id int64 1 141k | title stringlengths 15 150 | body stringlengths 43 35.6k | tags stringlengths 1 118 | label int64 0 1 |
|---|---|---|---|---|
367 | How can it be decidable whether $\pi$ has some sequence of digits? | <p>We were given the following exercise.</p>

<blockquote>
 <p>Let</p>
 
 <p>$\qquad \displaystyle f(n) = \begin{cases} 1 & 0^n \text{ occurs in the decimal representation of } \pi \\ 0 & \text{else}\end{cases}$</p>
 
 <p>Prove that $f$ is computable.</p>
</blockquote>
&... | computability undecidability | 1 |
368 | Counting binary trees | <p>(I'm a student with some mathematical background and I'd like to know how to count the number of a specific kind of binary trees.)</p>

<p>Looking at Wikipedia page for <a href="http://en.wikipedia.org/wiki/Binary_tree">Binary Trees</a>, I've noticed this assertion that the number of rooted binary trees of s... | combinatorics binary trees discrete mathematics | 1 |
374 | Languages accepted by modified versions of finite automata | <p>A deterministic finite automaton (DFA) is a state machine model capable of accepting all and only regular languages. DFAs can be (and usually are) defined in such a way that each state must provide some transition for all elements of the input alphabet; in other words, the transition function $\delta : Q \times \Sig... | formal languages automata finite automata | 1 |
376 | Is this language defined using twin primes regular? | <p>Let</p>

<p>$\qquad L = \{a^n \mid \exists_{p \geq n}\ p\,,\ p+2 \text{ are prime}\}.$</p>

<p>Is $L$ regular?</p>

<p>This question looked suspicious at the first glance and I've realized that it is connected with the <a href="https://en.wikipedia.org/wiki/Twin_prime">twin prime conjecture</... | formal languages automata regular languages finite automata | 1 |
386 | Path to formal methods | <p>It is not uncommon to see students starting their PhDs with only a limited background in mathematics and the formal aspects of computer science. Obviously it will be very difficult for such students to become theoretical computer scientists, but it would be good if they could become savvy with using formal methods a... | formal methods education | 0 |
390 | Proving closure under reversal of languages accepted by min-heap automata | <p><em>This is a follow-up question of <a href="https://cs.stackexchange.com/q/110/98">this one</a>.</em></p>

<p>In a previous question about <a href="https://cs.stackexchange.com/q/110/69">exotic state machines</a>, Alex ten Brink and Raphael addressed the computational capabilities of a peculiar kind of stat... | formal languages automata closure properties | 0 |
393 | Proving closure under complementation of languages accepted by min-heap automata | <p><em>This is a follow-up question of <a href="https://cs.stackexchange.com/q/110/98">this one</a></em>.</p>

<p>In a previous question about <a href="https://cs.stackexchange.com/q/110/69">exotic state machines</a>, Alex ten Brink and Raphael addressed the computational capabilities of a peculiar kind of stat... | formal languages automata closure properties | 1 |
394 | Computational power of deterministic versus nondeterministic min-heap automata | <p><em>This is a follow-up question of <a href="https://cs.stackexchange.com/q/110/98">this one</a>.</em></p>

<p>In a previous question about <a href="https://cs.stackexchange.com/q/110/69">exotic state machines</a>, Alex ten Brink and Raphael addressed the computational capabilities of a peculiar kind of stat... | formal languages automata nondeterminism | 1 |
396 | A DFA for recognizing comments | <p>The following DFA is a lexical analyzer which is supposed to recognize comments. The lexical analyzer will ignore the comment and goes back to the state one. I'm told that there's something wrong with it but I can't figure it out. What's the problem?</p>

<p><img src="https://i.stack.imgur.com/EeIdO.png" alt... | formal languages automata finite automata compilers | 1 |
401 | Is a function looking for subsequences of digits of $\pi$ computable? | <p><a href="https://cs.stackexchange.com/questions/367/how-can-it-be-decidable-whether-pi-has-some-sequence-of-digits">How can it be decidable whether $\pi$ has some sequence of digits?</a> inspired me to ask whether the following innocent-looking variation is computable:</p>

<p>$$f(n) = \begin{cases}
 1 ... | computability real numbers | 0 |
407 | Measuring the difficulty of SAT instances | <p>Given an instance of SAT, I would like to be able to estimate how difficult it will be to solve the instance.</p>
<p>One way is to run existing solvers, but that kind of defeats the purpose of estimating difficulty. A second way might be looking a the ratio of clauses to variables, as is done for phase transitio... | complexity theory satisfiability heuristics | 1 |
411 | How to find a superstar in linear time? | <p>Consider directed graphs. We call a node $v$ <em>superstar</em> if and only if no other node can be reached from it, but all other nodes have an edge to $v$. Formally:</p>

<p>$\qquad \displaystyle $v$ \text{ superstar } :\Longleftrightarrow \mathrm{outdeg}(v) = 0 \land \mathrm{indeg}(v) = n-1$</p>

... | algorithms graphs | 1 |
416 | Approximation of minimum bandwidth on binary trees | <p>Minimum bandwidth problem is to a find an ordering of graph nodes on integer line that minimizes the largest distance between any two adjacent nodes. </p>

<p>The decision problem is NP-complete even for binary trees. <a href="http://www.jstor.org/stable/10.2307/2100947">Complexity Results for Bandwidth Mini... | complexity theory np complete reference request approximation | 1 |
419 | Is there any concrete relation between Gödel's incompleteness theorem, the halting problem and universal Turing machines? | <p>I've always thought vaguely that the answer to the above question was affirmative along the following lines. Gödel's incompleteness theorem and the undecidability of the halting problem both being negative results about decidability and established by diagonal arguments (and in the 1930's), so they must somehow be t... | computability logic halting problem incompleteness | 1 |
421 | AVL trees are not weight-balanced? | <p>In a previous <a href="https://cs.stackexchange.com/questions/342/not-all-red-black-trees-are-balanced">question</a> there was a definition of weight balanced trees and a question regarding red-black trees. </p>

<p>This question is to ask the same question, but for <a href="http://en.wikipedia.org/wiki/AVL_... | data structures binary trees search trees balanced search trees avl trees | 1 |
423 | How hard is counting the number of simple paths between two nodes in a directed graph? | <p>There is an easy polynomial algorithm to decide whether there is a path between two nodes in a directed graph (just do a routine graph traversal with, say, depth-first-search).</p>

<p>However it seems that, surprisingly, the problem gets much harder if instead of testing for the existence we want want to <e... | algorithms complexity theory graphs | 1 |
427 | An indexing function for graphs | <p>Definition from wikipedia:</p>

<blockquote>
 <p>A graph is an ordered pair $G = (V, E)$ comprising a set $V$ of nodes together with a set $E$ of edges, which are two-element subsets of $V$.</p>
</blockquote>

<p>The set of all finite graphs (modulo isomorphism: we don't want nodes to have i... | algorithms graphs | 0 |
430 | Easy to state open problems in computability theory | <p>I was searching for interesting and easy to state open problems in computability (understandable by undergraduate students taking their first course in computability) to give examples of open problems (and obviously I want the students to be able to understand the problem without needing too much new definitions and... | computability | 1 |
431 | Break an authentication protocol based on a pre-shared symmetric key | <p>Consider the following protocol, meant to authenticate $A$ (Alice) to $B$ (Bob) and vice versa.</p>

<p>$$ \begin{align*}
 A \to B: &\quad \text{“I'm Alice”}, R_A \\
 B \to A: &\quad E(R_A, K) \\
 A \to B: &\quad E(\langle R_A+1, P_A\rangle, K) \\
\end{alig... | cryptography protocols authentication | 1 |
433 | Is it possible to derive a string in this rewriting system? | <p>Rewriting system is a set of rules in the form of $A \leftrightarrow B$. 
If we apply that rule to a string $w$ we replace any substring $A$ in $w$ with a substring $B$ and vice versa.</p>

<p>Given a starting string $AAABB$ can we derive $BAAB$ in the system with the following rules:</p>

<ul>&... | computability term rewriting | 1 |
435 | Fine-grained security models for XML data | <p>Access control models are typically very coarse-grained, allowing one access to a resource (possibly with some combination of read/write/execute permission) or exclude such access. Some models of database security allow access to be granted on a per row basis (though I don't have a reference handy).</p>

<p>... | security access control structured data | 0 |
438 | Is interaction more powerful than algorithms? | <p>I've heard the motto <a href="http://www.cs.brown.edu/people/pw/papers/ficacm.ps"><strong>interaction is more powerful than algorithms</strong></a> from <a href="http://www.cs.brown.edu/~pw/">Peter Wegner</a>. The basis of the idea is that a (classical) Turing Machine cannot handle interaction, that is, communicatio... | computability computation models | 1 |
439 | Which combinations of pre-, post- and in-order sequentialisation are unique? | <p>We know post-order,</p>

<pre><code>post L(x) => [x]
post N(x,l,r) => (post l) ++ (post r) ++ [x]
</code></pre>

<p>and pre-order</p>

<pre><code>pre L(x) => [x]
pre N(x,l,r) => [x] ++ (pre l) ++ (pre r)
</code></pre>

<p>and in-order traversal resp... | algorithms binary trees | 1 |
443 | termination of two concurrent threads with shared variables | <p>We're in a shared memory concurrency model where all reads and writes to integer variables are atomic. </p>

<ul>
<li><code>do:</code> $S_1$ <code>in parallel with:</code> $S_2$  means to execute $S_1$ and $S_2$ in separate threads, concurrently.</li>
<li><code>atomically(</code>$E$<code>)</co... | concurrency shared memory imperative programming | 1 |
444 | CCS process for a drink dispenser with two different prices | <p>A drink dispenser requires the user to insert a coin ($\bar c$), then press one of three buttons: $\bar d_{\text{tea}}$ requests a cup of tea $e_{\text{tea}}$, ditto for coffee, and $\bar r$ requests a refund (i.e. the machine gives back the coin: $\bar b$). This dispenser can be modeled by the following <a href="ht... | logic concurrency modelling process algebras ccs | 1 |
450 | Ray Tracing versus object-based rendering? | <p>Intro graphics courses usually have a project that asks you to build a ray tracer to render a scene. Many graphics students entering grad school say that they want to work on ray tracing. And yet it seems that ray tracing is a dead field in venues like SIGGRAPH etc. </p>

<p>Is ray tracing really the <em>bes... | graphics | 1 |
451 | Why are there so many programming languages? | <p>I'm pretty fluent in C/C++, and can make my way around the various scripting languages (awk/sed/perl). I've started using python a lot more because it combines some of the nifty aspects of C++ with the scripting capabilities of awk/sed/perl.</p>

<p>But why are there so many different programming languages ?... | programming languages | 1 |
454 | List of intro TCS books for those who don't know much about TCS | <p>If you have to recommend books for someone who wants to learn more about TCS at the introductory level such as automata theory, algorithmics, complexity theory, etc, what book(s) would you recommend for those who are interested and want to learn more about TCS, but not had any exposure to it?</p>
 | education | 0 |
455 | Why do neural networks seem to perform better with restrictions placed on their topology? | <p>Fully connected (at least layer to layer with more than 2 hidden layers) backprop networks are universal learners. Unfortunately, they are often slow to learn and tend to over-fit or have awkward generalizations. </p>

<p>From fooling around with these networks, I have observed that pruning some of the edges... | machine learning network topology neural networks | 0 |
465 | Similarities and differences in major process algebras | <p>To my knowledge, there are three major process algebras that have inspired a vast range of research into formal models of concurrency. These are:</p>

<ul>
<li>CCS and $\pi$-calculus both by Robin Milner</li>
<li>CSP by Tony Hoare and</li>
<li>ACP by Jan Bergstra and Jan Willem Klop</li>
</ul... | logic concurrency process algebras | 0 |
470 | How is the key in a private key encryption protocol exchanged? | <p>Windows NT used a point-to-point protocol where a client can communicate "securely" with a server by using a stream cipher to encrypt an array of messages with some key $k$. The server also encrypts its response with the same key $k$. But how is it aware of this key?</p>

<p>More general: if Alice and Bob us... | cryptography encryption | 1 |
473 | Decision problems vs "real" problems that aren't yes-or-no | <p>I read in many places that some problems are difficult to approximate (it is <a href="https://en.wikipedia.org/wiki/Hardness_of_approximation" rel="noreferrer"><strong>NP-hard</strong> to approximate</a> them). But approximation is not a decision problem: the answer is a real number and not Yes or No. Also for eac... | complexity theory time complexity np hard approximation | 1 |
477 | For what kind of data are hash table operations O(1)? | <p>From the answers to <a href="https://cs.stackexchange.com/questions/249/when-is-hash-table-lookup-o1">(When) is hash table lookup O(1)?</a>, I gather that hash tables have $O(1)$ worst-case behavior, at least amortized, when the data satisfies certain statistical conditions, and there are techniques to help make the... | data structures runtime analysis hash tables dictionaries | 1 |
481 | Break an authentication protocol based on a pre-shared symmetric key, with message numbers | <p>Consider the following protocol, meant to authenticate $A$ (Alice) to $B$ (Bob) and vice versa.</p>

<p>$$ \begin{align*}
 A \to B: &\quad \text{“I'm Alice”}, R_A \\
 B \to A: &\quad E(\langle 1, R_A\rangle, K) \\
 A \to B: &\quad E(\langle 2, R_A+1, P_A\rangle, K) \\
\end{align*}... | cryptography protocols authentication | 1 |
492 | Saving on array initialization | <p>I recently read that it is possible to have arrays which need not be initialized, i.e. it is possible to use them without having to spend any time trying to set each member to the default value. i.e. you can start using the array as if it has been initialized by the default value without having to initialize it. (So... | data structures arrays | 1 |
494 | Clustering of Songs (The Joe Walsh Problem) | <p>The Eagles are a rock supergroup from the 70s and 80s, responsible for such classics as <em>Hotel California</em>. They have two quite distinctive sounds, one where guitarist Joe Walsh is present (for example, in <em>Life in the Fast Lane</em>) and one where he is absent. The latter songs have a markedly more sombre... | machine learning modelling | 0 |
495 | Are generational garbage collectors inherently cache-friendly? | <p>A typical <a href="http://en.wikipedia.org/wiki/Garbage_collection_%28computer_science%29#Generational_GC_.28ephemeral_GC.29">generational garbage collector</a> keeps recently allocated data in a separate memory region. In typical programs, a lot of data is short-lived, so collecting young garbage (a minor GC cycle)... | programming languages computer architecture cpu cache garbage collection | 1 |
524 | Does there exist a priority queue with $O(1)$ extracts? | <p>There are a great many data structures that implement the priority-queue interface:</p>

<ul>
<li>Insert: insert an element into the structure</li>
<li>Get-Min: return the smallest element in the structure</li>
<li>Extract-Min: remove the smallest element in the structure</li>
</ul>

... | data structures amortized analysis priority queues | 1 |
525 | What is coinduction? | <p>I've heard of (structural) induction. It allows you to build up finite structures from smaller ones and gives you proof principles for reasoning about such structures. The idea is clear enough.</p>

<blockquote>
 <p>But what about coinduction? How does it work? How can one say anything conclusive about ... | terminology logic proof techniques formal methods coinduction | 1 |
539 | Visual Programming languages | <p>Most of us learned programming using "textual" programming languages like Basic, C/C++, and Java. I believe it is more natural and efficient for humans to think visually. Visual programming allows developers to write programs by manipulating graphical elements. I guess using visual programming should improve the qua... | programming languages | 1 |
540 | Notions of efficient computation | <p>A polynomial-time Turing machine algorithm is considered efficient if its run-time, in the worst-case, is bounded by a polynomial function in the input size. I'm aware of the strong Church-Turing thesis:</p>

<blockquote>
 <p>Any reasonable model of computation can be efficiently simulated on Turing mac... | complexity theory efficiency computation models | 1 |
541 | When are two simulations not a bisimulation? | <p>Given a <a href="http://en.wikipedia.org/wiki/State_transition_system">labelled transition system</a> $(S,\Lambda,\to)$, where $S$ is a set of states, $\Lambda$ is a set of labels, and $\to\subseteq S\times\Lambda\times S$ is a ternary relation. As usual, write $p \stackrel\alpha\rightarrow q$ for $(p,\alpha,q)\in\t... | programming languages formal methods semantics process algebras | 1 |
553 | How Do Common Pathfinding Algorithms Compare To Human Process | <p>This might border on computational cognitive science, but I am curious as to how the process followed by common pathfinding algorithms (such as <a href="http://en.wikipedia.org/wiki/A_star_search_algorithm">A*</a>) compares to the process humans use in different pathfinding situations (given the same information). A... | algorithms graphs artificial intelligence | 1 |
555 | Given a string and a CFG, what characters can follow the string (in the sentential forms of the CFG)? | <p>Let $\Sigma$ be the set of terminal and $N$ the set of non-terminal symbols of some context-free grammar $G$.</p>

<p>Say I have a string $a \in (\Sigma \cup N)^+$ such that $x a y \in \mathcal{S}(G)$ where $x,y\in (\Sigma \cup N)^*$ and $\mathcal{S}(G)$ are the sentential forms of $G$.</p>

<p>Given... | algorithms context free formal grammars compilers | 1 |
559 | Creating a Self Ordering Binary Tree | <p>I have an assignment where I need to make use a binary search tree and alter it to self order itself such that items that are accessed the most (have a higher priority) are at the top of the tree, the root being the most accessed node.</p>

<p>The professor gave me the BST and node struct to work with, but t... | algorithms data structures binary trees search trees | 1 |
561 | Why has research on genetic algorithms slowed? | <p>While discussing some intro level topics today, including the use of genetic algorithms; I was told that research has really slowed in this field. The reason given was that most people are focusing on machine learning and data mining. <br>
<strong>Update:</strong> Is this accurate? And if so, what advantages doe... | machine learning data mining evolutionary computing history | 1 |
569 | BigO, Running Time, Invariants - Learning Resources | <p>What are some good online resources that will help me better understand BigO notation, running time & invariants?</p>

<p>I'm looking for lectures, interactive examples if possible. </p>
 | algorithms landau notation education runtime analysis | 0 |
570 | Generating uniformly distributed random numbers using a coin | <p>You have one coin. You may flip it as many times as you want. </p>

<p>You want to generate a random number $r$ such that $a \leq r < b$ where $r,a,b\in \mathbb{Z}^+$. </p>

<p>Distribution of the numbers should be uniform. </p>

<p>It is easy if $b -a = 2^n$:</p>

<pre><code>r = a... | algorithms probability theory randomness random number generator | 0 |
576 | Expressing an arbitrary permutation as a sequence of (insert, move, delete) operations | <p>Suppose I have two strings. Call them $A$ and $B$. Neither string has any repeated characters.</p>

<p>How can I find the shortest sequence of insert, move, and delete operation that turns $A$ into $B$, where:</p>

<ul>
<li><code>insert(char, offset)</code> inserts <code>char</code> at the given ... | algorithms combinatorics string metrics | 0 |
578 | What is the complexity of these tree-based algorithms? | <p>Suppose we have a balanced binary tree, which represents a recursive partitioning of a set of $N$ points into nested subsets. Each node of the tree represents a subset, with the following properties: subsets represented by two children nodes of the same parent are disjoint, and their union is equal to the subset re... | algorithms time complexity binary trees | 1 |
580 | What combination of data structures efficiently stores discrete Bayesian networks? | <p>I understand the theory behind Bayesian networks, and am wondering what it takes to build one in practice. Let's say for this example, that I have a Bayesian (directed) network of 100 discrete random variables; each variable can take one of up to 10 values.</p>

<p>Do I store all the nodes in a DAG, and for ... | data structures machine learning | 1 |
581 | Algorithmic intuition for logarithmic complexity | <p>I believe I have a reasonable grasp of complexities like <span class="math-container">$\mathcal{O}(1)$</span>, <span class="math-container">$\Theta(n)$</span> and <span class="math-container">$\Theta(n^2)$</span>.</p>

<p>In terms of a list, <span class="math-container">$\mathcal{O}(1)$</span> is a constant ... | algorithms complexity theory time complexity intuition | 1 |
586 | Could quantum computing eventually be used to make modern day hashing trivial to break? | <p>Simply put, if one were to build a quantum computing device with the power of, say, 20 qubits, could such a computer be used to make any kind of modern hashing algorithm useless?</p>

<p>Would it even be possible to harness the power of quantum computing in a traditional computing application?</p>
 | cryptography quantum computing hash | 1 |
588 | Can every linear grammar be converted to Greibach form? | <p>Can every <a href="http://en.wikipedia.org/wiki/Linear_grammar" rel="nofollow">linear grammar</a> be converted to a linear <a href="http://en.wikipedia.org/wiki/Greibach_normal_form" rel="nofollow">Greibach normal form</a>, a form in which all productions look like $A \rightarrow ax$ where $a \in T$ and $x \in V \cu... | formal languages formal grammars | 1 |
594 | CPU frequency per year | <p>I know that since ~2004, Moore's law stopped working for CPU clock speed.
I'm looking for a graph showing this, but am unable to find it: most charts out there show the transistor count or the capacity per year.</p>

<p>Where can I find some data showing the CPU frequency of computers (anything is fine, ... | computer architecture empirical research data sets | 1 |
602 | Measuring one way network latency | <p>This is a puzzle about measuring network latency that I created. I believe the solution is that it's impossible, but friends disagree. I'm looking for convincing explanations either way. (Though it is posed as a puzzle I think it fits on this web site because of its applicability to the design and experience of comm... | computer networks protocols distributed systems | 1 |
614 | Late and Early Bisimulation | <p>This is a follow up to my earlier questions on <a href="https://cs.stackexchange.com/q/525/31">coinduction</a> and <a href="https://cs.stackexchange.com/q/541/31">bisimulation</a>.</p>

<p>A relation $R \subseteq S \times S$ on the states of an LTS is a <em>bisimulation</em> iff $\forall (p,q)\in R,$ 
$... | semantics formal methods process algebras pi calculus | 1 |
615 | Improve worst case time of depth first search on Euler graphs | <p>How to improve the worst case scenario for a depth first search on an Euler graph, starting at some point and ending at that same point?</p>

<p>I need to do the whole search but it is not fast enough for large amounts of data. I have tried <a href="https://en.wikipedia.org/wiki/Bidirectional_search" rel="no... | algorithms graphs graph traversal eulerian paths | 0 |
616 | How can I decide manually whether two CTL formulae are equivalent? | <p>Assume I have two formulae $\Phi$ and $\Psi$ (over the same set of atomic propositions $AP$) in <a href="http://en.wikipedia.org/wiki/Computation_tree_logic" rel="nofollow">CTL</a>. We have that $\Phi \equiv \Psi$ iff $Sat_{TS}(\Phi) = Sat_{TS}(\Psi)$ for all transition systems $TS$ over $AP$.</p>

<p>Given ... | logic model checking computation tree logic | 0 |
618 | How can solutions of a Diophantine equation be expressed as a language? | <p>I was given the question </p>

<blockquote>
 <p>Where does the following language fit in the Chomsky hierarchy?</p>
 
 <p>Nonnegative solutions $(x,y)$ to the Diophantine equation $3x-y=1$.</p>
</blockquote>

<p>I understand languages like $L = \{ 0^n1^n \mid n \ge 1\}$, but this l... | formal languages computability | 0 |
619 | How can I prove this language is not context-free? | <p>I have the following language</p>

<p>$\qquad \{0^i 1^j 2^k \mid 0 \leq i \leq j \leq k\}$</p>

<p>I am trying to determine which Chomsky language class it fits into. I can see how it could be made using a context-sensitive grammar so I know it is atleast context-sensitive. It seems like it wouldn't ... | formal languages context free formal grammars pumping lemma | 0 |
625 | Find shortest paths in a weighed unipathic graph | <p>A directed graph is said to be <em>unipathic</em> if for any two vertices $u$ and $v$ in
the graph $G=(V,E)$, there is at most one simple path from $u$ to $v$. </p>

<p>Suppose I am given a unipathic graph $G$ such that each edge has a positive or negative weight, but contains no negative weight cycles.<... | algorithms graphs | 1 |
627 | Decidablity of Languages of Grammars and Automata | <p><em>Note this is a question related to study in a CS course at a university, it is NOT homework and can be found <a href="http://www.cs.ucf.edu/%7Edmarino/ucf/transparency/cot4210/exam/" rel="noreferrer">here</a> under Fall 2011 exam2.</em></p>
<p>Here are the two questions I'm looking at from a past exam. They... | formal languages computability context free regular languages turing machines | 1 |
634 | What is beta equivalence? | <p>In the script I am currently reading on the lambda calculus, beta equivalence is defined as this:</p>

<blockquote>
 <p>The $\beta$-equivalence $\equiv_\beta$ is the smallest equivalence that contains $\rightarrow_\beta$.</p>
</blockquote>

<p>I have no idea what that means. Can someone expl... | logic terminology lambda calculus type theory | 1 |
636 | A Question relating to a Turing Machine with a useless state | <p>OK, so here is a question from a past test in my Theory of Computation class:</p>

<blockquote>
 <p>A useless state in a TM is one that is never entered on any input string. Let $$\mathrm{USELESS}_{\mathrm{TM}} = \{\langle M, q \rangle \mid q \text{ is a useless state in }M\}.$$
 Prove that $\mat... | computability undecidability formal methods turing machines | 1 |
640 | Language of the values of an affine function | <p>Write $\bar n$ for the decimal expansion of $n$ (with no leading <code>0</code>). Let $a$ and $b$ be integers, with $a > 0$. Consider the language of the decimal expansions of the multiples of $a$ plus a constant:</p>

<p>$$M = \{ \overline{a\,x+b} \mid x\in\mathbb{N} \}$$</p>

<p>Is $M$ regular? ... | formal languages context free regular languages integers | 1 |
641 | Language of the graph of an affine function | <p>Write $\bar n$ for the decimal expansion of $n$ (with no leading <code>0</code>). Let <code>:</code> be a symbol distinct from any digit. Let $a$ and $b$ be integers, with $a > 0$. Consider the language of solutions of the Diophantine equation $y=ax+b$:</p>

<p>$$L = \{ \bar{x} \mathtt: \bar{y} \mid y = a... | formal languages regular languages context free integers | 1 |
645 | Deciding on Sub-Problems for Dynamic Programming | <p>I have used the technique of dynamic programming multiple times however today a friend asked me how I go about defining my sub-problems, I realized I had no way of providing an objective formal answer. How do you formally define a sub-problem for a problem that you would solve using dynamic programming?</p>
 | algorithms dynamic programming | 1 |
653 | Is there a difference between $\lambda xy.xy$ and $\lambda x.\lambda y.xy$? | <p>I am currently learning the lambda calculus and was wondering about the following two different kinds of writing a lambda term. </p>

<ol>
<li>$\lambda xy.xy$ </li>
<li>$\lambda x.\lambda y.xy$</li>
</ol>

<p>Is there any difference in meaning or the way you apply beta reduction, or are ... | logic lambda calculus | 1 |
654 | Logarithmic vs double logarithmic time complexity | <p>In real world applications is there a concrete benefit when using $\mathcal{O}(\log(\log(n))$ instead of $\mathcal{O}(\log(n))$ algorithms ?</p>

<p>This is the case when one use for instance van Emde Boas trees instead of more conventional binary search tree implementations. 
But for example, if we take... | algorithms complexity theory binary trees algorithm analysis search trees | 1 |
663 | What are the uses of Markov Chains in CS? | <p>We all know that Markov Chains can be used for generating real-looking text (or real-sounding music). I've also heard that Markov Chains has some applications in the image processing, is that true? What are some other uses of MCs in CS?</p>
 | probability theory markov chains | 0 |
666 | Is there a 'string stack' data structure that supports these string operations? | <p>I'm looking for a data structure that stores a set of strings over a character set $\Sigma$, capable of performing the following operations. We denote $\mathcal{D}(S)$ as the data structure storing the set of strings $S$.</p>

<ul>
<li><code>Add-Prefix-Set</code> on $\mathcal{D}(S)$: given some set $T$ o... | data structures time complexity strings stacks | 0 |
669 | Are Turing machines more powerful than pushdown automata? | <p>I've came up with a result while reading some automata books, that Turing machines appear to be more powerful than pushdown automata. Since the tape of a Turing machine can always be made to behave like a stack, it'd seem that we can actually claim that TMs are more powerful. </p>

<p>Is this true?</p>
 | formal languages computability automata turing machines pushdown automata | 1 |
674 | Non-trivial tractable properties of triples | <p>Many intractable $NP$-complete problems can be modeled as deciding whether a set of triples, $F=${$t_1, t_2, ..., t_n$} where each triple $t_i$ is a subset of three elements over base set $U=${$a_1, a_2, ..., a_k$}, satisfy some non-trivial property. For example, 3-edge coloring of cubic graphs can be modeled as the... | complexity theory | 1 |
680 | How many edges can a unipathic graph have? | <p>A unipathic graph is a directed graph such that there is at most one simple path from any one vertex to any other vertex.</p>

<p>Unipathic graphs can have cycles. For example, a doubly linked list (not a circular one!) is a unipathic graph; if the list has $n$ elements, the graph has $n-1$ cycles of length ... | graphs combinatorics | 1 |
684 | Turing Completeness + Dataflow Unification = Arbitrarily invertible (pure, nonrecursive) functions? | <p>Assume we are working in a Turing-complete, referentially-transparent, higher-order language that supports arbitrary dataflow unification. Shouldn't it then be possible to construct the following function (using Haskell-like syntax, because that's what I'm most familiar with)?</p>

<pre><code>-- Takes an ar... | programming languages | 1 |
689 | How to convert an NFA with overlapping cycles into a regular expression? | <p>If I understand correctly, NFA have the same expressive power as regular expressions. Often, reading off equivalent regular expressions from NFA is easy: you translate cycles to stars, junctions as alternatives and so on. But what to do in this case: </p>

<p><img src="https://i.stack.imgur.com/yCGnv.png" al... | algorithms formal languages finite automata regular expressions | 1 |
699 | How does a two-way pushdown automaton work? | <p>Note that by "two-way pushdown automaton", I mean a pushdown automaton that can move its reading head both ways on the input tape.</p>

<p>I recently had the question of determining the computational power of two-way PDAs in the Chomsky hierarchy. I don't entirely understand two-way PDAs, but I can see how w... | formal languages computability automata pushdown automata | 1 |
701 | Decidable restrictions of the Post Correspondence Problem | <p>The <a href="http://en.wikipedia.org/wiki/Post_correspondence_problem" rel="noreferrer">Post Correspondence Problem</a> (PCP) is undecidable.</p>

<p>The <em>bounded version of the PCP</em> is $\mathrm{NP}$-complete and the <em>marked version of the PCP</em> (the words of one of the two lists are required to... | complexity theory computability reference request | 1 |
704 | How does a wifi password encrypt data using WEP and WPA? | <p>How does the password that we enter (to connect to a wireless network) encrypt the data on the wireless network?</p>

<p>Through my reading I am not sure if the password that we enter is the same as the passphrase. If that is right then how can the passphrase generate the four WEP keys?</p>

<p>I und... | cryptography computer networks encryption security | 0 |
706 | Finding exact corner solutions to linear programming using interior point methods | <p>The simplex algorithm walks greedily on the corners of a polytope to find the optimal solution to the linear programming problem. As a result, the answer is always a corner of the polytope. Interior point methods walk the inside of the polytope. As a result, when a whole plane of the polytope is optimal (if the obje... | algorithms optimization linear programming | 1 |
726 | Error-correcting rate is misleading | <p>In coding theory, 'how good a code is' means how many channel errors can be corrected, or better put, the maximal noise level that the code can deal with.</p>

<p>In order to get better codes, the codes are designed using a large alphabet (rather than binary one). And then, the code is good if it can deal wi... | information theory coding theory | 0 |
737 | Is Directed Graph a Graph? | <p>I came across an issue with the definition of a (directed) graph in Sipser's Introduction to the theory of computation, 2nd Ed.</p>

<p>On pp.10, An <strong>undirected graph</strong>, or simply a <strong>graph</strong>, is a set of points with lines connecting some of the points. The points are called nodes ... | graphs | 1 |
740 | Research on evaluating the performance of cache-obliviousness in practice | <p><a href="http://en.wikipedia.org/wiki/Cache-oblivious_algorithm">Cache-oblivious algorithms and data structures</a> are a rather new thing, introduced by Frigo et al. in <a href="http://userweb.cs.utexas.edu/~pingali/CS395T/2009fa/papers/coAlgorithms.pdf">Cache-oblivious algorithms, 1999</a>. Prokop's <a href="http:... | algorithms data structures computer architecture reference request cpu cache | 1 |
757 | Does every large enough string have repeats? | <p>Let $\Sigma$ be some finite set of characters of fixed size. Let $\alpha$ be some string over $\Sigma$. We say that a nonempty substring $\beta$ of $\alpha$ is a <em>repeat</em> if $\beta = \gamma \gamma$ for some string $\gamma$.</p>

<p>Now, my question is whether the following holds:</p>

<blockqu... | combinatorics strings word combinatorics | 1 |
764 | Are two elements always in a relation within a partially ordered set? | <p>In a partially ordered set, am I always able to order two arbitrary elements out of the set? Or is it possible that two elements within the set have no order relation to each other?</p>

<p>For example if there are three elements $\{a, b, c\}$ and $a \leq b$ and $a \leq c$, does either $b \leq c$ or $c \leq ... | terminology discrete mathematics order theory | 1 |
772 | Error in the use of asymptotic notation | <p>I'm trying to understand what is wrong with the following proof of the following recurrence </p>

<p>$$
T(n) = 2\,T\!\left(\left\lfloor\frac{n}{2}\right\rfloor\right)+n 
$$
$$
T(n) \leq 2\left(c\left\lfloor\frac{n}{2}\right\rfloor\right)+n \leq cn+n = n(c+1) =O(n)
$$</... | algorithms landau notation asymptotics recurrence relation | 1 |
783 | Reduction from 3-Partition problem to Balanced Partition problem | <p>The 3-Partition problem asks whether a set of $3n$ integers can be partitioned into $n$ sets of three integers such that each set sums up to some given integer $B$. The Balanced Partition problem asks whether $2n$ integers can be partitioned into two equal cardinality sets such that both sets have the same sum. Both... | complexity theory reductions np complete | 0 |
795 | Limitations of Stack Inspection | <p><em>This is a follow up to the <a href="https://cs.stackexchange.com/q/796/31">How does Stack Inspection work?</a> that explores the notion in more detail</em></p>

<p><a href="http://www.securingjava.com/chapter-three/chapter-three-6.html" rel="nofollow noreferrer">Stack inspection</a> is a mechanism for en... | security stack inspection | 1 |
796 | How does Stack Inspection work? | <p><em>This is precursor to my other, more advanced <a href="https://cs.stackexchange.com/q/795/31">question</a> about Stack Inspection.</em></p>

<p>Stack Inspection is a security mechanism introduced in the JVM to deal with running code originating from locations having different levels of trust. This is ques... | terminology security stack inspection | 1 |
802 | Are the Before and After sets for context-free grammars always context-free? | <p>Let $G$ be a context-free grammar. A string of terminals and nonterminals of $G$ is said to be a <em>sentential form</em> of $G$ if you can obtain it by applying productions of $G$ zero or more times to the start symbol of $S$. Let $\operatorname{SF}(G)$ be the set of sentential forms of $G$.</p>

<p>Let $\a... | formal languages context free formal grammars closure properties | 1 |
805 | Proving a binary tree has at most $\lceil n/2 \rceil$ leaves | <p>I'm trying to prove that a <a href="http://en.wikipedia.org/wiki/Binary_tree" rel="nofollow noreferrer">binary tree</a> with $n$ nodes has at most $\left\lceil \frac{n}{2} \right\rceil$ leaves. How would I go about doing this with induction?</p>

<p><em>For people who were following in the original question ... | data structures binary trees combinatorics graphs proof techniques | 1 |
808 | NP completeness proof of a spanning tree problem | <p>I am looking for some hints in a question asked by my instructor.</p>

<p>So I just figured out this decision problem is $\sf{NP\text{-}complete}$:</p>

<p>In a graph $G$, is there a spanning tree in $G$ that contain an exact set of $S=\{x_1, x_2,\ldots, x_n\}$ as leafs. I figured out we can prove th... | complexity theory graphs np complete | 1 |
809 | Logic gates from everyday materials | <p>Logic gates are an abstract device which can be implemented with electromagnetic relays, vacuum tubes, or transistors. These implemenations have been successful in computing in part because of various properties of chainability, durability, and size beyond their basic binary stability. They also work well because ... | computer architecture didactics | 1 |
811 | Proving that directed graph diagnosis is NP-hard | <p>I have a homework assignment that I've been bashing my head against for some time, and I'd appreciate any hints. It is about choosing a known problem, the NP-completeness of which is proven, and constructing a reduction from that problem to the following problem I'll call DGD (directed graph diagnosis).</p>
<h3>... | complexity theory np hard graphs | 1 |
813 | Are there other ways to describe formal languages other than grammars? | <p>I'm looking for mathematical theories that deal with describing formal languages (set of strings) in general and not just grammar hierarchies.</p>
 | formal languages formal grammars | 1 |
818 | How To Best Learn About Algorithms In Depth | <p>I have been reading this site with a great deal of interest, but I find a lot of it goes over my head. This has made me wish to learn a lot more about algorithms and CS in general. As far as I can tell from my research, there are 2 main ways of doing this. </p>

<ol>
<li><p>I can by a nice thick heavy bo... | algorithms education | 1 |
820 | Learning Automated Theorem Proving | <p><sup><em>I am learning <a href="http://en.wikipedia.org/wiki/Automated_theorem_proving" rel="noreferrer">Automated Theorem Proving</a> / <a href="http://en.wikipedia.org/wiki/Satisfiability_Modulo_Theories" rel="noreferrer">SMT solvers</a> / <a href="http://en.wikipedia.org/wiki/Proof_assistant" rel="noreferrer">Pro... | logic proof assistants automated theorem proving coq | 1 |
824 | Sorting functions by asymptotic growth | <p>Assume I have a list of functions, for example </p>

<p>$\qquad n^{\log \log(n)}, 2^n, n!, n^3, n \ln n, \dots$</p>

<p>How do I sort them asymptotically, i.e. after the relation defined by</p>

<p>$\qquad f \leq_O g \iff f \in O(g)$,</p>

<p>assuming they are indeed pairwise compara... | asymptotics landau notation reference question | 1 |
835 | Is the number of coin tosses of a probabilistic Turing machine a Blum complexity measure? | <p>I <a href="http://blog.computationalcomplexity.org/2004/04/blum-complexity-measures.html" rel="nofollow">read</a> that the number of coin tosses of a probabilistic Turing machine (PTM) is not a <a href="http://en.wikipedia.org/wiki/Blum_axioms" rel="nofollow">Blum complexity measure</a>. Why?</p>

<p>Clarifi... | computability complexity theory randomness probabilistic algorithms | 0 |
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