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

A new arXiv preprint (2608.07476v1) introduces a formal framework for constructing canonical interpretations from plural structure theories, classifying non-determinism into epistemic plurality (Type E) and structural plurality (Type S) and providing conditions for determinization. The authors show that pure inference-based completion reduces to a saturated closure operator under positive, non-retractive rules, and that Type S-strong theories achieve determinization via canonical selection. The framework also applies to LLM-assisted reasoning, where hallucination is viewed as unsupported canonicalization.

arXiv:2608.07476v1 Announce Type: new 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 . 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.