βˆ’ 1) is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then by the bounds on message delay and clock speed, for each correct (𝑀, β„“ βˆ’ 1), {𝑣,𝑀} ∈ 𝐸, its π‘˜-th pulse message is received at a local time 𝐻 ≀ 𝐻own + πœ—(Lβ„“βˆ’1 + 𝑒) ≀ 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ 3πœ… 2 < 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“), where the last step uses Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – (𝑣, β„“ βˆ’ 1) is faulty, implying that all (𝑀, β„“ βˆ’ 1), {𝑣,𝑀} ∈ 𝐸, are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then by the bounds on message delay and clock speed, for each correct (𝑀, β„“ βˆ’ 1), {𝑣,𝑀} ∈ 𝐸, its π‘˜-th pulse message is received at a local time 𝐻 ≀ 𝐻min + Ξ› βˆ’ 𝑑 βˆ’ 3πœ… 2 < 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“), where we use that in order to guarantee that Ξ› βˆ’ 𝑑 β‰₯ πœ—(2Lβ„“βˆ’1 + 𝑒) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', Equation (2)), this must also hold in an execution that differs by (𝑣, β„“ βˆ’ 1) being correct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' in such an execution, we have that max {𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} βˆ’ min {𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} ≀ max {𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} βˆ’ 𝑑𝑣,β„“βˆ’1 + 𝑑𝑣,β„“βˆ’1 βˆ’ min {𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} ≀ 2Lβ„“βˆ’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It remains to show that (𝑣, β„“) generates its pulse before receiving a (π‘˜ + 1)-th pulse message from a correct predecessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (𝑣, β„“ βˆ’ 1) is not faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then the earliest local time 𝐻 at which (𝑣, β„“) has received a π‘˜-th pulse from a correct predecessor is bounded from below by 𝐻 β‰₯ 𝐻own βˆ’ πœ—(Lβ„“βˆ’1 + 𝑒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As delays and clock speeds do not change, the earliest message reception time for a (π‘˜ + 1)- th pulse from a correct predecessor is Ξ› time later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, it is sufficient to show that 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“) ≀ 𝐻 + Ξ›.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish three subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – The inner loop terminated because 𝐻𝑣,β„“ (𝑑) = 𝐻max + πœ…/2 + πœ—πœ… and at local time 𝐻min a message from a correct predecessor (𝑀, β„“ βˆ’ 1), {𝑣,𝑀} ∈ 𝐸, was received by (𝑣, β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝐻own + πœ—(Lβ„“βˆ’1 + 𝑒) + 2πœ… β‰₯ 2𝐻own βˆ’ 𝐻min + 2πœ… β‰₯ 𝐻max + πœ… 2 + πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 44 Christoph Lenzen and Shreyas Srinivas and, by Equation (3), 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“) = 𝐻max + 3πœ… 2 + Ξ› βˆ’ 𝑑 ≀ 𝐻own + πœ—(Lβ„“βˆ’1 + 𝑒) + 2πœ… + Ξ› βˆ’ 𝑑 ≀ 𝐻own βˆ’ πœ—(Lβ„“βˆ’1 + 𝑒) ≀ 𝐻 + Ξ›.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – The inner loop terminated because 𝐻𝑣,β„“ (𝑑) = 𝐻max + πœ…/2 + πœ—πœ… and at local time 𝐻max a message from a correct predecessor (𝑀, β„“ βˆ’ 1), {𝑣,𝑀} ∈ 𝐸, was received by (𝑣, β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, 𝐻own + πœ—(Lβ„“βˆ’1 + 𝑒) β‰₯ 𝐻max and, by Equation (3), 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“) = 𝐻max + 3πœ… 2 + Ξ› βˆ’ 𝑑 ≀ 𝐻own + πœ—(Lβ„“βˆ’1 + 𝑒) + 3πœ… 2 + Ξ› βˆ’ 𝑑 ≀ 𝐻own βˆ’ πœ—(Lβ„“βˆ’1 + 𝑒) ≀ 𝐻 + Ξ›.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – The inner loop terminated because 𝐻𝑣,β„“ (𝑑) = 2𝐻ownβˆ’π»min+2πœ… and C𝑣,β„“ β‰₯ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Equation (3), then 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“) = 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ C𝑣,β„“ ≀ 𝐻own + Ξ› βˆ’ 𝑑 ≀ 𝐻 + Ξ›.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – The inner loop terminated because 𝐻𝑣,β„“ (𝑑) = 2𝐻own βˆ’ 𝐻min + 2πœ… and C𝑣,β„“ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then C𝑣,β„“ = 𝐻own βˆ’ 𝐻min + 3πœ… 2 and 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“) = 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ C𝑣,β„“ = 𝐻min βˆ’ 3πœ… 2 + Ξ› βˆ’ 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝐻min is bounded from above by the earliest local reception time of a message from a correct node (𝑀, β„“ βˆ’ 1), {𝑣,𝑀} ∈ 𝐸, we have that 𝐻min ≀ 𝐻own + πœ—(Lβ„“βˆ’1 + 𝑒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Equation (3), we conclude that 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“) ≀ 𝐻own + πœ—(Lβ„“βˆ’1 + 𝑒) βˆ’ 3πœ… 2 + Ξ› βˆ’ 𝑑 < 𝐻 + Ξ›.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (𝑣, β„“ βˆ’ 1) is faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝐻 = 𝐻min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Checking all cases in a similar fashion, we see that 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“) ≀ 𝐻max + 3πœ… 2 + Ξ› βˆ’ 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using that Equation (3) must also apply in an execution where (𝑣, β„“ βˆ’ 1) is not faulty and hence max{𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} βˆ’ min{𝑣,𝑀}∈𝐸{𝑑𝑀,β„“βˆ’1} ≀ 2Lβ„“βˆ’1, it follows that 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“) ≀ 𝐻max + 3πœ… 2 + Ξ› βˆ’ 𝑑 ≀ 𝐻min + 2πœ—(Lβ„“βˆ’1 + 𝑒) + 3πœ… 2 + Ξ› βˆ’ 𝑑 ≀ 𝐻min + Ξ› ≀ 𝐻 + Ξ›.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ We are now ready to show that Algorithm 3 is equivalent to Algorithm 1 in the absence of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 45 Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that for (𝑣, β„“) ∈ 𝑉ℓ, β„“ > 0, and the predecessors of (𝑣, β„“) are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then running Algorithm 1 instead of Algorithm 3 results in the same pulse times of node (𝑣, β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume towards a contradiction that the claim is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote by π‘‘π‘˜ 𝑣,β„“ and (π‘‘π‘˜ 𝑣,β„“)β€² the pulse times of Algorithm 1 and Algorithm 3 in executions with identical delays, clock speeds, and behavior of faulty nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', let π‘‘π‘˜ 𝑣,β„“ be minimal with the property that π‘‘π‘˜ 𝑣,β„“ β‰  (π‘‘π‘˜ 𝑣,β„“)β€².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consider the π‘˜-th loop iteration of Algorithm 3 at node (𝑣, β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish cases according to why the inner loop terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The inner loop terminated because 𝐻𝑣,β„“ (𝑑) = 𝐻max + πœ…/2 + πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then in Algorithm 1, we have that 𝐻own β‰₯ 𝐻max + πœ… 2 + πœ—πœ…, implying that Ξ” := min 𝑠 ∈N {max{𝐻own βˆ’ 𝐻max + 4π‘ πœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ…}} βˆ’ πœ… 2 β‰₯ min 𝑠 ∈N {𝐻own βˆ’ 𝐻max + 4π‘ πœ…} βˆ’ πœ… 2 β‰₯ 𝐻own βˆ’ 𝐻min βˆ’ πœ… 2 β‰₯ πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, Algorithm 1 computes C𝑣,β„“ = 𝐻own βˆ’ 𝐻max βˆ’ 3πœ… 2 and generates its π‘˜-th pulse at local time 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“) = 𝐻max + Ξ› βˆ’ 𝑑 βˆ’ C𝑣,β„“ = 𝐻max + 3πœ… 2 + Ξ› βˆ’ 𝑑 = 𝐻𝑣,β„“ ((π‘‘π‘˜ 𝑣,β„“)β€²), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The inner loop terminated because 𝐻𝑣,β„“ (𝑑) = 2𝐻own βˆ’π»min +2πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝐻min < ∞, also 𝐻own < ∞ for Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – In Algorithm 1, we have Ξ” := min 𝑠 ∈N {max{𝐻own βˆ’ 𝐻max + 4π‘ πœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ…}} βˆ’ πœ… 2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then the same holds in Algorithm 3, as there 𝐻max is either identical to that of Algorithm 1 of ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, both algorithms compute 𝐢𝑣,β„“ = min{𝐻own βˆ’π»min +3πœ…/2, 0} and subsequently 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“) = 𝐻own + Ξ› βˆ’ 𝑑 βˆ’ C𝑣,β„“ = 𝐻𝑣,β„“ ((π‘‘π‘˜ 𝑣,β„“)β€²), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – In Algorithm 1, we have Ξ” := min 𝑠 ∈N {max{𝐻own βˆ’ 𝐻max + 4π‘ πœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4π‘ πœ…}} βˆ’ πœ… 2 β‰₯ 0 Let 𝑠min ∈ N be such that Ξ” := max{𝐻own βˆ’ 𝐻max + 4𝑠minπœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4𝑠minπœ…} βˆ’ πœ… 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If Ξ” = 𝐻own βˆ’ 𝐻min βˆ’ 4𝑠minπœ… βˆ’ πœ…/2, the fact that 𝐻own and 𝐻min are identical in both algorithms, while 𝐻max is either also identical or βˆ’βˆž in Algorithm 3, again leads to the 46 Christoph Lenzen and Shreyas Srinivas contradiction 𝐻𝑣,β„“ (π‘‘π‘˜ 𝑣,β„“) = 𝐻𝑣,β„“ ((π‘‘π‘˜ 𝑣,β„“)β€²).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, suppose that Ξ” = 𝐻own βˆ’π»max + 4𝑠minπœ… βˆ’πœ…/2 in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, 0 ≀ Ξ” = 𝐻own βˆ’ 𝐻max + 4𝑠minπœ… βˆ’ πœ…/2 ≀ max{𝐻own βˆ’ 𝐻max + 4(𝑠min βˆ’ 1)πœ…, 𝐻own βˆ’ 𝐻min βˆ’ 4(𝑠min βˆ’ 1)πœ…} βˆ’ πœ… 2 = 𝐻own βˆ’ 𝐻min βˆ’ 4(𝑠min βˆ’ 1)πœ… βˆ’ πœ… 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 2𝐻own βˆ’ 𝐻min + 2πœ… β‰₯ 𝐻own + 4𝑠minπœ… βˆ’ 3πœ… 2 β‰₯ 𝐻max βˆ’ πœ… < 𝐻max βˆ’ πœ… 2 βˆ’ πœ—πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is a contradiction, as then the inner loop in Algorithm 3 would have terminated at an earlier time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='1 Self-Stabilization Making Algorithm 3 self-stabilizing follows standard techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Accordingly, we confine ourselves to a brief high-level discussion of how this is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The pulse propagation algorithm can be implemented in a self-stabilizing way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It stabilizes within 𝑂(βˆšπ‘›) pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The key observation is that self-stabilization can proceed layer by layer, where Corollary 6 shows that layer 0 stabilizes fast enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, we can assume that the correct nodes of the previous layer generate pulses at a stable frequency of Ξ› satisfying the skew bounds obtained in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This allows us to make sure that the timing of its listening loop aligns with the pulse signals from the previous layer: From all but one predecessor, the pulse signals must be received while the inner loop is running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, the inner loop will terminate within Ξ› time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Instead of restarting the inner loop dependent on the own generated pulse, we can instead start a loop iteration when receiving the first pulse after a quiet period of, say, Ξ›/10 (where too frequent pulses of a faulty predecessor are filtered out).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As such a quiet period must occur by Equation (2), this will align the loop correctly with the π‘˜-th pulses of correct predecessors for some π‘˜ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Once the inner loop terminates, we look for the next quiet period, and start a new instance of the inner loop on reception of the next pulse from a predecessor, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Whenever the inner loop terminates correctly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', not due to a timeout, we also compute the time to generate the next pulse as in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we do not wait until the pulse is generated before willing to start a new instance of the inner loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This way, we ensure that we do not miss the first pulse message of a correct predecessor for pulse π‘˜ + 1 in case the inner loop for pulse π‘˜ was misaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ C OBTAINING THE FINAL SKEW BOUNDS Recall that our model assumes that message delays and clock speeds do not vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If the behavior of faulty nodes is static, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the timing of their output pulse messages is identical in each pulse as well, a stable input frequency of 1/Ξ› results in repeating the exact same message pattern with the same timing every 1/Ξ› time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We can exploit this to bound Lβ„“,β„“+1 in terms of Lβ„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If faulty nodes do not change the timing of their output pulses, then L ∈ 𝑂(πœ… log 𝐷) with probability 1 βˆ’ π‘œ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 47 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Corollary 5, for correct (𝑣, β„“ + 1) ∈ 𝑉ℓ+1, β„“ ∈ N, min ((𝑀,β„“),(𝑣,β„“+1)) βˆˆπΈβ„“ (𝑀,β„“)βˆ‰πΉ {π‘‘π‘˜ 𝑀,β„“} + Ξ› βˆ’ 2πœ… ≀ π‘‘π‘˜ 𝑣,β„“+1 ≀ max ((𝑀,β„“),(𝑣,β„“)) βˆˆπΈβ„“ (𝑀,β„“)βˆ‰πΉ {π‘‘π‘˜ 𝑀,β„“} + Ξ› + 2πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Because the behavior of fault nodes does not change between pulses, a simple induction shows that π‘‘π‘˜+1 π‘₯,β„“β€² = π‘‘π‘˜ π‘₯,β„“β€² + Ξ› for all correct nodes (π‘₯, β„“β€²) ∈ 𝑉ℓ′, β„“β€² ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, min ((𝑀,β„“),(𝑣,β„“+1)) βˆˆπΈβ„“ (𝑀,β„“βˆ’1)βˆ‰πΉ {π‘‘π‘˜+1 𝑀,β„“ } βˆ’ 2πœ… ≀ π‘‘π‘˜ 𝑣,β„“+1 ≀ max ((𝑀,β„“),(𝑣,β„“+1)) βˆˆπΈβ„“ (𝑀,β„“)βˆ‰πΉ {π‘‘π‘˜+1 𝑀,β„“ } + 2πœ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Theorem 5, Lβ„“ ∈ 𝑂(πœ… log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that this bound applies uniformly over all executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, even if (𝑣, β„“) is faulty, using that its neighbors are within distance 2 of each other, it holds that min ((𝑀,β„“),(𝑣,β„“+1)) βˆˆπΈβ„“ (𝑀,β„“βˆ’1)βˆ‰πΉ {π‘‘π‘˜+1 𝑀,β„“ } βˆ’ max ((𝑀,β„“),(𝑣,β„“+1)) βˆˆπΈβ„“ (𝑀,β„“)βˆ‰πΉ {π‘‘π‘˜+1 𝑀,β„“ } ∈ 𝑂(πœ… log 𝐷), by virtue of comparing to an execution in which (𝑣, β„“) is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As (𝑣, β„“ + 1) was an arbitrary correct node, the claim of the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘ It remains to argue that some variation can be sustained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' With probability 1βˆ’π‘œ(1), L ∈ 𝑂(πœ… log 𝐷) even when in each pulse (i) a constant number of faulty nodes change their output behavior and timing, (ii) link delays vary by up to π‘›βˆ’1/2𝑒 log 𝐷, and (iii) hardware clock speeds vary by up to π‘›βˆ’1/2(πœ— βˆ’ 1) log 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The maximum length of a directed path in 𝐻 is bounded by 2βˆšπ‘›: at most 𝐷 ≀ βˆšπ‘› hops in layer 0, followed by at most βˆšπ‘› links from layer to layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, accumulating all changes in timing due to link delay and clock speed variation along a path results in a deviation of 𝑂((𝑒 + (πœ— βˆ’ 1)(Ξ› βˆ’ 𝑑)) log 𝐷 = 𝑂(πœ… log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is trivial for layer 0 and applies to pulse propagation through the layers as well, because our respective analysis relies on Corollary 5 and Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In order to take into account a constant number of faulty nodes with arbitrary behavior, we reason analogously to the proof of Theorem 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', rely on Corollary 5 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' β–‘' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'}