Hard symbolic-reasoning tasks such as Sudoku, maze pathfinding, and ARC remain challenging for LLMs due to their fixed-depth autoregressive reasoning, which limits systematic search, refinement, and backtracking. While recursive models such as Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM) address this limitation through iterative latent-state refinement, they are typically task-specific and do not leverage pretrained language priors. We propose R-Qwen, a recursive reasoning framework built upon a pretrained Qwen backbone. R-Qwen repeatedly refines a candidate solution through programmatic self-recursion and deep supervision, combining the structured iterative computation of recursive models with the linguistic and reasoning priors of pretrained LLMs. We further adapt Hierarchical Supervision Weighting (HSW) to autoregressive models by exponentially weighting losses across recursive steps. HSW reduces gradient variance by at least 50\%, improves the signal-to-noise ratio of stochastic gradients, and accelerates convergence. Across eight challenging benchmarks, R-Qwen consistently outperforms prior recursive reasoning models and substantially larger LLMs while using a comparable number of trainable parameters. Notably, on ARC-AGI dataset, our model achieves a 27.6\% improvement over the baseline, highlighting the effectiveness of recursive refinement for general symbolic reasoning. These results suggest that recursive reasoning mechanisms and pretrained language model priors are complementary approaches for improving symbolic puzzle-solving. Code and models will be released after acceptance.
Bohan Chen, Shivam N. Patel, Richard Hoffmann +2cs.AI
Tool calling allows large language models (LLMs) to invoke external computation during problem solving, a useful capability in various fields including AI for mathematics. We study this setting through weighted sum-of-squares (SOS) decomposition, a machine-checkable route to proving polynomial nonnegativity and hence polynomial inequalities. A candidate decomposition can be checked exactly, but finding one requires choosing among non-unique regroupings and coordinating multiple symbolic transformations. We develop an agent that combines algebraic task training, symbolic tools, and verifier-grounded optimization for this task. Rather than training only on the composite SOS task, we construct 1.35 million synthetic examples covering eight supporting polynomial tasks together with weighted-SOS decomposition. We first apply supervised fine-tuning (SFT) to direct algebra problems and simulated symbolic traces, and then use Group Relative Policy Optimization (GRPO) with task-specific symbolic rewards. The SFT corpus contains no native tool-calling messages; at evaluation, the agent uses native SymPy calls for expansion, collection, reordering, and factorization. Every final SOS answer is checked by exact expansion and coefficient comparison. On held-out, same-generator synthetic problems, the full SFT+GRPO+tools system is the strongest of four evaluated configurations, reaching 78.96% verified success on weighted SOS, compared with 44.73% for the base model with the same tools, and 91.75% macro accuracy across nine polynomial tasks. Within this controlled setting, our work provides a case study of combining domain-specific skill training, executable tools, and verifier feedback, and may inform the design of tool-calling agents in other domains with exactly checkable outputs.
Self-refinement often fails to strengthen few-shot inductive reasoning in large language models. Prompting a model to explicitly state its inferred rule does little on its own. What actually matters is a structurally enforced isolation between reasoning stages, so that information can only pass between them as a compressed symbolic state. We introduce \textbf{Hourglass reasoning}, which enforces strict context isolation between reasoning stages. The frozen LLM acts as a meta-constructor, building for each task a symbolic encoder--decoder: an Induction module compresses the support examples into a schema $φ$ (encoder) and a transient scaffold $z$; a Deduction module derives rule $T$ (decoder) from these and discards $z$; an Implementer compiles $(φ, T)$ into artifacts; an error-driven Refiner revises $(φ, T)$ and regenerates artifacts from scratch. Only $(φ, T)$ crosses stage boundaries, so all refinement stays anchored to the rule. We evaluate Hourglass across three benchmarks spanning visual abstraction, hardware synthesis, and textual rule induction, using GPT-5.5 and Gemini 3.1 Pro. On ARC-AGI-2, it raises best-of-5 accuracy by up to 14 points over an iterative-refinement baseline. On ChipBench, it nearly doubles Verilog synthesis accuracy with GPT-5.5, from 31\% to 58\%. BBEH-Linguini draws on puzzles from the International Linguistics Olympiad, a setting where prior work has shown that explicit verbalization can hurt performance. Hourglass mitigates this tendency, and on Gemini 3.1 Pro, it reverses the effect entirely. Ablations confirm that these gains come from the isolation between stages and the quality of the initial induction, not from prompt wording or the particular symbolic form used. It is how information flows through the reasoning process, rather than the language used to express it, that drives inductive reasoning in frozen LLMs.
Large language models can interpret natural-language chemistry questions, but their internal reasoning is difficult to inspect, constrain, and validate. This paper presents ChemOntoRule, a proof-of-concept symbolic core for AI-assisted school-level chemistry problem solving. The central design choice is task-centric ontology engineering: the ontology is constructed around the concepts, properties, relations, and executable procedures required by a defined collection of chemistry problems, rather than as a universal representation of chemistry. The implemented artifact combines a lightweight ontology serialized in JSON and RDF/Turtle with deterministic Python rules for electronic structure, periodic trends, oxidation states, oxide and hydride behavior, and related school-level reasoning patterns. A separate expert-coded fallback handles problem families not yet represented by general rules. The system was examined on 300 human-authored and manually validated chemistry problems. The complete system matched 296 of 300 reference answers (98.67%). The ontology-driven rule subset covered 269 problems and matched 266 references (98.88%); 31 problems were handled by task-specific expert-coded fallbacks, with 30 matches. Because the same collection informed ontology construction and evaluation, these results measure implemented coverage and internal consistency, not independent generalization. We analyze the four mismatches, distinguish structural validation from chemical correctness, and define a future architecture in which a language model acts primarily as a translator from user language into a normalized ontological task frame. Token efficiency is presented as a testable hypothesis for future controlled studies, not as a result of the current work.
Why do intelligent systems need to perform explicit symbolic reasoning? Computer science has traditionally regarded symbolic reasoning as a defining component of intelligence. Yet the remarkable success of modern foundation models raises a fundamental question: if increasingly capable AI systems can operate with little explicit symbolic reasoning, what role do symbolic methods actually play? This article argues that explicit symbolic reasoning is not a fundamental property of intelligence, but a computational consequence of operating on simplified models of reality. We propose the Compression Principle: every computational model is a simplified representation of reality, and explicit symbolic reasoning compensates for information omitted during model construction. From this principle, we derive the Modeling--Reasoning Trade-off: as computational models preserve richer representations of the world, the need for explicit symbolic reasoning correspondingly decreases. This perspective provides a unified explanation for both the historical success of symbolic methods and the remarkable effectiveness of modern foundation models. Paradoxically, the same development makes symbolic methods increasingly important for humans. As intelligent systems become more capable and more opaque, symbolic representations increasingly serve as interfaces through which humans specify requirements, verify behavior, regulate autonomous systems, and establish trust. We therefore argue that the future of symbolic methods lies not primarily as the computational engine of intelligent systems, but as the symbolic interface between increasingly capable AI systems and the humans who build, govern, and depend upon them.
Policy-grounded document review requires determining whether a target document complies with organization-specific policies, guidelines, or playbooks. While large language models can assist with policy interpretation and document analysis, end-to-end prompting leaves the applied policy logic implicit, making compliance decisions difficult to inspect, update, and test. We present PolicyGuard, a neuro-symbolic framework for policy-grounded document compliance review. PolicyGuard converts organizational policy guidance into an executable review engine consisting of typed relational logic rules and atom-level extraction questions. During review, LLMs answer these local questions using retrieved document evidence, and a symbolic evaluator applies the formal rules to detect non-compliance. We instantiate and evaluate PolicyGuard on company-specific NDA compliance review, where contract clauses must be checked against organization-specific negotiation policies. By separating policy formalization, local document interpretation, and symbolic compliance evaluation, PolicyGuard makes document review more explicit, maintainable, and systematically testable.
Logical reasoning is essential for reliable AI, yet existing benchmarks are largely first-order-logic-centric, focusing on object-level deduction over fixed predicates. This misses many realistic scenarios where models must reason over rules, predicates, functions, constraints, and decision procedures themselves. We introduce HOLMES (Higher-Order Logic Meets real-world Explainable Symbolic reasoning), the first real-world benchmark for higher-order symbolic reasoning in LLMs, containing 1379 instances. Built on higher-order logic, HOLMES pairs natural-language problems with HOL formalizations, ground-truth answers, verifiable reasoning traces, and fine-grained controllable reasoning factors across law and finance. Experiments show that current LLMs still struggle on HOLMES, with an average accuracy of only 50.64% and the best model reaching 59.54%. Our analyses further reveal that high final-answer accuracy can mask shortcut reasoning in conflict-resolution settings, while performance drops sharply under scope-conditioned and compositional reasoning. These findings identify higher-order symbolic reasoning as a key bottleneck for building reliable and verifiable LLMs. The project code and dataset are publicly available at https://github.com/wuyucheng2002/HOLMES.
We present BAss (BDD-based ADF symbolic solver), a novel analysis tool for Abstract Dialectical Frameworks (ADFs) based on Binary Decision Diagrams (BDDs). It supports the fully symbolic computation of all admissible, complete, and preferred interpretations, as well as two-valued and stable models of an ADFs. Our approach is inspired by the recently discovered equivalence between Boolean Networks (BNs) and ADFs by Heyninck et al. (2024) and Azpeitia et al. (2024), significantly extending current BDD-based tools bioLQM, AEON, and adf-bdd. We conducted experiments on a large-scale collection of real-world models from both the BN and ADF communities. Our results show that BAss dramatically outperforms previous BDD-based tools and is competitive (even significantly better in some cases) with state-of-the-art SAT/ASP-based methods, particularly in scenarios involving large solution spaces. Notably, BAss is able to enumerate all fixed points or minimal trap spaces of certain biological networks beyond the reach of existing tools, thereby enabling new analysis and case studies in systems biology. These results highlight the practical relevance of symbolic reasoning for complex real-world applications, particularly in systems biology and formal argumentation.