Symbolic Reasoning

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6 papers in the last four weeks, up 50% on the four weeks before. 0.1% of all new papers.

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Latest papers 55

Sep 28, 2026cs.AI

Training-Free Clinical Reasoning through Medical Ontologies and Cognitive Mapping: A Symbolic-Probabilistic Knowledge Graph Framework

Most clinical prediction systems learn patient-variable-outcome associations; we investigate a training-free diagnostic paradigm mapping patient observations to explicit medical knowledge. CKG Reasoner integrates candidate-specific Evidence Feature Nodes, patient-reference matching, a bounded Information Gate, knowledge-weighted evidence accumulation, disease similarity, and decisive clinical rules. Missing-aware normalization and coverage auditing distinguish absent from unavailable evidence. Candidate ranking is separate from outcome-label-independent K-means clustering, which uses four derived evidence coordinates (evidence strength, relative magnitude, directional similarity, and evidence completeness), not raw predictors or targets, to derive cohort-level assignments. Across six retrospective cohorts - four dengue (N = 1000, 1523, 989, 1018), malaria (N = 2190), and influenza (N = 4569) - a uniform, label-free, cohort-fitted K = 2 protocol yielded positive-class F1 scores of 0.996, 0.634, 0.936, 0.917, 0.695, and 0.842, and all-record accuracies of 0.996, 0.558, 0.914, 0.893, 0.707, and 0.906, respectively, with full partition-decision coverage using the frozen package and disease-specific knowledge representations. Neither scoring nor clustering uses outcome labels. Logistic regression provides a supervised baseline. Influenza incorporates confirmatory molecular PCR and is not independent pre-test prediction. Results characterize knowledge-grounded evidence separation, auditability, and sensitivity, not prospective clinical validity or comparative superiority. FOL/LLM-based clinical explanation remains unevaluated.

Surajit Das

Symbolic Reasoning Clinical Reasoning Clinical Prediction Knowledge Graphs

Sep 17, 2026cs.AI

PAA: The Probabilistic Allen Algebra: A Generative and Complete Probabilistic Extension of Allen's Interval Relations

Allen's interval algebra is a qualitative calculus for temporal relations, but its thirteen base relations are crisp predicates over exact interval boundaries. This is inadequate for temporal information from language, perception, databases, or uncertain histories, where times, durations, and boundaries are uncertain and expressions such as "just before" or "roughly during" have graded meaning. We develop the probabilistic Allen algebra (PAA): a generative and complete extension in which relation probabilities are derived from distributions over interval boundaries rather than assigned as scores. Time points are Gaussian; intervals have Gaussian midpoints and truncated-Gaussian durations. Every relation is a boundary-ordering predicate in one common probability space: point-point relations reduce to error functions, and point-interval and interval-interval relations to multivariate Gaussian orthant probabilities induced by linear inequalities. Contact relations (meets, starts, finishes, equals) receive positive measure through a tolerance band, and under a single tolerance the thirteen relations form a true partition that recovers crisp Allen as the tolerance vanishes. The construction derives Allen's taxonomy rather than positing it: coarse predicates such as precedence, overlap, and containment are unions of leaves whose probabilities are leaf sums, and this hierarchy is preserved as intervals collapse to points and thirteen relations reduce to five and then three. Each relation further decomposes into correlation-aware temporal primitives in the spirit of CIDOC CRM. The algebra is scale-invariant and separates graded expressions such as "shortly before" from contact relations. All results are Monte-Carlo validated and shipped as an open, tested Python package.

Julian Eggert

Honda Research Institute Europe, Offenbach, Germany · Honda Research Institute, Carl-Legien-Str. 30, 63073 Offenbach, Germany.

Probabilistic Logic Symbolic Reasoning Relational Reasoning

Sep 14, 2026cs.AI

Learning Symbolic Constraint Representations from Examples: A Neuro-Symbolic Approach

Learning user-defined concepts as constraint networks has been extensively studied in the constraint acquisition (CA) literature. However, existing approaches typically rely on intensive interactions with a human oracle, making the learning process costly in terms of time and number of queries. In this paper, we propose a neuro-symbolic framework for automatic CA that significantly reduces user involvement by introducing neural Oracle Transformer models which learn to emulate user responses and to generalize conceptual knowledge. Trained on previously available examples, the learned oracle interacts with a dedicated CA engine, FastCA, which systematically refines the oracle's responses into a sound, consistent, and interpretable constraint network. This neuro-symbolic interaction enables the recovery of structured symbolic models from data without prior domain knowledge. Our results demonstrate that this neuro-symbolic interplay effectively aligns data-driven pattern recognition with symbolic reasoning, offering a robust approach to automating model construction in combinatorial domains.

Nassim Belmecheri, Arnaud Gotlieb, Nadjib Lazaar +1

Simula Research Laboratory, Oslo, Norway · Microsoft, Norway · LISN, CNRS, Paris-Saclay University, France

Symbolic Reasoning Neurosymbolic Learning Constraint Programming

Sep 12, 2026cs.AI

A Multi-Stage Rule-Chaining Framework for Compositional and Interpretable Cognitive Reasoning

The Abstraction and Reasoning Corpus (ARC) benchmarks cognitive generalization, the ability to infer and apply abstract rules from limited examples. This paper presents a multi-stage rule-chaining framework that performs compositional reasoning across symbolic, structural, and conceptual levels. The framework integrates three complementary solvers: (1) a deterministic rule discovery module that induces atomic transformations through geometric, color, and object-based analysis; (2) a pattern-composition engine that reconstructs outputs via block merging, repetition, and spatial heuristics; and (3) a structural abstraction layer that infers hierarchical and nested relationships across grids. These solvers operate sequentially within a progressive fallback hierarchy, where each stage reuses prior reasoning traces to enhance interpretability and generalization. Training passed for 995 tasks out of 1000, further evaluated on 105 tasks out of 120 and solved 230 test tasks out of 240 ARC-AGI-2 tasks. The system achieved strong coverage across deterministic, compositional, and abstract categories, demonstrating an overall accuracy exceeding 95 percent. The proposed architecture bridges symbolic reasoning and pattern synthesis, providing interpretable insight into cognitive generalization. The results suggest that rule chaining and hierarchical composition can advance machine reasoning toward transparent, human-aligned abstraction without relying on task-specific tuning.

Deblina Kar

Indian Institute of Technology Kharagpur

Compositional Reasoning Interpretable ML Symbolic Reasoning

Sep 9, 2026cs.AI

From Symbolic Perception to Logical Deduction: A Framework for Guiding Language Models in Geometric Reasoning

Plane geometry remains a significant challenge in AI, requiring the integration of visual perception and mathematical reasoning. While Large Multimodal Models (LMMs) naturally handle visuo-linguistic inputs, they are often computationally intensive and opaque. We demonstrate that a pure Large Language Model (LLM), when equipped with specialized modules, can rival state-of-the-art LMMs on complex geometry problems. Our framework integrates a Geometric Vision Parser, which translates diagrams into symbolic form, with a Symbolic Solver that performs formal deductions, thereby mitigating hallucinations and promoting interpretable reasoning. To enable rigorous evaluation, we curate a benchmark of challenging problems from the 2025 Chinese Zhongkao examinations, ensuring data novelty and testing deeper deductive skills. Experiments demonstrate that our approach achieves performance comparable to Gemini 2.5 Pro while delivering clearer, human-like solutions.

Weichen Dai, Rafael Medeiros Cabral, Ziyi Shou +4

University of Science and Technology of China

Visual Reasoning Symbolic Reasoning LLM Mathematical Reasoning

Sep 7, 2026cs.AI

An Auditable Symbolic-RAG-Generative AI Architecture for Goal-Oriented Conversation Orchestration

Goal-oriented conversational systems must answer factual questions, understand visitor-provided information, and advance business objectives without becoming rigid questionnaires. This paper proposes a Symbolic-RAG-Generative architecture centered on the Goal-oriented Retrieval-Augmented Conversation Engine (GRACE). An instruction-constrained Business Goal Compiler transforms business intent into an immutable objective set, normalized priority vector, canonical questions, and initial state vector. At runtime, GRACE receives the complete conversation history, latest visitor message, current state, and grounded answer generated by a separate RAG component. It updates completion only from visitor-authored evidence and selects one contextually modulated follow-up. The core policy maximizes expected business progress subject to a minimum visitor-utility constraint. We formalize the state, monotonic transitions, source separation, question modulation, and constrained policy; present the reference architecture; and define an evaluation comprising 24 English real-estate and 10 Spanish professional-cleaning conversations, totaling 119 protocol-defined visitor turns. Across both domains, GRACE achieves 84.9% exact state-transition accuracy, 91.6% evidence precision, 89.6% evidence recall, 100% monotonicity, and 94.1% terminal-state accuracy. The evaluation establishes compelling symbolic-state performance across standard, multi-goal, RAG-detour, validation, refusal, and robustness scenarios.

Ramon Gonzalez, Antonio Diaz

Mentomy AI · Department of AI Research, Mentomy AI Almería, Spain

Retrieval-Augmented Generation Spoken Dialogue Systems Symbolic Reasoning Conversational Agents Task-Oriented Dialogue Systems

Sep 1, 2026cs.CL

A Certificate-Producing Cascade for Equational Implication: The SAIR EQT2 Stage 2 Solver

The SAIR Mathematics Distillation Challenge on Equational Theories asks a solver to classify whether one magma identity implies another and, for either verdict, to return a certificate accepted by a deterministic Lean judge. We present a single-file solver organized as a cheapest-first cascade. Its false branch combines coefficient tests over structured algebra families, bounded finite-model search, an explicit central-groupoid witness, and several infinite-carrier witnesses. Its true branch is a proof-producing ordered unit superposition procedure with Knuth-Bendix ordering, bidirectional demodulation, indexing, memoised substitution, and anytime size deepening. Search results remain outside the trusted base: successful derivations are replayed as small Lean terms, and countermodels are rechecked by the competition judge. The frozen solver is a 189,504-byte Python file with SHA-256 f2392533c9f4c03b.... In local runs through official judge revision 2848228, it produced accepted certificates for all 1,889 rows of the six public sets with no language-model calls. Separate measurements recorded full agreement on the 800 published Stage 1 evaluation-distribution problems, 100 accepted rows in the canonical Marathon manifest without tokens, and 200 accepted rows in the hosted playground. These are regression and playground measurements, not a leaderboard result and not evidence about a hidden set. All quantitative claims are tied to immutable result ledgers; the paper makes no completeness or comparative-superiority claim.

Haobo Ma, Wenlin Zhang, Manuel Israel Cázares

ChronoAI Pte. Ltd. · Singapore · National University of Singapore +2

Automated Theorem Proving Symbolic Reasoning

Aug 31, 2026cs.CL

Reactivating Test-Time Scaling for Plane Geometry Problem Solving

Plane geometry problem (PGP) solving has become a critical benchmark for multimodal reasoning because it requires accurate visual perception and precise multi-step symbolic deduction. Although test-time scaling (TTS) has demonstrated remarkable success in general mathematical reasoning, it fails to scale effectively under the symbolic-program paradigm for plane geometry. We identify two key obstacles: limited reasoning diversity induced by rigid symbolic programs and insufficient explicit visual grounding before symbolic deduction. To address these issues, we propose Multi-Trace Synthesis (MTS), which converts each symbolic program into heterogeneous reasoning traces, including executable Python scripts and CoT-augmented variants. We further propose Perception-Augmented (PA) training, which parses diagrams into structured semantic clauses before deduction, and Consensus-Guided Multi-Trace Ensemble (CG-MTE) for efficient self-adaptive inference. Experiments on three geometry benchmarks show that our method consistently improves PGP-solving across model scales and achieves strong performance against both general-purpose MLLMs and specialized geometry solvers. Under test-time scaling, CG-MTE achieves comparable accuracy to high-budget self-consistency while reducing sampling cost by up to 8x. Code and data are publicly available at https://github.com/Jason8Kang/ReTTS-PGPS.

Xiaoqiang Kang, Shengen Wu, Maizhen Ning +5

School of Advanced Technology, Xi’an Jiaotong-Liverpool University · The Hong Kong University of Science and Technology (Guangzhou) · Hithink Research +2

Spatial Reasoning Benchmarks Inference-Time Scaling Symbolic Reasoning Mathematical Reasoning

Aug 11, 2026cs.AI

Hypothesis Frontier: Verifier Guided LLM and Symbolic Search for First-Order Induction

First-order concept synthesis asks a system to infer one formula that classifies labeled objects consistently across several finite relational structures. Every candidate can be evaluated exactly, but quantified first-order formulas form a vast search space, and LLM outputs are often semantically promising without being fully correct. We introduce Hypothesis Frontier, a verifier-guided neurosymbolic framework that evaluates each LLM formula on every training object, retains the strongest verified hypothesis across rounds, and uses its remaining errors to guide subsequent generation. Symbolic processing repairs invalid formulas while remaining anchored to the LLM-generated hypothesis, and simplifies train-valid formulas without changing any training prediction. Under matched models, problem sets, and LLM-round budgets, Hypothesis Frontier solves substantially more problems than repeated original-prompt generation. After the final formulas are selected, exact simplification shortens many train-valid formulas while preserving every training prediction. Exact symbolic reasoning therefore helps both to solve more induction problems and to compress many of the resulting formulas.

Serafim Batzoglou

Independent Researcher

Symbolic Reasoning LLM-Based Program Synthesis Neuro-Symbolic Reasoning Inductive Logic Programming

Aug 11, 2026cs.AI

Reasoning Shortcuts and Value Symmetries: What Symmetry Permits, Architecture Realizes, and Optimization Selects

Reasoning shortcuts are solutions of a neurosymbolic system's rules that produce correct predictions through unintended concepts. A recent framework of Takemura, Inoue, and Nishino analyzes them through an automorphism group of value relabelings and asks, as its central open question, when rules pin concepts down. We first show that the framework's key definition, one shared permutation applied at every position, does not apply as stated to any of the four heterogeneous benchmarks it was evaluated on, and that the most direct embedding, padding domains to a common size, produces confident false pathology: 90.91% of solution pairs reported unexplained on CLE4EVR, where every well-defined member of the hierarchy we introduce reports 0%, and the padded verdict's content rotates with configuration-file ordering. Re-measuring eleven rule families under fifteen pre-specified predictions (thirteen confirmed), unexplained-pair rates span 0% to 99.9999% and track provable structure: six theorems give sufficient conditions for transitivity and its failure, including a Free Slot Lemma certifying Kandinsky's pathology from syntax alone. For circuit-given rules, deciding symmetry-inertness of a coordinate is coNP-complete; nontrivial-automorphism existence is coNP-hard under randomized reductions, lies in

Σ_2^p

, is not

Σ_2^p

-complete unless PH collapses, and on monotone circuits is coNP-complete outright. In the Boolean case transitivity is classified exactly: automorphisms explain everything iff the solution set is an affine coset. Weakly supervised models place all 94 observed shortcuts at the one level the componentwise theory flags and none at the 48 it certifies transitive; twelve typed-ambiguous levels produce none, separating what symmetry permits from what optimization selects, and a dual-head control replicates the geography. All numbers trace to released artifacts.

Xin Xu

Carnegie Mellon University · University of Pennsylvania

Symbolic Reasoning Neuro-Symbolic Reasoning

Aug 9, 2026cs.AI

SymDiag: Explainable Diagnosis for LLM Reasoning via Neuro-Symbolic Verification

Large language models (LLMs) increasingly serve as data-driven reasoners, yet their chains-of-thought (CoT) can be unfaithful even when final answers are correct. Most existing

verification'' signals are not diagnostic: answer matching observes only the outcome, LLM-as-judge provides subjective and non-verifiable critiques, and scalar rewards (e.g., PRMs/RMs) offer little insight into where a multi-step derivation fails.We propose \textbf{SymDiag}, a neuro-symbolic framework that \textbf{reframes reasoning verification as structured failure diagnosis}. SymDiag translates natural-language CoT into symbolic constraints and performs step-level satisfiability/entailment checks to (i) localize failing steps and (ii) produce verifiable diagnostic evidence, including counterexamples, inconsistency witnesses, and missing-premise indicators. A central challenge is that apparent

logic violations'' can be caused either by genuine reasoning defects or by neural-to-symbolic translation noise. SymDiag therefore incorporates a Self-Auditor that disentangles TranslationError from ReasoningError via dual symbolic encodings consistency checks, enabling robust diagnosis under partial observability. Across diverse mathematical, logical, scientific, and general reasoning benchmarks, SymDiag improves detection of unfaithful reasoning and provides substantially more effective feedback for multi-round reasoning repair than outcome-only verification and LLM-based judging, offering a principled foundation for trustworthy and scalable reasoning diagnosis.

Wenyao Cui, Huaping Zhang, Yongyi Huang +6

Beijing Institute of Technology Beijing, China · Zhongguancun Academy Beijing, China · Xinjiang Future Enterprise Incubator Co., Ltd. Xinjiang, China +1

CoT Faithfulness LLM Answer Verification Symbolic Reasoning Neuro-Symbolic Reasoning

Aug 4, 2026cs.AI

The RAIL Principles for Neurosymbolic AI: Reasoning, Assurances, Interfacing and Learning

Neurosymbolic AI systems that integrate machine learning and symbolic reasoning are rapidly gaining attention. They complement the data-intensive statistical approaches of neural networks and language models with symbolic reasoning algorithms to function in high-stakes domains or in low-data regimes that characterize many real-world applications. We argue that the neurosymbolic combination of machine learning and formal reasoning is not a niche approach within AI, but rather includes many already successful techniques that are of crucial importance to the development of reliable, efficient and, ultimately, trustworthy systems. This perspective prompts a re-examination of the design of current AI systems. We show that many leading AI systems, including some that are not traditionally considered as neurosymbolic, can be analysed from the perspective of four principles of neurosymbolic AI design: Reasoning, Assurances, Interfacing and Learning (RAIL). Applying the RAIL framework offers a unified view of seemingly disparate AI systems, ranging from physics-aware machine learning to neuro-guided search (such as Google DeepMind's Alpha-* suite), causal learning and tool-augmented Large Language Models. Importantly, the RAIL principles will enable engineers to make better-informed and more principled decisions about the design and deployment of production-level AI systems. In this article, we introduce the RAIL principles, examine how they can be applied across major areas of AI, and illustrate how they may guide practitioners to integrate neurosymbolic methods into next-generation AI technologies.

Agnese Chiatti, Michael Cochez, Cristina Cornelio +14

Politecnico di Milano · ELLIS Institute Finland & Abo Akademi University · Samsung AI +12

Symbolic Reasoning Neuro-Symbolic Reasoning Neuro-Symbolic AI Neurosymbolic Learning

Aug 4, 2026cs.AI

When Many Answers Are Valid, Voting Fails: Symbolic Verification for Best-of-K Causal Reasoning in LLMs

Self-consistency assumes the most frequent answer among sampled reasoning traces is the most reliable, but this can fail in causal reasoning: samples often repeat the same confounding error, and votes fragment across multiple valid answers, letting an invalid answer win despite a valid minority trace. We introduce CALVER (Causal Axiom-Level VERification), a training-free symbolic verifier that scores structured traces against Pearl's causal criteria, including -separation, backdoor adjustment, and intervention, and selects the highest-scoring candidate without consulting a reference answer. On CLEAR find-one-valid queries that admit multiple graph-valid answers, CALVER reaches 42.1% where plurality, a reward model, an LLM judge, and model confidence remain near 30% on identical frozen pools. Scaling the judge to 72B does not close the gap. In an audited clean-core subset, 11 of 21 graph-valid CALVER selections differ from the benchmark's listed answer while still satisfying the requested predicate. The advantage widens with the sampling budget and reproduces across ten published Bayesian networks, a second model family, and settings where the model must build the graph from text. CALVER also improves thresholded average-treatment-effect decisions against exact ground truth, generalizes to logic under a truth-table checker, and scores each candidate in milliseconds on CPU. CALVER needs only a causal structure, supplied outright or built from the text; wherever that holds, selection can aggregate via causal validity.

Omatharv Bharat Vaidya, Connor Thomas Jerzak, Zayne Rea Sprague +2

The University of Texas at Austin · New York University

Self-Consistency Decoding Symbolic Reasoning Causal Reasoning in Language Models Causal Inference

Aug 2, 2026cs.CV

Recursive Vision Language Models for General Symbolic Reasoning

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.

Omid Nejati Manzari, Guillaume Lajoie, Hassan Rivaz

Concordia University, Montreal, Canada · Mila — Quebec AI Institute · Université de Montréal

Symbolic Reasoning Implicit Reasoning in Language Models Algorithmic Reasoning LLM Reasoning

Jul 31, 2026cs.LO

Recovering Explanations from Transformed Rule-Based Ontologies

Datalog rules are often used to define ontologies over Knowledge Graphs. Rule reasoners routinely optimise such ontologies by rewriting their rules into a form that can be evaluated more efficiently. These transformations preserve the entailed facts, but not the structure of the underlying derivations. A proof tree under the rewritten rules explains why a fact holds, but does not readily yield an explanation in terms of the original rules. We study the problem of constructing, from a proof of entailment under the rewritten rules, a proof under the original ones: we establish its computational complexity and identify two practically relevant languages for specifying proof transformations.

Alex Ivliev, Markus Krötzsch, Maximilian Marx

Knowledge-Based Systems Group, TU Dresden, Dresden, Germany

Automated Theorem Proving Symbolic Reasoning Knowledge Representation

Jul 31, 2026cs.PL

Agentic Planning for Symbolic Execution

Symbolic execution seeks to explore feasible program paths, yet a practical run may exhaust its resources while much program behaviour remains unreached. We investigate a complementary way of extending its practical reach by reasoning about how the same tool is utilised from one bounded run to the next, while leaving ordinary state exploration to the underlying tool. We present Agolic, an agentic planning system that uses evidence from earlier runs to choose and configure later bounded symbolic execution (BSE) runs, which the underlying symbolic execution tool then carries out. The planning intelligence, available evidence and execution modes can be adapted to the symbolic execution tool and analysis objective. We evaluate one adaptation for branch-coverage exploration, in which an LLM-based agent reasons over source code, replayed coverage and earlier targeting attempts. We evaluate Agolic on several C and C++ programs. On every program, it extends the branch coverage obtained by continuous symbolic execution and covers more than

3\times

as many branches on average. It also covers more branches than each individual corpus from coverage-guided fuzzing and compiler-based concolic execution in our evaluation and reaches branches absent from all comparison corpora combined on six of the seven programs. Taken together, these results point to considerable untapped potential in existing symbolic execution tools, some of which may be realised by reasoning about how their capabilities are used across runs while leaving state selection during ordinary symbolic exploration to the underlying tool.

Daniel Koh Ji Yang, Yannic Noller, Corina S. Pasareanu +1

Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi, United Arab Emirates · Ruhr-Universität Bochum, Bochum, Germany · Carnegie Mellon University, Pittsburgh, PA, USA

Symbolic Reasoning Static Code Analysis LLM Agent Planning

Jul 25, 2026cs.AI

SymStep: Symbolic Step Verification for Logical Reasoning

Chain-of-thought (CoT) prompting can fail severely on constraint-dense logical reasoning tasks, where unverified errors accumulate silently across steps. We introduce SymStep: an LLM makes one atomic claim at a time (DEDUCE: Alice, pet, Cat), then a lightweight constraint propagator checks the claim for consistency with prior accepted deductions, rejects contradictions, and cascades implied facts automatically. SymStep+G additionally provides MRV guidance after each accepted step, directing the LLM toward the most constrained unresolved variable. On a 35-puzzle retained subset of ZebraLogicBench, a benchmark of 1,000 Einstein-style logic puzzles, Direct and CoT both achieve 0%, while SymStep+G reaches 97%. On AR-LSAT analytical reasoning problems, SymStep achieves 100% vs. CoT's 87%. On LGP-14, SymStep+G achieves 100% vs. 0% for CoT and Logic-LM, the strongest prior symbolic+LLM baseline we compare against. Ablation studies reveal that MRV guidance is a key mechanism for reducing directionless cycling, while consistency checking provides a safety net against explicit contradictions. Across six benchmarks spanning five task domains, SymStep variants match or exceed every baseline on constraint-dense and arithmetic tasks. Experiments on AQUA-RAT algebra confirm the advantage is constraint-density-specific.

Aida Usmanova, Rui Gao, Dilshod Azizov +2

Leuphana University of Lüneburg · Mohamed bin Zayed University of Artificial Intelligence

Symbolic Reasoning Structured Reasoning Neuro-Symbolic Reasoning Constraint Satisfaction LLM Reasoning

Jul 23, 2026cs.LO

Hybrid MKNF with Classical Negation in the Rule Component

Hybrid MKNF knowledge bases under the well-founded semantics integrate Description Logics with Logic Programming. However, they do not support classical negation in the rule component, limiting their ability to represent explicit negative knowledge. This limitation is particularly significant in safety-critical applications, where reasoning often requires explicit negative information rather than interpreting the absence of information as evidence of absence. To address this issue, we introduce an extension of Hybrid MKNF that supports classical negation in the rule component. We formally define the syntax and semantics of the extended language and present a general procedure for computing its well-founded model.

Arun Raveendran Nair Sheela, Christophe Rey, Florence De Grancey

Université Clermont Auvergne, LIMOS Laboratory, Thales · Université Clermont Auvergne, LIMOS · France +3

Logical Reasoning Symbolic Reasoning Nonmonotonic Reasoning

Jul 20, 2026cs.AI

Logical Judgments Under Pressure: Diagnosing Syllogistic Stability with Learned Soft Prefixes

To test how correct logical judgments respond to learned context, we prepend a soft prefix to an exactly labeled syllogistic reasoning benchmark while keeping the model fixed. Soft prefixes are opaque continuous vectors, so we characterize them through the behavior they induce across controlled variations in logical form and interface. By studying which prefixes succeed and how their effects generalize, we characterize how learned contextual pressure can override correct judgments and expose limits in a model's logical stability. Across Qwen3.6-35B-A3B MoE, Qwen3-8B, and Gemma 4 31B, learned prefixes redirect many correct answers and remain effective across unseen forms and interface changes. In repeated tests with Qwen3.6 MoE and Gemma, they outperform paired random controls in all 16 model--direction--split comparisons by 37 to 99 percentage points. Qwen3.6 MoE flip rates remain between 72% and 90% across wording and prompt changes, while Gemma validity prefixes retain 54% to 56% flip compared with less than 1% for matched random prefixes. Diagnostic tests show that the dominant effect is a broad preference for one answer meaning rather than fixed-symbol forcing or a logical operation that transfers reliably between tasks. The form of this bias differs across models. In both Qwen models, simple score models often predict which judgments will flip but not how far their margins will move, whereas Gemma's overall response is more closely approximated by the same models. These results show that the dominant behavioral effect of successful soft prefixes is a broad answer preference, while the remaining response reveals substantial model-specific differences in logical stability.

Brian K Chen

National University of Singapore

Logical Reasoning Symbolic Reasoning Prompt Tuning Language Model Robustness

Jul 17, 2026cs.AI

NeurOWL: An LLM-Based Neural-symbolic Framework for Incomplete OWL Ontology Reasoning

OWL ontologies provide a formal knowledge representation framework that enables semantic reasoning, and have been widely adopted across domains such as healthcare and bioinformatics. In practice, however, real-world ontologies are often incomplete, which pose challenges for reasoning. In this work, we focus on a fundamental subsumption reasoning problem: given an incomplete ontology and a candidate (non-entailed) subsumption, determine whether the subsumption is semantically plausible and, if so, providing a logically sound explanation containing potential missing axioms. This task unifies subsumption verification with ontology abduction, and generalizes the latter by removing the need for a predefined candidate set of missing axioms. To address this subsumption reasoning problem, we propose NeurOWL, an end-to-end neuro-symbolic framework that jointly performs verification and abduction, leveraging both formally defined semantics and textual semantics through Large Language Models and ontology embeddings. We evaluate NeurOWL on real-world ontologies across multiple domains, demonstrating strong and robust performance across different domains.

Hui Yang, Jiaoyan Chen, Yiping Song +2

The University of Manchester · Zhejiang University

Logical Reasoning Abductive Reasoning Symbolic Reasoning Neuro-Symbolic Reasoning

Jul 6, 2026cs.AI

The Changing Role of Symbolic Methods in Artificial Intelligence

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.

Jun Sun

Singapore Management University, Singapore

Logical Reasoning Symbolic Reasoning Human Oversight of AI

Jul 5, 2026cs.AI

Forethought: Verifiable Reasoning from Neurosymbolic Primitive Programming

Current agentic workflows usually involve decomposing user requests into sequences of tool calls with correctly resolved parameters, the results of which are processed through reasoning traces in the language model's context window. The prevailing route to improve such reasoning is test-time scaling, which trains models to search over long chains of thought; but the resulting capability is entangled in model weights, is not verifiable step-by-step, and is costly at inference. We present Forethought, a neurosymbolic reasoning system that instead treats reasoning as an explicit, verifiable program, that builds from a library of symbolic and neural primitives which are composed through a domain-specific language. The result are reasoning programs, which are concrete representations of the model's work, and as such can be inspected and modified before deployment. Instantiated as a tool-calling execution kernel and evaluated across five benchmarks, Forethought improves base-model accuracy by about 30% relative and outperforms vanilla prompting, reinforcement learning scaffolds, and prompt-evolution methods, enabling small models to match or exceed frontier models capabilities. In a direct comparison, a non-reasoning model augmented with Forethought competes with a dedicated reasoning model while requiring roughly three orders of magnitude less post-training investment, and remains model-agnostic and auditable.

Vishvesh Bhat, Jay Vaghasiya, Emmanuel Anaya Gonzalez

CoreThink AI · UC San Diego

Symbolic Reasoning Efficient Language Model Reasoning Agentic Reasoning LLM Tool Use Neuro-Symbolic Reasoning

Jul 2, 2026cs.AI

G-RRM: Guiding Symbolic Solvers with Recurrent Reasoning Models

In this work, we focus on SE-RRMs, a symbol-equivariant instantiation of RRMs that exhibits improved extrapolation to larger problem sizes. We propose a neuro-symbolic approach, ``Guiding with Recurrent Reasoning Models'' (G-RRM), which integrates SE-RRMs with symbolic solvers for constraint satisfaction problems. SE-RRMs act as neural solvers that generate full solution proposals and guide classical symbolic solvers, such as backtracking or SAT-based methods like Glucose 4.1 and CaDiCaL 3.0.0, that produce globally correct solutions. Centrally, we investigate when neural guidance with G-RRM improves the search efficiency of symbolic solvers. % Our experiments show that the efficacy of G-RRM depends on two conditions: first, the problem instances must have an expansive combinatorial search space to expose potential gains, and second, the solver architecture must be capable of dynamically overwriting its branching choices to recover when neural hints are imperfect. When these conditions hold, guidance drives median conflict counts to zero and yields significant wall-clock speedups: on

9\times9

Sudoku, where the SE-RRM correctly solves

91.1\%

of instances, backtracking accelerates by

33.3\times

and Glucose 4.1 by

1.70\times

(median,

p<0.001

), with Glucose 4.1 retaining a

1.17\times

speedup on perfect-hint

25\times25

grids. In contrast, CaDiCaL 3.0.0, whose runtime is overhead-dominated and which always respects the injected branching hints rather than overwriting them, shows no significant speedup (median

1.02\times

, n.s.) and even a small significant mean slowdown (

0.90\times

) on

9\times9

. These results delineate the regimes in which neural guidance translates into practical speedups.

Timo Bertram, Sidhant Bhavnani, Richard Freinschlag +3

ELLIS Unit Linz, LIT AI Lab & Institute for Machine Learning, Johannes Kepler University Linz, Austria · Institute for Symbolic Artificial Intelligence, Johannes Kepler University Linz, Austria · Clinical Research Institute for Medical AI, Johannes Kepler University Linz, Austria

Symbolic Reasoning Neuro-Symbolic Reasoning Constraint Satisfaction Inference-Time Guidance

Jul 1, 2026cs.CL

RusFinChain: A Russian Benchmark for Verifiable Chain-of-Thought Reasoning in Finance with Fuzzy-Aligned Evaluation

Multi-step symbolic reasoning is essential for robust financial analysis, yet most benchmarks neglect intermediate reasoning steps. FINCHAIN introduced verifiable Chain-of-Thought (CoT) evaluation but is limited to English. FINESSE-Bench includes a Russian block but relies on multiple-choice questions without step-level supervision. We present RusFinChain, the first Russian-language symbolic benchmark for verifiable CoT reasoning in finance. It spans 17 domains, 172 topics, and comprises 5,280 parameterized examples from executable Python templates, ensuring contamination-free evaluation. Each example includes a gold-standard reasoning chain with intermediate numeric values for automatic verification. We also introduce enhanced metrics: Fuzzy Numeric Alignment and Soft-Attention Alignment. We evaluate 8 open-weight LLMs on a stratified sample, generating 8,100 responses. Results reveal a substantial reasoning gap: models achieve Hard F1 of ~0.65 for step alignment, but only ~29% of final answers are correct. Our fuzzy and soft metrics show stronger correlation with final-answer correctness (Spearman rho approx 0.48) than the original ChainEval (rho approx 0.38-0.46), demonstrating superior diagnostic power. We release dataset, code, and evaluation framework to foster verifiable financial AI for the Russian-speaking community.

M. K. Arabov

Kazan Federal University · Institute of Computational Mathematics and Information Technologies · Department of Data Analysis +1

Logical Reasoning LLM Evaluation CoT Evaluation Symbolic Reasoning Finance

Jun 24, 2026cs.AI

Data-driven Machine Learning Cannot Reach Symbolic-level Logical Reasoning -- The Limit of the Scaling Law

Sphere neural networks have achieved symbolic level syllogistic reasoning without training data, raising the question of where the limit of the scaling law for logical reasoning lies, i.e., whether data-driven machine learning systems can achieve the same level by increasing training data and training time. We show two methodological limitations that prevent supervised deep learning from reaching the symbolic-level syllogistic reasoning: (1) training data can not distinguish all 24 types of valid syllogistic reasoning; (2) end-to-end mapping from premises to conclusion introduces contradictory training targets between neural components for pattern recognition and logical reasoning. Beside theoretical analysis, we experimentally illustrate that Euler Net cannot achieve rigorous syllogistic reasoning. We further challenge the most recent ChatGPTs (GPT-5-nano and GPT-5) to determine the satisfiability of syllogistic statements in four surface forms (patterns): words, double words, simple symbols, and long random symbols, showing that surface forms affect the reasoning performance and that ChatGPT GPT-5 may reach 100% accuracy but still provide incorrect explanations. As empirical training processes are stopped after achieving 100% accuracy, we conclude that supervised machine learning systems will not attain the rigour of symbolic logical reasoning.

Tiansi Dong, Mateja Jamnik, Pietro Liò

The Alan Turing Institute, 96 Euston Road, London, NW1 2DB, UK. · Department of Computer Science and Technology, University of Cambridge, 15 JJ Thomson Ave., Cambridge, CB3 0FD, UK.

Logical Reasoning Symbolic Reasoning Scaling Laws Neuro-Symbolic Reasoning LLM Reasoning

Jun 22, 2026cs.AI

HOLMES: Evaluating Higher-Order Logical Reasoning in LLMs

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.

Yucheng Wu, Jundong Xu, Mingzhen Ju +4

State Key Laboratory of Multimedia Information Processing, Peking University · School of Computer Science, Peking University · School of Computing, National University of Singapore +3

Logical Reasoning LLM Evaluation Symbolic Reasoning LLM Reasoning

Jun 17, 2026cs.AI

DeFAb: A Verifiable Benchmark for Defeasible Abduction in Foundation Models

A rule-based logic solver resolves every instance in our benchmark in under 50 microseconds with 100% accuracy; the best frontier language model reaches 65% at best and drops to 23.5% under rendering-robust evaluation (worst case over four surface renderings). We introduce DeFAb (Defeasible Abduction Benchmark), a dataset and generation pipeline that converts four decades of publicly funded knowledge bases into formally grounded instances for defeasible abduction: constructing hypotheses that explain anomalies by overriding defaults while preserving unrelated expectations. Because every hypothesis must pass polynomial-time checks for valid derivation, conservativity, and minimality, DeFAb makes logical rigor the instrument for measuring creativity and theoretical reasoning, scoring the disciplined construction of theory revisions rather than fluent but theory-destroying prose. The pipeline pairs taxonomic hierarchies (OpenCyc, YAGO, Wikidata) with behavioral property graphs (ConceptNet, UMLS) to produce 372,648+ instances across 33.75M materialized rules from 18 sources, in three levels with polynomial-time verifiable gold standards. Four frontier models do not reliably internalize defeasible reasoning: rendering-robust Level 2 accuracy is 7.8-23.5%; chain-of-thought variance (~36 pp) exceeds any inter-model gap; and a matched contamination control isolates a +19.4 pp Level 3 gap. We further release DeFAb-Hard (a 235-instance Level 3 difficulty variant; best model 53.3% vs 100% symbolic) and CONJURE (a kernel-verified transformative-creativity variant of 560 Lean 4/Mathlib instances whose gold answers are definitions the proof kernel did not previously contain, judge-free verifier; a pilot finds zero novel concepts). The same verifier doubles as an exact reward for preference optimization (DPO, RLVR/GRPO). Released under MIT at https://huggingface.co/datasets/PatrickAllenCooper/DeFAb.

Patrick Cooper, Alvaro Velasquez

University of Colorado Boulder

LLM Evaluation Abductive Reasoning Symbolic Reasoning Nonmonotonic Reasoning Synthetic Benchmark Generation Structured Reasoning

Jun 16, 2026cs.AI

A homotopy-type-theoretic generalization of neurosymbolic inference

A wide range of neurosymbolic (NeSy) systems compute one functional: a belief-weighted sum of a logical quantity over a space of

σ

-structures, of which weighted model counting, fuzzy logic, and probabilistic logic are special cases. This account is built on sets, and a set deliberately forgets two things that are important for NeSy: when two

σ

-structures are the same up to a symmetry of the theory, and how many distinct proofs witness a query. Types, in the sense of homotopy type theory, preserve this information and turn the functional into a belief-weighted homotopy cardinality, a notion of size that counts each object in inverse proportion to its symmetries. We develop the framework from scratch for NeSy systems, prove a conservativity theorem that recovers the classical functional when symmetries are trivial, and show that the symmetry our framework exposes is exactly the one behind reasoning shortcuts. The payoff is concrete: the shortcut-aware concept posterior that recent methods reach by ensembling or expressive density estimation is the only symmetry-invariant point of the confusion-set simplex, computable in closed form by averaging a single model over the symmetry group. On MNIST reasoning-shortcut benchmarks this single-model wrapper is better calibrated than a diversity-trained ensemble, while leaving label accuracy and identifiable concepts untouched. Code is freely available at https://github.com/bio-ontology-research-group/hott-nesy.

Fernando Zhapa-Camacho, Robert Hoehndorf

Computer, Electrical and Mathematical Sciences & Engineering Division, King Abdullah University of Science and Technology, 4700 KAUST, 23955, Thuwal, Saudi Arabia · KAUST Center of Excellence for Smart Health (KCSH) · KAUST Center of Excellence for Generative AI

Symbolic Reasoning Neuro-Symbolic Reasoning Neuro-Symbolic AI

Jun 15, 2026cs.AI

Exploiting Search in Symbolic Numeric Planning with Patterns

In this paper, we present a procedure for numeric planning based on Symbolic Pattern Planning (SPP). Given a numeric planning problem

Π

, a pattern

\prec

is a sequence of actions used to define a formula encoding the subsequences of

\prec

executable from a starting state

S

. Cardellini, Giunchiglia, and Maratea (2024a) follow the Planning as Satisfiability approach by defining, at each step

n \ge 0

, a formula

Π^\prec_n

in which

(i)

the pattern

\prec

is computed only for

n=0

in the initial state

I

of

Π

, and then exploited at each step

n

,

(ii)

the starting state

S

is set to

I

, and

(iii)

the set

G

of goals is required to hold in the last state that can be reached by one of the subsequences of

\prec

concatenated

n

times. The procedure begins with

n=0

, terminates as soon as

Π^\prec_n

is satisfiable, and otherwise proceeds by incrementing

n

. In this paper, possibly at each step,

(i)

we symbolically search for an intermediate state

P

reachable from

I

, closer to a goal state,

(ii)

dynamically recompute the pattern

\prec_h

-- to be used in the next step -- in

P

,

(iii)

refine the pattern

\prec_g

used to reach

P

, and

(iv)

start the new search from the state

S

which can be either the initial state

I

or the last computed intermediate state

P

, exploiting the computed patterns

\prec_g

and

\prec_h

to define the pattern

\prec

to be used in the search. In particular, at each step, we define a formula

Π^{\prec}_{S,P}

encoding the existence of a state

P'

closer than

P

to a goal state, with

P'

reachable from the starting state

S

when using the pattern

\prec

. We present different techniques for producing such formulas, each corresponding to a different strategy for exploring the search space. We prove their correctness and completeness, the latter under certain conditions.

Matteo Cardellini, Enrico Giunchiglia

DIBRIS, University of Genoa, Italy

Symbolic Planning Symbolic Reasoning

Jun 12, 2026cs.AI

PrologMCP: A Standardized Prolog Tool Interface for LLM Agents

Frontier reasoning-tuned language models still fail on deductive tasks at depth, and the cost of improved performance through extended internal reasoning scales poorly. Symbolic delegation offers a complementary route: a language model translates the problem, while a solver performs the inference. However, current autoformalization pipelines for logic programming are typically bespoke integrations tied to particular tasks or agents. We introduce PrologMCP, a task-agnostic, open-source server that exposes Prolog as a stateful tool through the Model Context Protocol (MCP). Its compact tool interface, structured error reporting, and per-session isolation make the translate-run-inspect-repair loop a reusable primitive for MCP-capable agents. We evaluate a formalizer agent enhanced with PrologMCP against standard and reasoning LLMs (Claude Sonnet 4.6, GPT-4.1, and o4-mini) on two subsets of PARARULE-Plus: a general-purpose sample and a more challenging one targeting a specific failure mode of natural-language reasoning. On the general sample, the formalizer matches or exceeds reasoning LLMs (accuracy 1.00 vs.\ 1.00 / 0.998), with the largest gains over standard models (0.762 for GPT-4.1). On the challenging subset, the formalizer remains near-perfect (1.00 / 0.99) while reasoning LLMs drop to 0.95 / 0.94. These results suggest that delegating inference to Prolog via MCP is a robust and inspectable alternative to extended natural-language reasoning.

Agnieszka Mensfelt, Adarsh Prabhakaran, Adrian Haret +2

Department of Computer Science, Royal Holloway, University of London, UK

Logic Programming Symbolic Reasoning Agentic Reasoning Neuro-Symbolic Reasoning Autoformalization LLM Reasoning