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

结构理论中的确定化：通过闭包、可比性和联合可采性实现的统一框架

Abstract: We develop a formal framework for constructing canonical interpretations from plural structure theories. A structure theory is a triple T = ({\Sigma}, 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).

摘要： 我们开发了一个用于从多元结构理论构建规范解释的形式化框架。结构理论是一个三元组 T = ({\Sigma}, A, I)，由签名、公理和推理策略组成，其可采解释族收集了所有结构结论的全局一致赋值。我们区分了三个层级的规范化：闭包稳定（基于种子的收敛）、全局完备（与种子无关的收敛）和确定化（唯一的确定性可采解释）。

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.

非确定性被分为认知多元性（E型）和结构多元性（S型），其中细分的“强S型”子类以缺乏公共上界为特征。由此产生了两种规范化机制：基于算子的完备和基于选择器的构建。我们提供了这些机制存在的充分结构条件，并证明在正向、非回溯规则及附加可靠性条件下，纯粹的基于推理的完备可简化为饱和闭包算子。

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.

对于E型理论，我们确立了闭包稳定性，而完全确定化则取决于一个尚待解决的全局汇合性属性。对于强S型理论，确定化通过规范选择实现。我们进一步证明，多级规范化通过分阶段算子形成了一个结构上非交换的系统，并提供了一个条件分类定理，将理论内在机制归约为闭包或选择。该框架也适用于大模型辅助推理，其中“幻觉”可被视为一种缺乏支持的规范化。