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

Computer Science > Artificial Intelligence [Submitted on 27 Apr 2026] Determinization in Structure Theories: A Unified Framework via Closure, Comparability, and Joint Admissibility Hai Hai Fu 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. Comments: Formal framework paper on canonicalization and determinization in structure theories; version v2.16.4; 29 pages Subjects: Artificial Intelligence (cs.AI); Logic in Computer Science (cs.LO) Cite as: arXiv:2608.07476 [cs.AI]   (or arXiv:2608.07476v1 [cs.AI] for this version)   https://doi.org/10.48550/arXiv.2608.07476 Focus to learn more Submission history From: Hai Hai Fu [view email] [v1] Mon, 27 Apr 2026 10:10:55 UTC (33 KB) Access Paper: HTML (experimental) view license Current browse context: cs.AI < prev   |   next > new | recent | 2026-08 Change to browse by: cs cs.LO References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)