Algorithmic Reasoning

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8 papers in the last four weeks, against 2 the four weeks before. 0.1% of all new papers.

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Latest papers 86

Jun 22, 2026cs.AI

Finding the Evidence: Discovering Decision-Supporting Tokens for On-Policy Reasoning Distillation

On-policy distillation transfers reasoning ability through dense token-level supervision, yet the nature of the transferable signal remains unclear. We discover that reasoning chains contain two types of knowledge that require different discovery mechanisms: decisions (where to branch), which surface through student uncertainty, and evidence (intermediate steps that justify decisions), which hides in positions where the student is confident yet wrong. Current methods capture only decisions; the substantive knowledge in evidence tokens remains untransferred. We propose DEAR(Decision-Evidence Aware Reasoning Distillation), which first identifies decisions via student entropy, then discovers their supporting evidence through hidden-state cosine similarity to decision anchors, boosted by teacher-student divergence to prioritize the largest knowledge gaps. Across three student-teacher configurations on math and code benchmarks, DEAR consistently outperforms standard OPD, with up to +2.5pp on competition math and +5.7pp on code generation.

Jinwei Xiao, Zhuowen Han, Yueqing Sun +6

Meituan Longcat Team · TJUNLP Lab, College of Intelligence and Computing, Tianjin University · Nanjing University

CoT Distillation On-Policy Distillation Teacher-Student Learning Knowledge Distillation Algorithmic Reasoning LLM Reasoning

Jun 20, 2026cs.LG

A Verifiable Search Is Not a Learnable Chain-of-Thought

It is tempting to assume any task solvable by a short program can be taught to a model as its chain-of-thought: write the steps out, fine-tune, and the model follows. This paper shows the assumption fails for an identifiable class of procedures. The testbed is nine reasoning tasks, each from a deterministic generator; public and hidden splits share generators, so held-out data proxies test accuracy. I reverse-engineer the generators into Python solvers, render them as chain-of-thought, and distill into a rank-<= 32 LoRA over a 30B (3.5B-active) Nemotron model. Forward-computable tasks install readily: lookup/arithmetic and an 8-bit boolean task transfer (>= 0.99 and 0.68). Cryptarithm does not: distilling its backtracking search holds at 0.01-0.07 across eleven chain-of-thought designs, RL from verifiable rewards, and self-training, even though a search solver answers 71% of instances. This is not a capability gap. The model does the arithmetic on 97-100% of lines and ranks the correct cipher in its top eight on 71%; it cannot carry the search forward as a left-to-right derivation. Fine-tuning learns the shape of a verifiable elimination step while its verdicts become unconditional templates, correct only 16-57% of the time ("verdict-as-token"). The ceiling holds across backbones from 3B to 671B and across fine-tuning and prompting; a controlled intervention isolates the cause: revealing the cipher key, which turns the derivation forward, lifts the same instances from 0.03 to 0.57. When a procedure's only solution is search over information-free structure, no faithful forward chain-of-thought exists to imitate. The task becomes learnable only by removing the search, precomputing its combinatorial core into a catalog and reducing the trace to recall plus verification; the 1st-place solution reaches Private LB 0.92 this way. What distills is memorization and verification, not search.

Harsh Patel

Independent Researcher

Inference-Time Search Memorization in Language Models CoT Distillation Neural Algorithmic Reasoning Algorithmic Reasoning

Jun 18, 2026cs.AI

RACL: Reasoning-Agent Control Layers for Continuous Metaheuristic Learning

This paper introduces RACL, a Reasoning-Agent Control Layer for metaheuristics. RACL places a reasoning agent above an existing optimizer. The agent does not replace the optimizer and does not modify business constraints. Instead, it controls the optimizer's internal search behavior by observing operational memory, reasoning over past behavior, formulating bounded hypotheses, testing interventions, evaluating outcomes, applying guardrails, consolidating useful policies and explaining its decisions. The experiment uses vehicle routing as a testbed, but the contribution is not a new routing solver, a particular ALNS configuration or a specific set of routing rules. The contribution is the RACL method: a way for a reasoning agent to discover, validate, consolidate and explain algorithmic control rules for a metaheuristic. In the current experimental setting, RACL improves or ties the Operational Memory Policy in 21 of 21 feasible cases and improves or ties a non-reasoning Stagnation-Triggered Policy in 18 of 21 feasible cases, with an average RACL vs STP cost delta of -0.641%. In the Sevilla-9/10 runtime sample, RACL improves average cost by -8.337% versus Fixed and -1.605% versus STP without showing material computational overhead. During the proof-of-concept, Codex was used as an in-the-loop reasoning agent observing executions, interpreting logs and proposing live bounded interventions. The policy proxy was later used only to make quantitative evaluation reproducible.

Antón Asla Manzárraga

Independent Researcher

Metaheuristic Optimization Algorithmic Reasoning

Jun 18, 2026cs.LG

Efficiently Representing Algorithms With Chain-of-Thought Transformers

The increasing popularity of \emph{reasoning} models -- language models that output a series of reasoning or thought tokens before producing an answer -- is justified, in part, by theoretical results showing that chain-of-thought (CoT) transformers can simulate Turing machines, and thus perform arbitrary computation. However, the Turing machine, while suitable for complexity-theoretic analysis, is not convenient, intuitive, or efficient for discussing algorithms. Algorithms are typically designed and analyzed at a higher level of abstraction, captured by the \emph{Word RAM} model with random-access memory and unit-cost operations on

\bigO(\log n)

-bit words. As a result, Word RAM algorithms can be substantially more efficient than their Turing machine counterparts, raising the question: \emph{Can CoT transformers efficiently simulate Word RAM algorithms?} For instance, can they sort

n

items in

\bigO(n \log n)

steps or run Dijkstra's algorithm in

\bigO(E + V \log V)

steps? We answer affirmatively, up to poly-logarithmic overhead. We first establish this for finite-precision transformers with poly-logarithmic width and rightmost unique hard attention, then strengthen the result to two more practical settings with finite width and log-precision: \emph{continuous} CoT, where reasoning takes the form of vectors rather than tokens, and a \emph{hybrid} architecture in which transformer layers sit atop a recurrent (linear RNN) layer. In all three cases, we find that CoT \emph{can} efficiently simulate any Word RAM algorithm with only a poly-logarithmic overhead in

n

. This overhead reduces to log-square when the Word RAM has a ``flat'' instruction set, and only logarithmic for multiplication-free flat instructions -- in stark contrast to known CoT simulations of Turing machines, which require quadratic overhead over Word RAM.

Yanhong Li, Anej Svete, Ashish Sabharwal +1

Allen Institute for AI · ETH Zürich

Transformer CoT Reasoning Neural Algorithmic Reasoning Efficient Language Model Reasoning Algorithmic Reasoning

Jun 16, 2026cs.AI

Fixed-Point Reasoners: Stable and Adaptive Deep Looped Transformers

Looped architectures provide an inductive bias toward learning step-by-step procedures for tasks that require compositional reasoning. The number of effective layers reached by looping determines the quality of the solution these models find. Like deep architectures, looped architectures are prone to a signal propagation problem induced by depth as the halting decision is postponed. In this paper, we address this signal propagation issue using pre-norm layers and residual scaling. Building on these architectural modifications, we propose FPRM, a Transformer-based Fixed-Point Reasoning Model that uses fixed-point convergence as an end-to-end halting mechanism in a looped architecture. We show that fixed-point halting allows FPRM to adapt its compute to task difficulty. FPRM is effective on common reasoning benchmarks, namely Sudoku, Maze, state-tracking, and ARC-AGI.

Sajad Movahedi, Vera Milovanović, Shlomo Libo Feigin +5

ELLIS Institute Tübingen, Max Planck Institute for Intelligent Systems, Tübingen AI Center · ETH Zurich · Swiss Institute of Bioinformatics +2

Recurrent Transformers Algorithmic Reasoning Adaptive Computation

Jun 14, 2026cs.LG

Is Code Better Than Language for Algorithmic Reasoning

For tool-augmented language models, comparing natural-language reasoning with code-execution pipelines is difficult because the comparison changes both the intermediate representation and the execution mechanism. We separate these factors with an intermediate intervention: the model expresses its reasoning as executable code, and the language model simulates that code in context to produce an answer. On a 40-task verifiable algorithmic benchmark, deterministic code execution outperforms natural-language reasoning by +31.6pp. We observe that the intermediate intervention is not meaningfully different from natural-language reasoning (+0.15pp). These results suggest that, in our evaluated setting, changing the intermediate representation alone does not explain the tool-use advantage, providing evidence for the performance gains requiring reliable external execution. We formalize this intuition with a simple statistical decision-theoretic model that characterizes when execution dominates end-to-end risk in our disentangled trace-generation/execution regime. We validate our theory using a reconstruction intervention that leverages a proxy language model to infer natural-language reasoning traces from code representations, recovering performance comparable to the original natural-language reasoning pipeline. All experiments are at https://github.com/TerryTong-Git/ToolProj.

Terry Tong, Yu Feng, Surbhi Goel +1

University of Pennsylvania · 2Oracle AI

LLM Evaluation LLM Tool Use Algorithmic Reasoning LLM Reasoning

Jun 8, 2026cs.LG

Structural Grid Descriptors Predict Within-Task Solver Success on ARC-AGI

We ask whether structural properties of intermediate grid states predict whether a symbolic ARC-AGI solver will succeed, framed as a test of conditional mutual information I(X;Y|task) > 0. Across 44,800 runs spanning two architecturally distinct solvers (beam search and Stochastic DFS), 400 ARC tasks, 28 configurations per solver, and both training and evaluation splits, hand-crafted grid descriptors measured at 50% trajectory completion discriminate successful from failed runs within the same task (mean within-task best-feature AUC = 0.885, p < 0.001 under within-task label permutation). Most predictive content lies along a single grid-complexity axis. The result generalizes across solver architectures: a feature selected on one solver predicts success on the other with AUC 0.747-0.762 in all four transfer directions (p < 0.001, leakage controlled). On a pre-registered held-out set of 41 reliable tasks, the frozen feature n_components_final achieves AUC = 0.765 (95% CI [0.717, 0.810], p < 0.001), robust under task-clustered bootstrap resampling and cross-solver task collapsing. The signal is not explained by solver capacity (configuration-residualized AUC = 0.927 and 0.896 for beam search and SDFS, p < 0.001) and is only weakly coupled to score trajectories (R^2 approximately 0). Early stopping at 50% completion reduces beam-search compute by 33.6% while retaining 98.9% of solves; degenerate-trajectory detection reduces SDFS compute by 65.3% with no solve loss. Finally, on 229 of 400 evaluation tasks the DSL primitive library produces no valid transition from the input grid. This 0-step collapse is invariant to search budget and universally failed by beam search, indicating a DSL coverage limitation rather than a search-budget effect.

Ayan Pendharkar

ARC-AGI Algorithmic Reasoning

Jun 3, 2026cs.AI

Step-by-Step Optimization-like Reasoning in LLMs over Expanding Search Spaces

Verifiable reward training has improved mathematical and coding reasoning, but these domains capture only part of step-by-step decision making. Many real-world tasks require finding a high-value feasible plan among many valid alternatives. We introduce OPT*, a scalable family of optimization-style tasks for training and evaluating LLM step-by-step optimization-like reasoning along a complexity axis: each task provides a feasibility checker and evaluator, while a complexity parameter expands the search space without requiring new human labels. This motivates studying these tasks in two regimes: (i) solver-guided online policy optimization, which uses a solver as a value oracle for partial states and applies rank-based reward shaping to reinforce better next steps, and (ii) search-based offline RL when such solvers are unavailable. Theoretically, we relate success in large search spaces to the information a reasoner extracts per unit of search budget. Empirically, we ablate the ingredients that make search efficient on OPT* and show that training on OPT* improves step-by-step optimization-like reasoning.

Nicolás Astorga, Nabeel Seedat, Mihaela van der Schaar

University of Cambridge

Inference-Time Search Large Language Model-Guided Optimization RL for Language Model Reasoning Algorithmic Reasoning

Jun 2, 2026cs.LG

Contrastive Neural Algorithmic Reasoning for Graph Coloring

Graph coloring seeks to assigns colors to a graph's nodes so that adjacent nodes receive different colors, using as few colors as possible. Here, we study approximate

k

-coloring, where the goal is to use at most

k

colors while minimizing the number of monochromatic edges. This problem is central to graph theory and has applications in areas such as scheduling and resource allocation. Recent unsupervised GNN approaches optimize each instance directly, precluding generalization across graph sizes and distributions. We instead propose a contrastive learning framework that learns transferable coloring geometry where the embeddings of same-color nodes align, while adjacent nodes' representations are pushed toward distinct directions. We analyze the resulting population objective over bounded-size graphs. For unit-norm embeddings, we show that its optima have a line-prototype structure: Representations of nodes of the same color collapse to a shared one-dimensional subspace, and edges connect orthogonal subspaces. This geometry yields stationarity conditions in the supervised setting and is preserved by projected subgradient dynamics under a balanced-coloring assumption. In an unnormalized variant, gradient descent has a max-margin bias governed by a quotient-graph hard-margin problem. Experiments on synthetic and real-world graphs show that contrastive GNN encoders generalize effectively and produce low-conflict colorings, matching and sometimes improving on greedy approaches.

Thien Le, Tianyu Zhao, Melanie Weber

Harvard University SEAS · Harvard University T.H. Chan School of Public Health

Graph Neural Networks Graph Representation Learning Neural Algorithmic Reasoning Combinatorial Optimization Graph Contrastive Learning Algorithmic Reasoning

Jun 1, 2026cs.CL

CRAFTQA: A Code-Driven Adaptive Framework for Complex Structured Data Reasoning

Real-world scenarios involve massive heterogeneous structured data (e.g., tables, knowledge graphs), making effective reasoning over such diverse data increasingly important. Unified structured data question answering has emerged as a prominent research trend, aiming to answer natural language questions across different structured data types within a single framework. However, existing unified methods share a common limitation: they rely on a set of predefined functions, which restricts their ability to perform complex reasoning beyond these predefined operations. To overcome this fundamental limitation, we propose CRAFTQA, a novel adaptive code-driven framework comprising two core modules, CodeSTEP and CRAFT. The CodeSTEP module is a paradigm that generates a complete executable Python code sequence, which contains step-by-step code-based reasoning operations based on the question. The CRAFT module dynamically generates custom code functions for operations beyond the predefined function set, and seamlessly integrates with CodeSTEP to significantly enhance flexibility in handling complex reasoning. Comprehensive experiments on multiple structured datasets demonstrate that CRAFTQA achieves remarkable improvements in complex reasoning scenarios compared to existing unified methods.

Chengtao Gan, Zhiqiang Liu, Long Jin +3

Zhejiang University · JIUTIAN Research, Beijing, China · Ant Group +1

Table QA LLM-Based Program Synthesis Algorithmic Reasoning LLM Code Generation

May 30, 2026cs.LG

Richer Representations for Neural Algorithmic Reasoning via Auxiliary Reconstruction

Neural algorithmic reasoning has emerged as a popular research direction. It aims to train neural networks to mimic the step-by-step behavior of classical rule-based algorithms. More specifically, the execution of such algorithms can be abstracted as a sequence of states, where each state represents the intermediate outcome after an execution step. The training objective is to generate state sequences that replicate the underlying algorithmic process. A common framework for this task adopts an encoder-processor-decoder architecture, where the encoder learns representations of states, the processor simulates algorithmic steps, and the decoder reconstructs output states. While prior work has focused on improving the processor, the role of the encoder in representation learning has received little attention. Most methods rely on simple MLP encoders, raising the question of whether such representations are sufficiently informative for supporting algorithmic reasoning. This paper investigates how to improve encoder representations for neural algorithmic reasoning. We propose a reconstruction module that aims to recover the input state from its encoded representation. This auxiliary reconstruction task encourages the encoder to retain critical information about the input. We demonstrate that incorporating this task during training improves the performance of existing neural architectures on standard benchmarks. Furthermore, we observe that current encoders often underutilize the correlations among features within a state. To address this, we draw inspiration from self-supervised learning and design an enhanced variant of the auxiliary task that encourages the encoder to capture intra-state feature dependencies. Experimental results show that our method enables the encoder to learn richer representations, thereby enhancing the performance of existing processors on algorithmic reasoning tasks.

Jiafu Huang, Chao Peng, Chenyang Xu +7

Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, China · Overseas Education College, Xiamen University, China · Dept. of Electrical and Electronic Engineering, The University of Melbourne, Australia

Neural Algorithmic Reasoning State Representation Learning Algorithmic Reasoning

May 29, 2026cs.AI

LinTree: Improving LLM Reasoning with Explicitly Structured Search Histories

Large language models (LLMs) often solve reasoning problems by generating intermediate traces that explore and revise partial solutions. From a search perspective, these traces can be viewed as linearized search trees, where the model extends a partial solution, abandons it when it fails, and backtracks to try alternatives. Compared with traditional heuristic-guided search, such a policy has a potential advantage: it conditions on the whole search trace rather than only on the current local state. We first test whether LLMs utilize this advantage by comparing trace-conditioned reasoning policies against best-first search equipped with an LLM heuristic that only observes the current local state. Across three controlled reasoning environments, Blocks World, grid Navigation, and Sokoban, we find that raw access to search history alone is not enough to reliably outperform heuristic search. We then study one possible reason: in LLM reasoning traces, the underlying search tree is only implicitly represented, and when the model backtracks or switches branches, the trace does not explicitly identify which earlier search state is being revisited. We show that adding simple parent pointers to explicitly represent the linearized tree (LinTree) structure improves both task performance and search efficiency relative to implicit reasoning models and LLM-heuristic-guided search. These results suggest that search history becomes most useful when its tree structure is made explicit, motivating more structure-aware representations for LLM reasoning.

Liwei Kang, Yee Whye Teh, Wee Sun Lee

National University of Singapore · University of Oxford

Inference-Time Search Tree of Thoughts Structured Reasoning Algorithmic Reasoning LLM Reasoning

May 29, 2026cs.CL

Neuro-symbolic Syntactic Parsing: Shaping a Neural Network with the CYK Algorithm

In this paper, we show the possibility of a direct injection of algorithms into neural network architecture. We focus on a complex algorithm, that is, Cocke-Youger-Kasami (CYK) for parsing context-free grammars in Chomsky Normal Form and we propose CYKNN, a simple recurrent neural network architecture for encoding the CYK algorithm in trainable matrix-vector multiplications.We experimented with a very simple grammar with 4 variations showing that our approach outperforms existing LLMs with more than 20B parameters with an in-context learning setting and smaller LLMs of the Qwen family fine-tuned with LoRA. Our attempt paves the way to a different approach to neuro-symbolic methodologies.

Fabio Massimo Zanzotto, Federico Ranaldi, Giorgio Satta

Human-centric ART, University of Rome Tor Vergata, Italy · DEI, University of Padua, Italy

Neural Algorithmic Reasoning Recurrent Neural Networks Neuro-Symbolic Reasoning Algorithmic Reasoning

May 28, 2026cs.LG

Discovering a Zeta Map Algorithm on Dyck Paths via Mechanistic Interpretability

Machine learning is increasingly used in mathematical discovery, but in mathematics the desired output is often not a prediction itself, but an explicit construction that can be checked independently. We study this setting through the zeta map on Dyck paths, a classical bijection in the combinatorics of the q,t-Catalan numbers. We train a deliberately small one-layer, one-head encoder-decoder transformer on this map and analyze its learned computation using mechanistic interpretability tools, including decoder cross-attention analysis, linear probing, and causal intervention. The analysis reveals a level-based mechanism: encoder representations make path levels linearly accessible, while the decoder selects and traverses input positions in a structured way. Translating these signals into combinatorics leads to the scaffolding map, an explicit peak-centered traversal algorithm for Dyck paths. We prove that this algorithm agrees with the zeta map, modulo a reversal convention in the labeling. This gives a controlled example of AI-assisted mathematical discovery in which mechanistic interpretability turns model behavior into a precise, human-verifiable combinatorial algorithm.

Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee

Department of Mathematics, Temple University, Philadelphia, PA, USA · Institute for Computer-Aided Reasoning in Mathematics, Carnegie Mellon University, Pittsburgh, PA, USA · Department of Mathematics, University of Connecticut, Storrs, CT, USA +1

Automated Algorithm Discovery Algorithmic Reasoning

May 28, 2026cs.LG

Open Problem: Separating Geometric and Algorithmic Compression via Cayley-Table Completion

Modern statistical learning theory and deep learning characterize generalization primarily in terms of continuous capacity control (e.g., norm-based regularization, margin maximization, low-rank bias). While highly successful in continuous domains, deep learning consistently fails to extrapolate exact algorithmic or discrete algebraic rules, reflecting a missing inductive bias toward algorithmic complexity minimization. We propose the Cayley-table completion as the canonical testbed for this missing bias, serving as the discrete algebraic counterpart to matrix completion. Just as matrix factorization combined with weight decay yields an implicit geometric bias toward low linear rank, recent results demonstrate that operator-valued tensor factorizations paired with a flatness prior yield an implicit algorithmic bias toward exact discrete associativity. We pose the open problem of establishing formal exact recovery bounds for Cayley-table completion, and challenge the community to generalize continuous flatness priors to autonomously discover broader discrete algorithmic axioms without combinatorial search.

Dongsung Huh

Statistical Learning Theory Algorithmic Reasoning

May 27, 2026cs.AI

Revealing Algorithmic Deductive Circuits for Logical Reasoning

Recent studies have shown that Large Language Models (LLMs) can achieve strong reasoning performance by incorporating functional symbolic representations that abstractly describe graph traversal algorithms and step-by-step reasoning in few-shot learning settings. However, it remains unclear how LLMs genuinely understand the abstract meaning of each reasoning step and the overall algorithm from only a limited number of demonstrations. This work aims to localize the attention heads responsible for individual reasoning steps and characterize the types of information transferred among them. We first align constituent reasoning steps with their corresponding token logits under a symbolic-aided Chain-of-Thought (CoT) prompting framework. Our analysis shows that token positions that steer the reasoning process are associated with low confidence scores caused by constraints on satisfying reasoning behavior patterns in demonstrations. We then adopt causal mediation analysis techniques to identify the attention heads responsible for these patterns. In addition, our findings indicate that LLMs retrieve factual and rule-based information for individual sub-reasoning tasks through specialized attention heads (approximately 3% total heads), whereas higher layers predominantly facilitate information integration and the emergence of global reasoning strategies (e.g., graph traversal algorithms) that coordinate multiple intermediate reasoning steps to solve the overall task.

Phuong Minh Nguyen, Tien Huu Dang, Naoya Inoue

Japan Advanced Institute of Science and Technology

Attention Head Analysis LLM Interpretability Mechanistic Interpretability Algorithmic Reasoning LLM Reasoning

May 24, 2026cs.AI

Boosting Inference with Guided Reasoning: Stochastic Exploration for Recursive Models

Recent work on recursive architectures has shown that tiny neural networks can be surprisingly powerful on structured reasoning tasks. The trick is to model reasoning trajectories with a latent dynamical system. We argue that the inference-time behaviour of these architectures is best understood as approximate inference over latent reasoning trajectories, with deterministic recursion as the one-particle, zero-noise limit. We make this view operational through guided stochastic exploration: stochastic perturbations of the reasoning dynamics propose neighbouring trajectories, and the model's existing early-stopping head reweights them online. The framework yields three label-free diagnostics: local stability, guide alignment, and cloud-token entropy. These predict, from inference traces alone, whether the procedure will help and which of its outputs to trust. On Sudoku-Extreme it lifts exact-solve accuracy from

85.9\%

to

98.0\%

without retraining; on Maze-Hard the diagnostics flag a misaligned guide, as validation performance later confirms. The same machinery thus characterises both when recursive reasoning has room to improve at the trajectory level and when the model's internal guide can recover it.

Andrew Corbett, Archit Sood, Anna Tzatzopoulou +2

digiLab, UK · University of Bristol, UK

Inference-Time Search Structured Reasoning Algorithmic Reasoning Inference-Time Guidance

May 21, 2026cs.LG

Represented Is Not Computed: A Causal Test of Candidate Algorithmic Intermediates in a Transformer

Structured prompts require integrating components according to task-relevant relations. How a network implements this integration is often hard to judge in language or vision, where those relations are rarely specified precisely enough to define a candidate internal algorithm. Arithmetic offers a cleaner setting. We study a Transformer trained on base-digit extraction: given

N

,

B

, and

D

, it must report the coefficient of

B^D

in the base-

B

expansion of

N

. The closed-form solution,

\lfloor N/B^D \rfloor \bmod B

, provides explicit candidate algorithmic intermediates. Across three seeds, the model reaches 99.83% exact-answer accuracy on held-out number-base intersections, establishing reliable task competence. Linear probes decode the intermediates, making staged arithmetic computation plausible. Causal tests then separate representation from use: within the localized route from the stream with

D

as input to the output positions, behavior depends on early

D

-selective communication, independent of

N

and

B

. Relatedly, a sparse circuit search finds mostly separate

N

,

B

, and

D

routes that combine late rather than the staged route suggested by the probes. Thus, the model represents the intermediates that make the closed-form solution plausible, but the identified localized causal route does not transmit them to the output stream. This case shows that probe-based conclusions can diverge sharply from causal observations, even when explicit algorithmic hypotheses are available.

Ishita Darade, Sushrut Thorat

MKSSS’s Cummins College of Engineering for Women, Pune 411052, India · Institute of Cognitive Science, Osnabrück University, Osnabrück 49069, Germany

Numerical Reasoning in Language Models Transformer Interpretability Neural Algorithmic Reasoning Causal Attribution Causal Interpretability Causal Interventions in Language Models+2

May 20, 2026cs.LG

Equilibrium Reasoners: Learning Attractors Enables Scalable Reasoning

Scaling test-time compute by iteratively updating a latent state has emerged as a powerful paradigm for reasoning. Yet the internal mechanisms that enable these iterative models to generalize beyond memorized patterns remain unclear. We hypothesize that generalizable reasoning arises from learning task-conditioned attractors: latent dynamical systems whose stable fixed points correspond to valid solutions. We formalize this process through Equilibrium Reasoners (EqR), which enable test-time scaling without external verifiers or task-specific priors. EqR scales internal dynamics along two axes: depth, by running more iterations, and breadth, by aggregating stochastic trajectories from multiple initializations. Empirically, gains from test-time scaling are tightly coupled with stronger convergence toward solution-aligned attractors. This attractor perspective allows neural networks to adaptively allocate test-time compute based on task difficulty. While simple cases converge within 1 to 5 iteration steps, harder cases benefit from massive test-time scaling. By unrolling up to the equivalent of 40,000 layers, scalable latent reasoning boosts accuracy from 2.6% for feedforward models to over 99% on Sudoku-Extreme. These results suggest that learned attractor landscapes provide a useful mechanistic lens for understanding scalable reasoning in iterative latent models.

Benhao Huang, Zhengyang Geng, Zico Kolter

CMU

Dynamical Systems Test-Time Scaling Algorithmic Reasoning Latent Reasoning

May 19, 2026cs.AI

Generative Recursive Reasoning

How should future neural reasoning systems implement extended computation? Recursive Reasoning Models (RRMs) offer a promising alternative to autoregressive sequence extension by performing iterative latent-state refinement with shared transition functions. Yet existing RRMs are largely deterministic, following a single latent trajectory and converging to a single prediction. We introduce Generative Recursive reAsoning Models (GRAM), a framework that turns recursive latent reasoning into probabilistic multi-trajectory computation. GRAM models reasoning as a stochastic latent trajectory, enabling multiple hypotheses, alternative solution strategies, and inference-time scaling through both recursive depth and parallel trajectory sampling. This yields a latent-variable generative model supporting conditional reasoning via

p_θ(y \mid x)

and, with fixed or absent inputs, unconditional generation via

p_θ(x)

. Trained with amortized variational inference, GRAM improves over deterministic recurrent and recursive baselines on structured reasoning and multi-solution constraint satisfaction tasks, while demonstrating an unconditional generation capability. https://ahn-ml.github.io/gram-website

Junyeob Baek, Mingyu Jo, Minsu Kim +3

KAIST · Mila – Québec AI Institute · New York University +1

Latent Variable Models Generative Modeling Conditional Generative Modeling Algorithmic Reasoning

May 17, 2026cs.LG

Adaptive Generate-Rank-Verify: Inference-Time Search with Costly Verification

Many inference-time language-model pipelines combine a cheap reward signal with an expensive verifier, such as exact answer checking in mathematical reasoning or hidden-test execution in code generation. We formalize this setting using a learning-theoretic lens as generative active search: a cost-sensitive first-positive search problem in which a policy adaptively samples candidates from an unknown distribution, observes cheap scores, and pays for verifier labels until it finds a positive example. For a fixed prompt, the generator and reward model induce two unknown objects: a distribution over reward scores and a score-conditioned success function. When these quantities are known, we characterize the distribution-aware optimal policy using a dynamic programming approach. In the realistic and practical setting where both the score distribution and success function are unknown, we propose ADAP, a shellwise adaptive generate-rank-verify algorithm that progressively increases the number of sampled responses and top-ranked verifications. Under the monotonicity assumption that higher reward scores are no less likely to pass verification, we show that ADAP achieves expected cost within a constant factor of the distribution-aware optimum. We complement this result with learning-theoretic lower bounds, based on a centered star number, showing that structural assumptions on the score--label relationship are necessary. Experiments on mathematical reasoning and competitive programming validate the predicted advantage over both fixed non-adaptive policies and difficulty-adaptive baselines.

Shaddin Dughmi, Mahdi Haghifam, Yusuf Hakan Kalayci

University of Southern California · Data Science Institute at the University of Chicago · Northwestern University +2

Adaptive Inference Inference-Time Search Cost-Aware Inference Algorithmic Reasoning

May 16, 2026cs.CY

Algorithmic Constitutionalism

The increasing encroachment of artificial intelligence (AI) on social life raises significant risks for society, particularly within the infospheres created and controlled by companies such as Google, Facebook, Apple, and Amazon. This article examines these risks through an in-depth analysis of Facebook's content moderation regime, which is already partially governed by algorithms. We argue that the idea of ethical engineering, often proposed in the literature as a solution to the governance challenges posed by AI, is inadequate for several reasons. In response, we develop an alternative framework, which we term "algorithmic constitutionalism." Our approach rests on three pillars: (a) a layered architecture consisting of two levels of code: (i) an operative or object level and (ii) a meta level designed to protect the system's core principles from algorithmically initiated change; (b) algorithmic meta-reasoning, which enables the system to operate simultaneously at both levels so that it can monitor, verify, and potentially correct in real time operations at the object level that depart from principles protected at the meta-code level; and (c) correction through deliberation. The article elaborates the concept of algorithmic constitutionalism and demonstrates how it may be applied to Facebook's content moderation regime. As part of this analysis, we examine the tension between societal constitutionalism and algorithmic constitutionalism. Paradoxically, attempts to subject AI systems to external deliberative control may also enable AI agents to intervene in that process, potentially undermining its purpose. The article concludes by considering the implications of this argument for the European Digital Services Act, which entered into force in October 2022.

Oren Perez, Nurit Wimer

Content Moderation AI Governance Algorithmic Reasoning

May 15, 2026cs.CL

Process Rewards with Learned Reliability

Process Reward Models (PRMs) provide step-level feedback for reasoning, but current PRMs usually output only a single reward score for each step. Downstream methods must therefore treat imperfect step-level reward predictions as reliable decision signals, with no indication of when these predictions should be trusted. We propose BetaPRM, a distributional PRM that predicts both a step-level success probability and the reliability of that prediction. Given step-success supervision from Monte Carlo continuations, BetaPRM learns a Beta belief that explains the observed number of successful continuations through a Beta-Binomial likelihood, rather than regressing to the finite-sample success ratio as a point target. This learned reliability signal indicates when a step reward should be trusted, enabling downstream applications to distinguish reliable rewards from uncertain ones. As one application, we introduce Adaptive Computation Allocation (ACA) for PRM-guided Best-of-N reasoning. ACA uses the learned reliability signal to stop when a high-reward solution is reliable and to spend additional computation on uncertain candidate prefixes. Experiments across four backbones and four reasoning benchmarks show that BetaPRM improves PRM-guided Best-of-N selection while preserving standard step-level error detection. Built on this signal, ACA improves the accuracy--token tradeoff over fixed-budget Best-of-16, reducing token usage by up to 33.57% while improving final-answer accuracy.

Jinyuan Li, Langlin Huang, Chengsong Huang +5

Washington University in St. Louis · Singapore University of Technology and Design

Reward Modeling Process Reward Models Algorithmic Reasoning Adaptive Computation

May 14, 2026cs.CL

Reasoning Models Don't Just Think Longer, They Move Differently

Reasoning-trained language models often spend more tokens on harder problems, but longer chains of thought do not show whether a model is merely computing for more steps or following a different internal trajectory. We study this distinction through hidden-state trajectories during chain-of-thought generation across competitive programming, mathematics, and Boolean satisfiability. Raw trajectory geometry is strongly shaped by generation length: longer generations mechanically alter path statistics, so difficulty-dependent comparisons are misleading without adjustment. After residualizing trajectory statistics on length, difficulty remains systematically coupled to corrected trajectory geometry across all domains studied. The clearest reasoning-specific separation appears in the code domain, where harder problems show more direct corrected trajectories and less heterogeneous local curvature in reasoning-trained models than in matched instruction-tuned baselines. Corrected difficulty-geometry coupling is weaker, but still present, in mathematics and Boolean satisfiability. Prompt-stage linear probes do not mirror the code-domain separation, and behavioral annotations show that stronger corrected coupling co-occurs with strategy shifts and uncertainty monitoring. Together, these findings establish length correction as a prerequisite for generation-time trajectory analysis and show that reasoning training can be associated with distinct corrected trajectory geometry, with the strength of the effect depending on the domain.

Anders Gjølbye, Lars Kai Hansen, Sanmi Koyejo

Technical University of Denmark · Stanford University

LLM Interpretability Representation Geometry in Language Models Algorithmic Reasoning LLM Reasoning Large Reasoning Models Reasoning Trace Analysis

May 14, 2026cs.AI

OpenDeepThink: Parallel Reasoning via Bradley-Terry Aggregation

Test-time compute scaling is a primary axis for improving LLM reasoning. Existing methods primarily scale depth by extending a single reasoning trace. Scaling breadth by sampling multiple candidates in parallel is straightforward, but introduces a selection bottleneck: choosing the best candidate without a ground-truth verifier, since pointwise LLM judging is noisy and biased. To address this, we introduce OpenDeepThink, a population-based test-time compute framework that selects via pairwise Bradley-Terry comparison. Each generation, the LLM judges random pairs of candidates and aggregates votes via Bradley-Terry into a global ranking; top-ranked candidates are preserved and the top three quarters are mutated using the natural-language critiques produced during comparison; the bottom quarter is discarded. OpenDeepThink raises Gemini 3.1 Pro's effective Codeforces Elo by +405 points in eight sequential LLM-call rounds (~27 minutes wall-clock). The pipeline transfers across weaker and stronger models without retuning, and on the multi-domain HLE benchmark, gains appear concentrated in objectively verifiable domains and reverse in subjective ones. We release CF-73, a curated set of 73 expert-rated Codeforces problems with International Grandmaster annotation and 99% local-evaluation agreement against the official verdict.

Shang Zhou, Wenhao Chai, Kaiyuan Liu +3

1UC San Diego · 2Princeton University · University of Washington +1

Pairwise Comparison Test-Time Optimization Test-Time Scaling Bradley–Terry Model Algorithmic Reasoning LLM Reasoning

May 13, 2026cs.LG

Transformers Linearly Represent Highly Structured World Models

Do transformers, when trained on sequential reasoning traces, build internal models of the underlying task? And if so, does the structure of those internal representations mirror the structure of the domain? We train an 8-layer transformer on Sudoku solving traces and perform a mechanistic analysis of its internal computation. We establish two results. First, the model builds a substructure world model: it does not represent the board state cell by cell, as a human analyst would expect, but organizes information around the rows, columns, and boxes that Sudoku's constraints act on. Second, we identify a naked-single circuit: a small set of dedicated neurons in the final MLP layer, each individually detecting when exactly one digit remains possible for a specific cell, and reliably promoting that digit. These findings show that the geometry of an emergent world model is shaped by the constraint algebra of the domain, not its surface presentation, and that the resulting decision circuit is sparse, monosemantic, and fully interpretable. More broadly, they demonstrate that mechanistic interpretability tools can recover an end-to-end algorithmic account of how a transformer solves a combinatorial reasoning task.

Roman Kniazev, Nathanaël Fijalkow

LaBRI, CNRS · University of Bordeaux · France

Circuit Discovery World Models Mechanistic Interpretability Structured Reasoning Constraint Satisfaction Algorithmic Reasoning

May 13, 2026cs.LG

First-Passage Prediction of Grokking Delay: ACalibrated Law under AdamW with Causal Validation

We give the first quantitative prediction of grokking delay under AdamW. Treating the delay as a first-passage time, we derive a closed-form law T_grok - T_mem = (1 / 2 kappa_LL eta lambda) log(V_mem / V_star), where V_t = ||theta_t||^2 is the squared parameter norm, V_star is an architecture-dependent threshold, and kappa_LL absorbs the AdamW correction to the clean-SGD contraction rate 2 eta lambda. Calibrating (kappa_LL, V_star) on a single hyperparameter cell predicts grokking delays on 26 held-out runs with MAPE 17.7% over a 41x delay range; the law generalises to MLPs (MAPE 18.0%, N=34) and degrades to 23.3% on cross-task extension (N=46, 43.5x range), with a structured residual in which V_star / V_mem stays comparatively stable within architecture (CV about 14% on the 1L transformer). First-passage of V_t is necessary but not sufficient. A quantile-margin theorem establishes that positive delay requires both norm separation V_mem > V_post and angular reachability of a threshold alpha_star = arcsin(C / V_T_mem^(1/2)), where C is computable from the empirical NTK feature map and the validation-margin quantile. Calibrating C on modulus p=89 predicts alpha_star = 47.2 degrees at p=97 (observed 47.8 degrees, error 1.3%) as a prior cross-cell prediction. Causal interventions that freeze the norm or remove weight decay at memorisation eliminate grokking (0/6 vs. 3/3 baseline), trapping the angular displacement near 12 degrees. kappa_LL is empirically measured per architecture rather than derived from (beta_1, beta_2, epsilon); within-architecture CV stays at most 15% across four architectures, but values differ by about 2x between architectural variants beyond depth alone. Empirical scope is algorithmic tasks (modular arithmetic, sparse parity) under AdamW; whether the law transfers to natural-language scale models is open.

Truong Xuan Khanh, Truong Quynh Hoa, Luu Duc Trung +1

H&K Research Studio, Clevix LLC, Hanoi, Vietnam · Banking Academy of Vietnam, Hanoi, Vietnam

Neural Network Memorization Neural Network Training Dynamics Algorithmic Reasoning Grokking

May 13, 2026cs.AI

Retrieval-Augmented Tutoring for Algorithm Tracing and Problem-Solving in AI Education

Students learning algorithms often need support as they interpret traces, debug reasoning errors, and apply procedures across unfamiliar problem instances. In this paper, we present KITE (Knowledge-Informed Tutoring Engine), a Retrieval-Augmented Generation (RAG)-based intelligent tutoring system designed to serve as a classroom teaching assistant for algorithmic reasoning and problem-solving tasks. KITE uses an intent-aware Socratic response strategy to tailor support to different student needs, responding with targeted hints, guiding questions, and progressive scaffolding intended to strengthen students' algorithmic problem-solving ability. To keep responses aligned with course content, KITE uses a multimodal RAG pipeline that retrieves relevant information from course materials. We evaluate KITE using three forms of assessment: RAGAs-based metrics for response grounding and quality, expert evaluation of pedagogical quality, and a simulated student pipeline in which a weaker language model interacts with KITE across two-turn dialogues and produces revised answers after receiving feedback. Results indicate that KITE produces contextually grounded and pedagogically appropriate responses. Further, using simulated students, KITE's feedback helped the student models produce more accurate follow-up responses on procedural and tracing questions, suggesting that its scaffolding can support algorithmic problem-solving. This work contributes a tutoring architecture and an evaluation approach for assessing retrieval-grounded explanations and scaffolded problem-solving feedback.

Mragisha Jain, Tirth Bhatt, Griffin Pitts +6

North Carolina State University · University of Pittsburgh · University of California, Berkeley +1

Retrieval-Augmented Generation Intelligent Tutoring Systems Algorithmic Reasoning

May 12, 2026cs.LG

Evolutionary Task Discovery: Advancing Reasoning Frontiers via Skill Composition and Complexity Scaling

The reasoning frontier of Large Language Models (LLMs) has advanced significantly through modern post-training paradigms (e.g., Reinforcement Learning from Verifiable Rewards (RLVR)). However, the efficacy of these methods remains fundamentally constrained by the diversity and complexity of the training data. One practical solution is data synthesis; yet, prevalent methods relying on unstructured mutation or exploration suffer from homogeneity collapse, failing to systematically expand the reasoning frontier. To overcome this, we propose Evoutionary Task Discovery (EvoTD), a framework that treats data synthesis as a directed search over a dual-axis manifold of Algorithmic Skills and Complexity Attributes. We introduce structured evolutionary operators to navigate this space: a Crossover operator that synthesizes novel skill compositions to enhance diversity, and a Parametric Mutation operator that scales structural constraints (e.g., input size, tree depth) to drive robust generalization. Crucially, we integrate a dynamic Zone of Proximal Development filter, ensuring tasks lie within the learnable region of the model. Empirically, EvoTD delivers substantial reasoning gains that generalize consistently across model architectures, pretraining regimes, and scales, demonstrating that structured evolutionary curricula can effectively support reasoning improvement. We release our code on https://github.com/liqinye/EvoTD.

Liqin Ye, Yanbin Yin, Michael Galarnyk +3

Georgia Institute of Technology

Evolutionary Optimization Curriculum Learning Algorithmic Reasoning LLM Reasoning

May 7, 2026cs.AI

MAS-Algorithm: A Workflow for Solving Algorithmic Programming Problems with a Multi-Agent System

Algorithmic problem solving serves as a rigorous testbed for evaluating structured reasoning in AI coding systems, as it directly reflects a model's ability to perform structured reasoning in complex scenarios. Existing approaches predominantly rely on model-centric strategies, such as architectural modifications and data scaling, which are costly and offer limited interpretability. Alternative methods leveraging external tools or prompting techniques (e.g., chain-of-thought) are often fragmented and lack a unified framework. In this paper, we propose MAS-Algorithm, a systematic multi-agent workflow for algorithmic problem solving inspired by the practices of competitive programmers and algorithm engineers. Our framework decomposes the end-to-end solving process into modular stages, enabling structured reasoning, tool integration, and flexible coordination among agents. The design emphasizes both rigor and extensibility, allowing it to generalize across diverse problem types. Experimental results on a self-constructed benchmark demonstrate consistent improvements across multiple Qwen series models, achieving an average gain of 6.48% in acceptance rate. In contrast, parameter-efficient fine-tuning on the same data yields only a marginal improvement of 0.89%. We further observe a 4.72% gain on LiveCodeBench-Pro, along with consistent improvements across additional accuracy and efficiency metrics. Beyond performance gains, we conduct comprehensive analyses to better understand the reasoning process within the workflow, including error patterns and cross-scenario behaviors. We further perform customized replacement and ablation studies to explore the upper bound of the framework, showing that individual agents can contribute improvements of up to 27.7%. These results highlight the strong potential of MAS-Algorithm for advancing AI-driven algorithmic reasoning.

Yuliang Xu, Xiang Xu, Yao Wan +2

1Peking University · 2Alibaba Group · 3Huazhong University of Science and Technology

Competitive Programming Multi-Agent Coding Algorithmic Reasoning