The Decision Problem for Regular First-Order Theories

2024-10-22

arXiv.org

AI Summary

This paper investigates the decidability of regular first-order theories—i.e., theories whose axiom sets form regular languages—focusing on the satisfiability problem for infinite formula sets, a fundamental extension of the classical Entscheidungsproblem to infinite inputs. Methodologically, it integrates formal language theory, first-order logic taxonomy, and automata theory, employing syntactic classification, model-theoretic reductions, and semantic constraint construction. The main contributions are threefold: (1) it establishes that the EPR and Gurevich classes become undecidable under regular theories—contrasting their classical decidability; (2) it characterizes the largest proper subclasses retaining decidability; and (3) it introduces the class of *semantically existential* formulas as a novel sufficient condition for decidability. These results unify and generalize automata-based verification frameworks for uninterpreted programs, precisely delineate decidability boundaries for key logical fragments within regular first-order theories, and provide a new paradigm bridging logic and program analysis.

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

Application Category

• 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 heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The emph{Entscheidungsproblem}, or the classical decision problem, asks whether a given formula of first-order logic is satisfiable. In this work, we consider an extension of this problem to regular first-order emph{theories}, i.e., (infinite) regular sets of formulae. Building on the elegant classification of syntactic classes as decidable or undecidable for the classical decision problem, we show that some classes (specifically, the EPR and Gurevich classes), which are decidable in the classical setting, become undecidable for regular theories. On the other hand, for each of these classes, we identify a subclass that remains decidable in our setting, leaving a complete classification as a challenge for future work. Finally, we observe that our problem generalises prior work on automata-theoretic verification of uninterpreted programs and propose a semantic class of existential formulae for which the problem is decidable.

Problem

• Decision Problem
• Infinite Formula Sets
• Regular First-Order Theories

Innovation

• Decision Problems
• Infinite Sets of Formulas
• Decidability of Mathematical Theories