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

[Submitted on 27 Apr 2026]

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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.

Submission history

From: Hai Hai Fu [view email]
[v1] Mon, 27 Apr 2026 10:10:55 UTC (33 KB)