用代数不变量让大模型推理更可靠，防止错误传播。
Structured Abductive-Deductive-Inductive Reasoning for LLMs via Algebraic Invariants
- 将溯因、演绎、归纳三类推理转化为可执行协议
- 通过五条代数不变式约束推理链，最强约束确保结论不高于最弱前提
- 适合追求严谨逻辑推理的AI研究者与系统开发者
大语言模型在结构化逻辑推理中存在系统性缺陷：混淆假设生成与验证，无法区分猜想与已证实知识，且允许薄弱推理步骤在多步推演中无限传播。本文提出一种符号推理框架，将皮尔士的三重推理（溯因、演绎、归纳）转化为大模型辅助推理的显式规程。该框架通过五个代数不变量（伽马五元组）强制逻辑一致性，其中最强约束——最弱环节界限——确保推理链中任意结论的可靠性不超过其最薄弱前提。这一原则在可能性逻辑中独立成立，并经实证验证适用于思维链推理，有效防止逻辑矛盾在多步推理中累积。我们通过包含100项性质和16种模糊测试的属性测试套件，在超过10^5个生成案例上验证了所有不变量，提供了一个经过验证的参考实现，可作为未来推理基准的基础。
原文摘要 · Abstract (English)
Large language models exhibit systematic limitations in structured logical reasoning: they conflate hypothesis generation with verification, cannot distinguish conjecture from validated knowledge, and allow weak reasoning steps to propagate unchecked through inference chains. We present a symbolic reasoning scaffold that operationalizes Peirce's tripartite inference -- abduction, deduction, and induction -- as an explicit protocol for LLM-assisted reasoning. The framework enforces logical consistency through five algebraic invariants (the Gamma Quintet), the strongest of which -- the Weakest Link bound -- ensures that no conclusion in a reasoning chain can exceed the reliability of its least-supported premise. This principle, independently grounded as weakest link resolution in possibilistic logic and empirically validated for chain-of-thought reasoning, prevents logical inconsistencies from accumulating across multi-step inference. We verify all invariants through a property-based testing suite of 100 properties and 16 fuzz tests over 10^5+ generated cases, providing a verified reference implementation of the invariants suitable as a foundation for future reasoning benchmarks.
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