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

Wenyao Cui, Huaping Zhang, Yongyi Huang, Qiuchi Li, Jian Xu, Cheng-Lin Liu, Chunxiao Gao, Juan Wang, Baohua Zhang

arXiv:2608.08786v1cs.AI

TL;DR

LLM chains-of-thought can be unfaithful despite correct final answers, while existing verification signals provide limited diagnostic insight. SymDiag translates reasoning into symbolic constraints, checks steps, disentangles translation artifacts from reasoning errors, and improves faithfulness detection and multi-round repair.

  • Problem

    Existing verification signals often cannot localize failures or provide auditable guidance for repairing invalid intermediate reasoning despite correct outcomes.

  • Method

    SymDiag compiles natural-language reasoning into symbolic constraints, performs step-level satisfiability and entailment checks, and uses self-auditing to separate translation artifacts from reasoning errors.

  • Results

    70.7 overall F1; SymDiag achieves the best faithfulness-detection performance across all datasets and significantly improves multi-round reasoning correction.

  • Takeaways & Limitations

    SymDiag provides an explainable diagnostic system that localizes, attributes, and repairs reasoning failures with verifiable evidence across domains.

Abstract

from arXiv · show

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.

1 Introduction

LLMs demonstrate strong multi-step reasoning and broad downstream adoption, yet their chains-of-thought can be unfaithful because intermediate steps may be inconsistent or rely on problematic reasoning. The introduction positions SymDiag alongside answer matching and RM/LLM-as-Judge approaches.

  • Motivation: LLMs show strong multi-step reasoning and have been adopted across diverse downstream domains.The listed applications span perception, image editing, motion understanding, action counting, speech, and medical analysis.
  • Motivation: Chains-of-thought can be unfaithful because intermediate steps may be inconsistent or rely on unsupported reasoning.The supplied passage explicitly identifies unfaithful intermediate steps as a limitation despite strong reasoning capabilities.
  • Verification approaches: The introduction contrasts LLM-as-Judge, RM, Answer Matching, and SymDiag as reasoning-verification approaches.These approaches are listed together in the supplied comparison labels.

Q LLM Rollout Q LLM Rollout Q LLM Rollout

SymDiag reframes LLM reasoning verification as structured failure diagnosis rather than outcome or scalar scoring. It uses step-level symbolic checks and self-auditing to localize and attribute failures, provide verifiable repair evidence, and evaluate errors across reasoning domains.

  • Diagnosis-centric verification: SymDiag moves beyond answer matching, subjective judging, and scalar rewards by treating verification as diagnosis of localized reasoning failures.These existing signals provide limited insight into where derivations fail or how to repair them.
  • Diagnosis-centric verification: Step-level symbolic checks localize CoT failure points and produce counterexamples, inconsistency witnesses, and missing-premise indicators for targeted repair.The framework returns checkable evidence rather than only a score or corrected trace.
  • Self-audited diagnosis: The Self-Auditor distinguishes TranslationError from ReasoningError using dual symbolic encodings, cross-branch consistency checks, and text-consistent reformulations.This reduces spurious misdiagnosis from underspecified or ambiguous natural-language translations.
  • Diagnosis-and-repair pipeline: SymDiag forms an evidence-grounded diagnosis-and-repair loop that attributes failures and guides targeted multi-round reasoning correction.Its diagnostic outputs include counterexamples, unsat cores, and missing premises.
  • Cross-domain evaluation: The diagnostic benchmark spans mathematics, logical, scientific, and general reasoning, including AIME, AR-LSAT, GPQA, and MMLU.It evaluates diverse reasoning errors within a shared symbolic diagnostic framework.

2 Related Work

Prior verification methods provide outcome labels, subjective critiques, scalar rewards, or formal checks, but often fail to diagnose and localize reasoning errors. SymDiag reframes verification as failure diagnosis with verifiable evidence and self-audited attribution of translation versus reasoning defects.

  • Outcome-only evaluation cannot detect unfaithful-but-correct traces or localize errors in multi-step derivations.
  • LLM-as-Judge methods produce subjective, inconsistent, and difficult-to-audit critiques that may favor fluent but logically flawed reasoning.Attention-based explanations provide only coarse, non-verifiable step-level signals, making reliable repair actions difficult.
  • Process reward models score intermediate steps and can improve best-of-n sampling and faithfulness, but typically return scalar rewards rather than structured violation explanations.
  • Symbolic and formal-verification systems strengthen reasoning with solvers, logic representations, provers, and machine-checkable guarantees, yet often focus on solving or scoring rather than diagnosing failures.Some mathematical systems require domain-specific libraries, while computation tools do not directly address general logical entailment.
  • SymDiag addresses persistent gaps by localizing failing steps, attributing failure types, and returning counterexamples, inconsistency witnesses, missing-premise indicators, and self-audited TranslationError-versus-ReasoningError diagnoses.

3 Methodology · 3.1 Problem Setup and Outputs · 3.2 Stage I: Neuro-Symbolic Compilation and Verification

SymDiag diagnoses LLM reasoning by compiling natural-language chains of thought into symbolic programs, verifying each step, and producing localized, checkable failure evidence. Its dual encodings and Self-Auditor distinguish translation artifacts from genuine reasoning errors before diagnosis-guided repair.

  • 3 Methodology: SymDiag combines symbolic compilation, step-level satisfiability and entailment checks, failure localization, error attribution, and actionable feedback for reasoning repair.The framework is designed as a two-stage pipeline for explainable failure diagnosis and diagnosis-guided repair.
  • 3.1 Problem Setup and Outputs: Given a problem and CoT, SymDiag outputs step verdicts, failing-step indices, taxonomy-based error labels, and symbolic evidence such as counterexamples or missing-premise indicators.It can flag unfaithful-but-correct traces whose final answers are correct despite invalid intermediate entailments.
  • 3.2 Stage I: Neuro-Symbolic Compilation and Verification: Stage I represents reasoning as symbolic states containing accumulated premises, intended inferences, and constraints, compiled into solver-executable Prolog fragments.Constraints include domain, type, boundary, and task-specific conditions.
  • 3.2.2 Neuro-Symbolic Generator: The generator creates two independent encodings: a formal translation and a critical restatement that explicitly scopes claims to expose hidden assumptions.The formal branch also normalizes mathematical and scientific quantities, units, and relations, while modeling quantifiers and scope for logic tasks.
  • 3.2.2 Neuro-Symbolic Generator: Deterministic preprocessing checks signatures, variable groundability, constraint normalization, and solver compatibility, routing syntax failures to TranslationError.The dual encodings reduce single-path translation bias and improve robustness to paraphrase variability.
  • 3.2.4 Self-Auditor: Translation vs. Reasoning Error Disentanglement: The Self-Auditor compares branch-level facts, constraints, and entailment targets, then uses symbolic sanity tests and canonical rewrites to separate TranslationError from ReasoningError.Failures that disappear under text-consistent rewrites are attributed to translation, and only Approved states reach the verifier.
  • 3.2.5 Step-Level Symbolic Verification: Each approved step fails when its premises and constraints are inconsistent or when its claimed inference is not locally entailed; failed entailments can yield counterexample assignments.The full CoT is faithful only when all steps pass in at least one approved branch, with conservative handling when both branches fail.
  • 3.2.6 Symbolic Diagnosis: Failure Localization and Evidence.: For every failing step, SymDiag emits an error label, supporting evidence, and repair scope indicating whether to patch locally or rewrite surrounding reasoning.Evidence may include inconsistency witnesses and other solver-derived artifacts.

3.3 Stage II: Diagnosis-Guided Repair

Stage II turns symbolic diagnoses into structured, evidence-backed feedback and uses them to repair the original reasoning trace. Repair is localized for contained defects but rewrites the trace from the earliest failure when errors propagate.

  • Stage II: Diagnosis-Guided Repair: For each failing step, SymDiag verbalizes the error label and attaches checkable evidence such as counterexamples, inconsistency witnesses, or missing-premise indicators.This structured feedback converts diagnosis into actionable repair guidance.
  • Stage II: Diagnosis-Guided Repair: Localized defects prompt minimal edits to the offending step while preserving the surrounding reasoning context.Examples include arithmetic or algebra errors and invalid equivalences confined to one step.
  • Stage II: Diagnosis-Guided Repair: When a failure propagates through later steps, the model rewrites the reasoning from the earliest failing step onward.Propagation can result from an early missing premise or a type mismatch affecting subsequent steps.

3.4 Dataset Construction

SymDiag constructs a unified CoT faithfulness dataset labeled as Faithful or Unfaithful based on verifiable reasoning defects. A two-stage pipeline builds the corpus, while evaluation uses a manually verified 240-instance gold set.

  • Labeling: Instances receive Faithful or Unfaithful labels based on final-answer correctness and verifiable CoT defects such as missing premises or invalid equivalences.Unfaithful traces may also contain ignored boundary conditions or defects leading to incorrect final answers.
  • Corpus construction: The two-stage pipeline samples CoTs and answers from base LLMs, labels incorrect answers Unfaithful, and uses conservative multi-judge voting to retain high-precision Faithful traces.The voting stage applies specifically to instances with correct answers.
  • Evaluation set: 240 manually verified instances form the evaluation gold set, sampled from a full automatically constructed corpus of 437,792 instances across four reasoning domains.The core experimental dataset manually audits 30 examples from each dataset.

4 Experiments

SymDiag is evaluated on a manually audited, four-domain dataset for faithfulness diagnosis and diagnosis-guided iterative reasoning repair. Experiments compare its step-level, evidence-grounded feedback with answer matching, LLM judging, and scalar logical rewards across open-weight models of different capacities.

  • Experimental Setup: Core experiments use 240 manually audited instances spanning mathematics, logical, science, and general reasoning, each labeled Faithful or Unfaithful.The dataset is a curated subset with reasoning-trace annotations.
  • Evaluation: SymDiag is evaluated on Faithfulness Detection and Feedback and Repair, measuring trace classification and the effectiveness of iterative reasoning correction.Faithfulness Detection reports F1 scores, while Repair Effectiveness evaluates standard benchmark metrics after each repair round and learning curves.
  • Baselines: The comparison includes Answer Matching, LLM-as-Judge, and LogicReward, representing final-answer, natural-language judging, and scalar theorem-prover-based supervision signals.Answer Matching supplies 1-bit correctness feedback; LLM-as-Judge evaluates answers and CoT; LogicReward uses scalar signals and Isabelle outputs.
  • Model Coverage: Experiments assess robustness across Llama-3.2-1B, Qwen3-1.7B, Qwen3-8B, and GPTOSS-20B base reasoners.The models’ CoTs are verified, diagnosed, and repaired by SymDiag rather than treating the models themselves as verifiers.
  • SymDiag Procedure: SymDiag translates each CoT into symbolic step records, verifies them step-by-step, distinguishes TranslationError from ReasoningError, and returns localized evidence-grounded feedback for patching or rewriting.The two-branch diagnosis addresses failures arising from either translation noise or reasoning defects.

5 Results

On a manually audited diagnostic dataset, SymDiag outperforms outcome-only, scalar-reward, and LLM-judging baselines in detecting unfaithful reasoning and guiding iterative repair. Its localized, verifiable evidence enables faster, more sustained correction gains across datasets.

  • Faithfulness detection: SymDiag achieves the highest overall faithfulness-detection F1 score, 70.7, outperforming answer matching, LLM-as-Judge, and reward-based baselines.Results are evaluated on the manually audited diagnostic subset, with the overall score computed across all instances.
  • Iterative reasoning repair: SymDiag produces faster and larger accuracy gains across repair rounds than all baselines, with the strongest final performance across datasets.Figure 4 attributes these gains to localized, verifiable error evidence that supports targeted correction.
  • Iterative reasoning repair: Answer Matching provides minimal repair improvement because it lacks localized guidance, while Reward Model and LogicReward offer noisy, non-localizable scalar signals.The limitations are especially pronounced in open-domain settings for the scalar-reward methods.
  • Iterative reasoning repair: LLM-as-Judge delivers moderate early repair gains but quickly saturates because its feedback is not sufficiently grounded in localized premises and constraint violations.SymDiag instead uses such evidence to enable sustained improvements.

6 Analysis

The analysis attributes SymDiag’s gains to the interaction of translation, symbolic verification, and diagnosis, while showing that its error patterns and supervision needs vary with model scale. Iterative Self-Auditor feedback reduces translation and execution failures, increasing the pipeline’s pass rate, though remaining errors are primarily logic-level.

  • Ablation analysis: Removing any SymDiag component consistently degrades faithfulness-detection F1, indicating that its gains arise from interacting translation, verification, and diagnosis modules.The ablation study identifies no single heuristic as sufficient for the system’s overall performance.
  • Ablation analysis: Step-level symbolic verification is the primary driver of accurate failure detection, while the Self-Auditor prevents translation artifacts from being misclassified as reasoning errors.Removing Formal Translation or Critical Restatement also reduces performance by weakening exposure of hidden assumptions.
  • Error-type analysis: Reasoning failures shift with model capacity: smaller models often make arithmetic errors and omit premises, whereas larger models show more high-level structural failures.The findings motivate scale-aware supervision: constraint enforcement and premise completion for smaller models, and safeguards against rule hallucination and overgeneralization for larger models.
  • Iterative auditing: The Passed rate rises from 53.4% to 81.4% through iterative Self-Auditor feedback, as Translation Errors fall from 20.1% to near zero and Execution Failures from 5.5% to below 1%.Before auditing, the total error rate is 46.6%; gains diminish across rounds, and remaining errors are dominated by irreducible logic-level critique failures.
  • Model-scale analysis: Scaling the generator and auditor from GPTOSS-120B to GPTOSS-20B reduces overall F1 across all methods, showing that stronger base models benefit translation quality and diagnostic precision.The comparison evaluates Answer Matching, Reward Model, and SymDiag under both model scales.

7 Conclusion

SymDiag reframes LLM reasoning verification as explainable failure diagnosis by translating chains-of-thought into symbolic constraints and performing step-level verification. The paper identifies future directions in stronger verification, language robustness, and diagnosis-aware supervision.

  • Core contribution: SymDiag reframes LLM reasoning verification as an explainable failure diagnosis problem.The framework is presented as neuro-symbolic.
  • Core contribution: Translating chains-of-thought into symbolic constraints and performing step-level verification localizes faulty inferences and produces verifiable diagnostic evidence.
  • Future directions: Future work includes stronger or hybrid symbolic verifiers, improved robustness to underspecified language, and diagnosis-aware reward models and reasoning supervisors.The passage also mentions efficient and meta-learning paradigms for training such models and supervisors.

A Error Taxonomy

SymDiag uses a domain-agnostic taxonomy to characterize common LLM reasoning failures across mathematical, logical, scientific, and general reasoning settings. The taxonomy distinguishes missing premises, invalid transformations, neglected constraints, misapplied rules, arithmetic errors, and type or entity mismatches.

  • A Error Taxonomy: The taxonomy is designed to cover common LLM failure modes across math, logic, science, and general reasoning.It is explicitly domain-agnostic.
  • A Error Taxonomy: Failure categories include premise omission or missing assumptions, invalid inference, and constraint or boundary neglect.Examples include underivable claims, non-entailing transformations, and ignored domain, case, sign, or unit constraints.
  • A Error Taxonomy: The taxonomy also identifies rule misuse or hallucinated rules, arithmetic or algebra errors, and type or entity mismatches.These cover inapplicable theorems or principles, verified computational mismatches, and mismatched types or entities.
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