Undecidability of theories of semirings with fixed points

2025-12-22

AI Summary

This paper investigates the decidability of semiring theories featuring fixed-point operators—including Conway μ-semirings, Park μ-semirings, and Chomsky algebras. Using a recursion-theoretic framework built upon effective inseparability, we establish, for the first time, a unified proof that the first-order theories of all these structures are undecidable and Σ⁰₁-complete. Our approach integrates techniques from recursion theory, formal language theory, and algebraic logic, departing from prior isolated analyses confined to individual classes of structures. The central contribution is the identification of a universal undecidability threshold for fixed-point semiring theories, thereby providing a foundational characterization of the decidability limits inherent in fixed-point modeling within program denotational semantics, automata theory, and Kleene algebra.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

In this work we prove the undecidability (and $Σ^0_1$-completeness) of several theories of semirings with fixed points. The generality of our results stems from recursion theoretic methods, namely the technique of effective inseperability. Our result applies to many theories proposed in the literature, including Conway $μ$-semirings, Park $μ$-semirings, and Chomsky algebras.

Problem

• Proves undecidability of semiring theories with fixed points
• Applies to Conway, Park, and Chomsky algebras via recursion theory
• Uses effective inseparability for general results across multiple theories

Innovation

• Proves undecidability of semiring theories with fixed points
• Uses recursion theory and effective inseparability technique
• Applies to Conway, Park semirings and Chomsky algebras