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

1. Problem Statement

Large Language Models (LLMs) have demonstrated strong multi-step reasoning capabilities, yet their Chains-of-Thought (CoT) often suffer from a critical gap between outcome correctness and process validity. A model may arrive at the correct final answer through unfaithful intermediate steps, including inconsistent logic, hidden assumptions, or non-entailing transformations.

Existing verification methods fail to provide diagnostic insights necessary for targeted repair:

• Outcome-only verification (Answer Matching): Can only detect if the final answer is correct or incorrect, failing to localize errors in long derivations or identify unfaithful-but-correct traces.
• LLM-as-Judge: Provides natural language critiques that are subjective, inconsistent, and difficult to audit or convert into specific repair actions.
• Scalar Rewards (PRMs/RMs): Reduce rich failure modes to a single score, offering little guidance on where or why a derivation failed.
• Neuro-Symbolic Approaches: While methods like Logic-LM or SymbCoT introduce symbolic structures, they often struggle to distinguish between genuine reasoning defects and neural-to-symbolic translation artifacts (e.g., parsing noise). Without this distinction, verifiers may "misdiagnose" translation errors as logical failures, leading to unreliable feedback.

The core challenge is to move beyond scoring or subjective judging toward a structured failure diagnosis that localizes errors, attributes their causes, and produces verifiable evidence for repair.

2. Methodology: SymDiag Framework

SymDiag is a two-stage neuro-symbolic framework designed to reframe reasoning verification as an explainable failure diagnosis problem.

Stage I: Explainable Failure Diagnosis

This stage transforms natural language CoT into verifiable symbolic constraints and performs step-level validation.

1. Neuro-Symbolic Generator (Dual-Branch Encoding):
   To mitigate translation brittleness, the system generates two independent symbolic encodings for each CoT step:

   • Formal Translation Branch: Compiles the input problem and CoT into a formal representation (using Prolog as the backend), normalizing entities, predicates, and relations.
   • Critical Restatement Branch: Re-expresses the CoT step into a stricter, explicitly scoped statement to expose hidden assumptions before compiling it into a second symbolic program.
   • Goal: Dual encodings reduce single-path translation bias and increase robustness to paraphrase variability.

2. Self-Auditor (Error Disentanglement):
   A critical component that distinguishes between Translation Errors (artifacts of the neural-to-symbolic conversion) and Reasoning Errors (genuine logical defects).

   • Translation Consistency Check: Compares facts and constraints between the two branches. Large divergences suggest translation ambiguity.
   • Logical Critique Check: Runs lightweight sanity tests (e.g., immediate contradictions). If a failure disappears under minimal, text-consistent rewrites (e.g., variable renaming), it is attributed to translation rather than reasoning.
   • Output: A binary attribution ($TranslationError$ vs. $ReasoningError$) that filters out spurious misdiagnoses.

3. Step-Level Symbolic Verification:
   Approved symbolic states undergo rigorous checks:

   • Satisfiability: Checks if the accumulated premises and constraints are satisfiable ($SAT(P_i \land C_i)$).
   • Local Entailment: Verifies if the current step's claim logically follows from prior premises ($P_{i-1} \land C_{i-1} \models \phi_i$). This is operationalized by checking the unsatisfiability of the negation.
   • Evidence Generation: Upon failure, the system generates verifiable evidence such as counterexamples, inconsistency witnesses (unsat cores), and missing-premise indicators.

Stage II: Diagnosis-Guided Repair

The diagnostic output is converted into actionable feedback for the LLM:

• Localization: Identifies specific failing indices in the CoT.
• Attribution: Assigns error labels (e.g., arithmetic error, invalid inference, missing premise).
• Repair Strategy:
  • Local Patch: If the defect is isolated (e.g., a calculation error), the model is prompted to minimally edit the specific step.
  • Global Rewrite: If the failure propagates (e.g., a missing premise affecting later steps), the model is prompted to rewrite the reasoning from the earliest failing point.

3. Key Contributions

1. Diagnosis-Centric View: Formalizes reasoning verification as a failure diagnosis problem, aiming to localize errors, attribute causes, and support targeted repair, rather than merely scoring outcomes.
2. Self-Audited Disentanglement: Introduces a mechanism to explicitly separate translation artifacts from genuine reasoning failures using dual symbolic encodings and cross-branch consistency checks, enabling reliable diagnosis under partial observability.
3. Evidence-Grounded Pipeline: Proposes a closed-loop system that produces verifiable diagnostic evidence (counterexamples, unsat cores) to guide iterative repair, significantly improving multi-round reasoning correction.
4. Cross-Domain Diagnostic Benchmark: Constructs a unified dataset spanning mathematics (AIME), logical reasoning (AR-LSAT), scientific reasoning (GPQA), and general reasoning (MMLU), demonstrating that diverse reasoning errors can be analyzed within a shared symbolic framework.

4. Experimental Results

Experiments were conducted on a manually audited gold set of 240 instances across four domains.

• Faithfulness Detection: SymDiag achieved the highest overall F1 score (70.7) in detecting unfaithful reasoning, significantly outperforming Answer Matching, LLM-as-Judge, Reward Models, and LogicReward. The performance gap was particularly pronounced in logical and general reasoning benchmarks where correct answers often mask invalid intermediate steps.
• Reasoning Repair: In multi-round repair tasks, SymDiag demonstrated faster and more sustained accuracy gains compared to baselines. While Answer Matching provided minimal improvement due to a lack of localized guidance, and LLM-as-Judge feedback saturated quickly due to its coarseness, SymDiag's step-level, evidence-grounded diagnoses enabled targeted corrections.
• Ablation Studies: Removing the Self-Auditor or step-level symbolic verification caused significant performance drops, confirming that the system's success relies on the interaction between translation robustness and explicit logical checking.
• Error Attribution Analysis: SymDiag revealed distinct failure modes across model scales. Smaller models (e.g., 1B parameters) frequently committed arithmetic errors and omitted premises, while larger models (e.g., 20B) showed fewer low-level errors but higher incidences of hallucinated rules and type mismatches.

5. Significance and Claims

The paper positions SymDiag not merely as a "better verifier" or a "better reward signal," but as an explainable neuro-symbolic diagnostic system. Its primary significance lies in:

• Trustworthiness: By providing verifiable evidence (counterexamples, inconsistency witnesses) rather than subjective scores, it offers a principled foundation for diagnosing LLM reasoning.
• Scalability: The framework is designed to handle diverse reasoning domains (math, logic, science, general) within a unified symbolic diagnostic framework.
• Actionability: Unlike previous methods that return a score or a corrected trace, SymDiag localizes failures and attributes them to specific causes (translation vs. reasoning), enabling targeted repairs that are more effective for iterative improvement.

The authors conclude that while stronger base models improve translation quality, the structured neuro-symbolic pipeline of SymDiag remains robust across model scales, offering a scalable path toward trustworthy reasoning diagnosis.