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Delta1 with LLM: symbolic and neural integration for credible and explainable reasoning

Yang Xu, Jun Liu, Shuwei Chen, Chris Nugent, Hailing Guo

Abstract

Neuro-symbolic reasoning increasingly demands frameworks that unite the formal rigor of logic with the interpretability of large language models (LLMs). We introduce an end to end explainability by construction pipeline integrating the Automated Theorem Generator Delta1 based on the full triangular standard contradiction (FTSC) with LLMs. Delta1 deterministically constructs minimal unsatisfiable clause sets and complete theorems in polynomial time, ensuring both soundness and minimality by construction. The LLM layer verbalizes each theorem and proof trace into coherent natural language explanations and actionable insights. Empirical studies across health care, compliance, and regulatory domains show that Delta1 and LLM enables interpretable, auditable, and domain aligned reasoning. This work advances the convergence of logic, language, and learning, positioning constructive theorem generation as a principled foundation for neuro-symbolic explainable AI.

Delta1 with LLM: symbolic and neural integration for credible and explainable reasoning

Abstract

Neuro-symbolic reasoning increasingly demands frameworks that unite the formal rigor of logic with the interpretability of large language models (LLMs). We introduce an end to end explainability by construction pipeline integrating the Automated Theorem Generator Delta1 based on the full triangular standard contradiction (FTSC) with LLMs. Delta1 deterministically constructs minimal unsatisfiable clause sets and complete theorems in polynomial time, ensuring both soundness and minimality by construction. The LLM layer verbalizes each theorem and proof trace into coherent natural language explanations and actionable insights. Empirical studies across health care, compliance, and regulatory domains show that Delta1 and LLM enables interpretable, auditable, and domain aligned reasoning. This work advances the convergence of logic, language, and learning, positioning constructive theorem generation as a principled foundation for neuro-symbolic explainable AI.
Paper Structure (52 sections, 12 equations, 1 figure, 3 tables)

This paper contains 52 sections, 12 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Contributions:
  3. Related Works
  4. Logic, Learning, and Neuro-Symbolic Reasoning.
  5. Symbolic Theorem Proving, Automated Discovery, and Explainable Integration.
  6. Minimal Unsatisfiable Subset (MUS) Extraction.
  7. Position within the Field.
  8. Formal–Neural Integration: $\Delta_{1}$ + LLM Architecture
  9. Architecture Overview
  10. End-to-End $\Delta_{1}$ + LLM Pipeline Stages
  11. Design Rationale
  12. Outcome: Explainability by Construction
  13. Theoretical Framework of $\Delta_{1}$
  14. Foundational Setting
  15. Constructive Logic of $\Delta_1$
  16. ...and 37 more sections

Figures (1)

  • Figure 1: Architecture of the $\Delta_{1}$ + LLM pipeline. Natural‐language policy text is first processed by the front-end LLM to extract atomic predicates. The $\Delta_{1}$ generator deterministically constructs all minimal unsatisfiable clause sets and corresponding theorems ($S\setminus\{C\}\vdash\neg C$), producing explicit proof traces. A back-end LLM module then translates each theorem into natural‐language explanations and remediation suggestions, enabling explainability by construction: formal correctness from $\Delta_{1}$ and interpretive clarity from the LLM.