Daily Papers

Hard attention Chain-of-Thought (CoT) transformers are known to be Turing-complete. However, it is an open problem whether softmax attention Chain-of-Thought (CoT) transformers are Turing-complete. In this paper, we prove a stronger result that length-generalizable softmax CoT transformers are Turing-complete. More precisely, our Turing-completeness proof goes via the CoT extension of the Counting RASP (C-RASP), which correspond to softmax CoT transformers that admit length generalization. We prove Turing-completeness for CoT C-RASP with causal masking over a unary alphabet (more generally, for letter-bounded languages). While we show this is not Turing-complete for arbitrary languages, we prove that its extension with relative positional encoding is Turing-complete for arbitrary languages. We empirically validate our theory by training transformers for languages requiring complex (non-linear) arithmetic reasoning.

A universal word (u-word) for d-dimensional permutations of length n is a 2-dimensional word with d-1 rows, any size n window of which is order-isomorphic to exactly one permutation of length n, and all permutations of length n are covered. It is known that u-words (in fact, even u-cycles, a stronger claim) for d-dimensional permutations exist. In this paper, we use the idea of incomparable elements to prove that u-words of length (n!)^{d-1}+n-1-i(n-1), for dgeq 2 and $0leq ileq 2^{d-1}{n-1}left[(1+(n-1)!)^{d-1}-left(1+(n-1)!{2}right)^{d-1}right], for d-dimensional permutations of length n exist, which generalizes the respective result of Kitaev, Potapov and Vajnovszki for ``usual'' permutations (d=2$).

We contribute to the study of formal languages that can be recognized by transformer encoders. We focus on two self-attention mechanisms: (1) UHAT (Unique Hard Attention Transformers) and (2) AHAT (Average Hard Attention Transformers). UHAT encoders are known to recognize only languages inside the circuit complexity class {sf AC}^0, i.e., accepted by a family of poly-sized and depth-bounded boolean circuits with unbounded fan-ins. On the other hand, AHAT encoders can recognize languages outside {sf AC}^0), but their expressive power still lies within the bigger circuit complexity class {sf TC}^0, i.e., {sf AC}^0-circuits extended by majority gates. We first show a negative result that there is an {sf AC}^0-language that cannot be recognized by an UHAT encoder. On the positive side, we show that UHAT encoders can recognize a rich fragment of {sf AC}^0-languages, namely, all languages definable in first-order logic with arbitrary unary numerical predicates. This logic, includes, for example, all regular languages from {sf AC}^0. We then show that AHAT encoders can recognize all languages of our logic even when we enrich it with counting terms. We apply these results to derive new results on the expressive power of UHAT and AHAT up to permutation of letters (a.k.a. Parikh images).

The Omega numbers-the halting probabilities of universal prefix-free machines-are known to be exactly the Martin-L{\"o}f random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-L{\"o}f random left-c.e. real alpha, a universal prefix-free machine U whose halting probability is alpha. We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real alpha, one cannot uniformly produce a left-c.e. real beta such that alpha -- beta is neither left-c.e. nor right-c.e.

Yor\`ub\'a is a widely spoken West African language with a writing system rich in orthographic and tonal diacritics. They provide morphological information, are crucial for lexical disambiguation, pronunciation and are vital for any computational Speech or Natural Language Processing tasks. However diacritic marks are commonly excluded from electronic texts due to limited device and application support as well as general education on proper usage. We report on recent efforts at dataset cultivation. By aggregating and improving disparate texts from the web and various personal libraries, we were able to significantly grow our clean Yor\`ub\'a dataset from a majority Bibilical text corpora with three sources to millions of tokens from over a dozen sources. We evaluate updated diacritic restoration models on a new, general purpose, public-domain Yor\`ub\'a evaluation dataset of modern journalistic news text, selected to be multi-purpose and reflecting contemporary usage. All pre-trained models, datasets and source-code have been released as an open-source project to advance efforts on Yor\`ub\'a language technology.

Computational replication of Chinese calligraphy remains challenging. Existing methods falter, either creating high-quality isolated characters while ignoring page-level aesthetics like ligatures and spacing, or attempting page synthesis at the expense of calligraphic correctness. We introduce UniCalli, a unified diffusion framework for column-level recognition and generation. Training both tasks jointly is deliberate: recognition constrains the generator to preserve character structure, while generation provides style and layout priors. This synergy fosters concept-level abstractions that improve both tasks, especially in limited-data regimes. We curated a dataset of over 8,000 digitized pieces, with ~4,000 densely annotated. UniCalli employs asymmetric noising and a rasterized box map for spatial priors, trained on a mix of synthetic, labeled, and unlabeled data. The model achieves state-of-the-art generative quality with superior ligature continuity and layout fidelity, alongside stronger recognition. The framework successfully extends to other ancient scripts, including Oracle bone inscriptions and Egyptian hieroglyphs. Code and data can be viewed in https://github.com/EnVision-Research/UniCalli{this URL}.

Abugida refers to a phonogram writing system where each syllable is represented using a single consonant or typographic ligature, along with a default vowel or optional diacritic(s) to denote other vowels. However, texting in these languages has some unique challenges in spite of the advent of devices with soft keyboard supporting custom key layouts. The number of characters in these languages is large enough to require characters to be spread over multiple views in the layout. Having to switch between views many times to type a single word hinders the natural thought process. This prevents popular usage of native keyboard layouts. On the other hand, supporting romanized scripts (native words transcribed using Latin characters) with language model based suggestions is also set back by the lack of uniform romanization rules. To this end, we propose a disambiguation algorithm and showcase its usefulness in two novel mutually non-exclusive input methods for languages natively using the abugida writing system: (a) disambiguation of ambiguous input for abugida scripts, and (b) disambiguation of word variants in romanized scripts. We benchmark these approaches using public datasets, and show an improvement in typing speed by 19.49%, 25.13%, and 14.89%, in Hindi, Bengali, and Thai, respectively, using Ambiguous Input, owing to the human ease of locating keys combined with the efficiency of our inference method. Our Word Variant Disambiguation (WDA) maps valid variants of romanized words, previously treated as Out-of-Vocab, to a vocabulary of 100k words with high accuracy, leading to an increase in Error Correction F1 score by 10.03% and Next Word Prediction (NWP) by 62.50% on average.

We demonstrate that it is feasible to diacritize Hebrew script without any human-curated resources other than plain diacritized text. We present NAKDIMON, a two-layer character level LSTM, that performs on par with much more complicated curation-dependent systems, across a diverse array of modern Hebrew sources.

Intermediate reasoning or acting steps have successfully improved large language models (LLMs) for handling various downstream natural language processing (NLP) tasks. When applying LLMs for code generation, recent works mainly focus on directing the models to articulate intermediate natural-language reasoning steps, as in chain-of-thought (CoT) prompting, and then output code with the natural language or other structured intermediate steps. However, such output is not suitable for code translation or generation tasks since the standard CoT has different logical structures and forms of expression with the code. In this work, we introduce the universal code (UniCode) as the intermediate representation. It is a description of algorithm steps using a mix of conventions of programming languages, such as assignment operator, conditional operator, and loop. Hence, we collect an instruction dataset UniCoder-Instruct to train our model UniCoder on multi-task learning objectives. UniCoder-Instruct comprises natural-language questions, code solutions, and the corresponding universal code. The alignment between the intermediate universal code representation and the final code solution significantly improves the quality of the generated code. The experimental results demonstrate that UniCoder with the universal code significantly outperforms the previous prompting methods by a large margin, showcasing the effectiveness of the structural clues in pseudo-code.

The Markov numbers are positive integers appearing as solutions to the Diophantine equation x^2 + y^2 + z^2 = 3xyz. These numbers are very well-studied and have many combinatorial properties, as well as being the source of the long-standing unicity conjecture. In 2018, Canakc{\i} and Schiffler showed that the Markov number m_{a{b}} is the number of perfect matchings of a certain snake graph corresponding to the Christoffel path from (0,0) to (a,b). Based on this correspondence, Schiffler in 2023 introduced two orderings on lattice paths. For any path omega, associate a snake graph G(omega) and a continued fraction g(omega). The ordering <_M is given by the number of perfect matchings on G(omega), and the ordering <_L is given by the Lagrange number of g(omega). In this work, we settle two conjectures of Schiffler. First, we show that the path omega(a,b) = RRcdots R UU cdots U is the unique maximum over all lattice paths from (0,0) to (a,b) with respect to both orderings <_M and <_L. We then use this result to prove that sup L(omega) over all lattice paths is exactly 1+sqrt5.

Reverse-mode differentiation is used for optimization, but it introduces references, which break the purity of the underlying programs, making them notoriously harder to optimize. We present a reverse-mode differentiation on a purely functional language with array operations. It is the first one to deliver a provably efficient, purely functional, and denotationally correct reverse-mode differentiation. We show that our transformation is semantically correct and verifies the cheap gradient principle. Inspired by PROPs and compilation to categories, we introduce a novel intermediate representation that we call 'unary form'. Our reverse-mode transformation is factored as a compilation scheme through this intermediate representation. We obtain provably efficient gradients by performing general partial evaluation optimizations after our reverse-mode transformation, as opposed to manually derived ones. For simple first-order programs, the obtained output programs resemble static-single-assignment (SSA) code. We emphasize the modularity of our approach and show how our language can easily be enriched with more optimized primitives, as required for some speed-ups in practice.

We study the matrix models pi:C(S_N^+)to M_N(C(X)) which are flat, in the sense that the standard generators of C(S_N^+) are mapped to rank 1 projections. Our first result is a generalization of the Pauli matrix construction at N=4, using finite groups and 2-cocycles. Our second result is the construction of a universal representation of C(S_N^+), inspired from the Sinkhorn algorithm, that we conjecture to be inner faithful.

We study whether an aperiodic hierarchy can provide a structural advantage for lossless compression over periodic alternatives. We show that Fibonacci quasicrystal tilings avoid the finite-depth collapse that affects periodic hierarchies: usable n-gram lookup positions remain non-zero at every level, while periodic tilings collapse after O(log p) levels for period p. This yields an aperiodic hierarchy advantage: dictionary reuse remains available across all scales instead of vanishing beyond a finite depth. Our analysis gives four main consequences. First, the Golden Compensation property shows that the exponential decay in the number of positions is exactly balanced by the exponential growth in phrase length, so potential coverage remains scale-invariant with asymptotic value Wvarphi/5. Second, using the Sturmian complexity law p(n)=n+1, we show that Fibonacci/Sturmian hierarchies maximize codebook coverage efficiency among binary aperiodic tilings. Third, under long-range dependence, the resulting hierarchy achieves lower coding entropy than comparable periodic hierarchies. Fourth, redundancy decays super-exponentially with depth, whereas periodic systems remain locked at the depth where collapse occurs. We validate these results with Quasicryth, a lossless text compressor built on a ten-level Fibonacci hierarchy with phrase lengths {2,3,5,8,13,21,34,55,89,144}. In controlled A/B experiments with identical codebooks, the aperiodic advantage over a Period-5 baseline grows from 36{,}243 B at 3 MB to 11{,}089{,}469 B at 1 GB, explained by the activation of deeper hierarchy levels. On enwik9, Quasicryth achieves 225{,}918{,}349 B (22.59%), with 20{,}735{,}733 B saved by the Fibonacci tiling relative to no tiling.

This article provides an introduction to surface code quantum computing. We first estimate the size and speed of a surface code quantum computer. We then introduce the concept of the stabilizer, using two qubits, and extend this concept to stabilizers acting on a two-dimensional array of physical qubits, on which we implement the surface code. We next describe how logical qubits are formed in the surface code array and give numerical estimates of their fault-tolerance. We outline how logical qubits are physically moved on the array, how qubit braid transformations are constructed, and how a braid between two logical qubits is equivalent to a controlled-NOT. We then describe the single-qubit Hadamard, S and T operators, completing the set of required gates for a universal quantum computer. We conclude by briefly discussing physical implementations of the surface code. We include a number of appendices in which we provide supplementary information to the main text.

Handwritten text recognition in low resource scenarios, such as manuscripts with rare alphabets, is a challenging problem. The main difficulty comes from the very few annotated data and the limited linguistic information (e.g. dictionaries and language models). Thus, we propose a few-shot learning-based handwriting recognition approach that significantly reduces the human labor annotation process, requiring only few images of each alphabet symbol. The method consists in detecting all the symbols of a given alphabet in a textline image and decoding the obtained similarity scores to the final sequence of transcribed symbols. Our model is first pretrained on synthetic line images generated from any alphabet, even though different from the target domain. A second training step is then applied to diminish the gap between the source and target data. Since this retraining would require annotation of thousands of handwritten symbols together with their bounding boxes, we propose to avoid such human effort through an unsupervised progressive learning approach that automatically assigns pseudo-labels to the non-annotated data. The evaluation on different manuscript datasets show that our model can lead to competitive results with a significant reduction in human effort. The code will be publicly available in this repository: https://github.com/dali92002/HTRbyMatching

Most undeciphered lost languages exhibit two characteristics that pose significant decipherment challenges: (1) the scripts are not fully segmented into words; (2) the closest known language is not determined. We propose a decipherment model that handles both of these challenges by building on rich linguistic constraints reflecting consistent patterns in historical sound change. We capture the natural phonological geometry by learning character embeddings based on the International Phonetic Alphabet (IPA). The resulting generative framework jointly models word segmentation and cognate alignment, informed by phonological constraints. We evaluate the model on both deciphered languages (Gothic, Ugaritic) and an undeciphered one (Iberian). The experiments show that incorporating phonetic geometry leads to clear and consistent gains. Additionally, we propose a measure for language closeness which correctly identifies related languages for Gothic and Ugaritic. For Iberian, the method does not show strong evidence supporting Basque as a related language, concurring with the favored position by the current scholarship.

The question of whether there exists a finite group of order at least three in which every element except one is a commutator has remained unresolved in group theory. In this article, we address this open problem by developing an algorithmic approach that leverages several group theoretic properties of such groups. Specifically, we utilize a result of Frobenius and various necessary properties of such groups, combined with Plesken and Holt's extensive enumeration of finite perfect groups, to systematically examine all finite groups up to a certain order for the desired property. The computational core of our work is implemented using the computer system GAP (Groups, Algorithms, and Programming). We discover two nonisomorphic groups of order 368,640 that exhibit the desired property. Our investigation also establishes that this order is the minimum order for such a group to exist. As a result, this study provides a positive answer to Problem 17.76 in the Kourovka Notebook. In addition to the algorithmic framework, this paper provides a structural description of one of the two groups found.

Chomsky and others have very directly claimed that large language models (LLMs) are equally capable of learning languages that are possible and impossible for humans to learn. However, there is very little published experimental evidence to support such a claim. Here, we develop a set of synthetic impossible languages of differing complexity, each designed by systematically altering English data with unnatural word orders and grammar rules. These languages lie on an impossibility continuum: at one end are languages that are inherently impossible, such as random and irreversible shuffles of English words, and on the other, languages that may not be intuitively impossible but are often considered so in linguistics, particularly those with rules based on counting word positions. We report on a wide range of evaluations to assess the capacity of GPT-2 small models to learn these uncontroversially impossible languages, and crucially, we perform these assessments at various stages throughout training to compare the learning process for each language. Our core finding is that GPT-2 struggles to learn impossible languages when compared to English as a control, challenging the core claim. More importantly, we hope our approach opens up a productive line of inquiry in which different LLM architectures are tested on a variety of impossible languages in an effort to learn more about how LLMs can be used as tools for these cognitive and typological investigations.

Multi-turn problem solving is critical yet challenging for Large Reasoning Models (LRMs) to reflect on their reasoning and revise from feedback. Existing Reinforcement Learning (RL) methods train large reasoning models on a single-turn paradigm with verifiable rewards. However, we observe that models trained with existing RL paradigms often lose their ability to solve problems across multiple turns and struggle to revise answers based on contextual feedback, leading to repetitive responses. We ask: can LRMs learn to reflect their answers in a multi-turn context? In this work, we find that training models with multi-turn RL using only unary feedback (e.g., "Let's try again") after wrong answers can improve both single-turn performance and multi-turn reasoning. We introduce Unary Feedback as Observation (UFO) for reinforcement learning, which uses minimal yet common unary user feedback during iterative problem solving. It can be easily applied to existing single-turn RL training setups. Experimental results show that RL training with UFO keeps single-turn performance and improves multi-turn reasoning accuracy by up to 14%, enabling language models to better react to feedback in multi-turn problem solving. To further minimize the number of turns needed for a correct answer while encouraging diverse reasoning when mistakes occur, we design reward structures that guide models to produce careful and deliberate answers in each turn. Code: https://github.com/lichengliu03/unary-feedback

A new information theoretic condition is presented for reconstructing a discrete random variable X based on the knowledge of a set of discrete functions of X. The reconstruction condition is derived from Shannon's 1953 lattice theory with two entropic metrics of Shannon and Rajski. Because such a theoretical material is relatively unknown and appears quite dispersed in different references, we first provide a synthetic description (with complete proofs) of its concepts, such as total, common and complementary informations. Definitions and properties of the two entropic metrics are also fully detailed and shown compatible with the lattice structure. A new geometric interpretation of such a lattice structure is then investigated that leads to a necessary (and sometimes sufficient) condition for reconstructing the discrete random variable X given a set { X_1,ldots,X_{n} } of elements in the lattice generated by X. Finally, this condition is illustrated in five specific examples of perfect reconstruction problems: reconstruction of a symmetric random variable from the knowledge of its sign and absolute value, reconstruction of a word from a set of linear combinations, reconstruction of an integer from its prime signature (fundamental theorem of arithmetic) and from its remainders modulo a set of coprime integers (Chinese remainder theorem), and reconstruction of the sorting permutation of a list from a minimal set of pairwise comparisons.

In this paper, we introduce UniSent universal sentiment lexica for 1000+ languages. Sentiment lexica are vital for sentiment analysis in absence of document-level annotations, a very common scenario for low-resource languages. To the best of our knowledge, UniSent is the largest sentiment resource to date in terms of the number of covered languages, including many low resource ones. In this work, we use a massively parallel Bible corpus to project sentiment information from English to other languages for sentiment analysis on Twitter data. We introduce a method called DomDrift to mitigate the huge domain mismatch between Bible and Twitter by a confidence weighting scheme that uses domain-specific embeddings to compare the nearest neighbors for a candidate sentiment word in the source (Bible) and target (Twitter) domain. We evaluate the quality of UniSent in a subset of languages for which manually created ground truth was available, Macedonian, Czech, German, Spanish, and French. We show that the quality of UniSent is comparable to manually created sentiment resources when it is used as the sentiment seed for the task of word sentiment prediction on top of embedding representations. In addition, we show that emoticon sentiments could be reliably predicted in the Twitter domain using only UniSent and monolingual embeddings in German, Spanish, French, and Italian. With the publication of this paper, we release the UniSent sentiment lexica.

We show that transformer-based large language models are computationally universal when augmented with an external memory. Any deterministic language model that conditions on strings of bounded length is equivalent to a finite automaton, hence computationally limited. However, augmenting such models with a read-write memory creates the possibility of processing arbitrarily large inputs and, potentially, simulating any algorithm. We establish that an existing large language model, Flan-U-PaLM 540B, can be combined with an associative read-write memory to exactly simulate the execution of a universal Turing machine, U_{15,2}. A key aspect of the finding is that it does not require any modification of the language model weights. Instead, the construction relies solely on designing a form of stored instruction computer that can subsequently be programmed with a specific set of prompts.

The class of tree-adjoining languages can be characterized by various two-level formalisms, consisting of a context-free grammar (CFG) or pushdown automaton (PDA) controlling another CFG or PDA. These four formalisms are equivalent to tree-adjoining grammars (TAG), linear indexed grammars (LIG), pushdown-adjoining automata (PAA), and embedded pushdown automata (EPDA). We define semiring-weighted versions of the above two-level formalisms, and we design new algorithms for computing their stringsums (the weight of all derivations of a string) and allsums (the weight of all derivations). From these, we also immediately obtain stringsum and allsum algorithms for TAG, LIG, PAA, and EPDA. For LIG, our algorithm is more time-efficient by a factor of O(n|N|) (where n is the string length and |N| is the size of the nonterminal set) and more space-efficient by a factor of O(|Gamma|) (where |Gamma| is the size of the stack alphabet) than the algorithm of Vijay-Shanker and Weir (1989). For EPDA, our algorithm is both more space-efficient and time-efficient than the algorithm of Alonso et al. (2001) by factors of O(|Gamma|^2) and O(|Gamma|^3), respectively. Finally, we give the first PAA stringsum and allsum algorithms.

We report upon the results of a research and prototype building project Worldly~OCR dedicated to developing new, more accurate image-to-text conversion software for several languages and writing systems. These include the cursive scripts Farsi and Pashto, and Latin cursive scripts. We also describe approaches geared towards Traditional Chinese, which is non-cursive, but features an extremely large character set of 65,000 characters. Our methodology is based on Machine Learning, especially Deep Learning, and Data Science, and is directed towards vast quantities of original documents, exceeding a billion pages. The target audience of this paper is a general audience with interest in Digital Humanities or in retrieval of accurate full-text and metadata from digital images.

In an unpublished preprint batanin, Batanin conjectures that it is possible to take `slices' of a globular operad, thereby isolating the algebraic structure in each dimension. It was further hypothesised that the slices of a globular operad for some theory of higher category contain essential information about those higher categories, namely whether or not they are equivalent to the fully weak variety. In this paper, we use the theory of presentations for globular operads developed in Me to provide a concrete definition of slices, and calculate the slices for several key theories of n-category.