arXiv:2608.07476cs.AIcs.LO2026-08

提出统一框架解决结构理论中的非确定性问题,实现从多解释到唯一标准解释的转化。

Determinization in Structure Theories: A Unified Framework via Closure, Comparability, and Joint Admissibility

  • 通过闭包、可比性和联合可接受性构建标准解释的三重机制。
  • 区分认知多样与结构多样两类非确定性,证明强结构类可唯一确定解释。
  • 适用于大模型推理中的幻觉分析,揭示错误生成的本质是无效标准化。

我们构建了一个形式化框架,用于从多重结构理论中生成标准解释。结构理论T = (Σ, A, I)由签名、公理和推理策略构成,其可接受解释族包含所有全局一致的结构结论赋值。我们区分了三个层次的标准化:闭包稳定(基于种子的收敛)、全局完成(与种子无关的收敛)和确定化(唯一可接受解释)。非确定性分为认知多样性(类型E)和结构多样性(类型S),其中类型S-强类在无共同上界时具有特殊性质。两种标准化机制浮现:基于算子的完成与基于选择器的构造。我们给出了这些机制存在的充分结构条件,并证明仅依赖推理的完成在正向且不可回溯规则下退化为饱和闭包算子。对类型E理论,闭包稳定成立;而完全确定化依赖于全局收敛性,尚待证明。对类型S-强理论,可通过规范选择实现确定化。我们进一步发现多层次标准化构成非交换结构系统,通过分阶段算子实现;并提供一个条件分类定理,将理论内机制归约为闭包或选择。该框架还可应用于大模型辅助推理,其中幻觉被视为未受支持的标准解释。

原文摘要 · Abstract (English)

We develop a formal framework for constructing canonical interpretations from plural structure theories. A structure theory is a triple T = (Σ, A, I) consisting of a signature, axioms, and an inference policy, whose admissible interpretation family collects all globally consistent assignments of structural conclusions. We distinguish three levels of canonicalization: closure stabilization (per-seed convergence), global completion (seed-independent convergence), and determinization (a unique admissible interpretation). Non-determinism is classified into epistemic plurality (Type E) and structural plurality (Type S), with a refined Type S-strong subclass characterized by the absence of common upper bounds. Two canonicalization mechanisms arise: operator-based completion and selector-based construction. We provide sufficient structural conditions under which these mechanisms exist, and show that pure inference-based completion reduces to a saturated closure operator under positive, non-retractive rules with an additional soundness condition. For Type E theories, closure stabilization is established, while full determinization depends on a global confluence property that remains open. For Type S-strong theories, determinization is achieved via canonical selection. We further show that multi-level canonicalization forms a structurally non-commutative system via staged operators, and provide a conditional classification theorem reducing theory-intrinsic mechanisms to closure or selection. The framework also applies to LLM-assisted reasoning, where hallucination can be viewed as unsupported canonicalization.

逻辑推理形式化框架大模型幻觉结构理论

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