Decidability Results for Fragments of First-Order Logic via a Symbolic Model Property

2026-05-07

AI Summary

This study addresses the decidability of certain fragments of first-order logic obtained by relaxing restrictions on quantifier alternation. The authors introduce and generalize the notion of the symbolic model property—stipulating that every satisfiable formula within the fragment admits a symbolic model—to arbitrary base theories with standard models, thereby proposing a general construction methodology. By integrating symbolic structures, linear integer arithmetic, and model-checking techniques, they establish that an extended stratified fragment, which includes self-loop functions under specific syntactic constraints, enjoys the symbolic model property. Consequently, this fragment is shown to be decidable, effectively broadening the boundary of known decidable fragments in first-order logic.

Abstract

Recently, symbolic structures were proposed as finite representations of potentially infinite first-order structures, where Linear Integer Arithmetic terms and formulas define the domain and interpretations of a structure. We generalize symbolic structures to use any base theory that admits a standard model. Symbolic structures induce a symbolic model property, which holds for a fragment of first-order logic if every satisfiable formula in the fragment has a symbolic model. The symbolic model property implies decidability, since the model-checking problem for symbolic structures is decidable. We use the symbolic model property to prove decidability for several fragments that extend the fragment of stratified formulas, relaxing the quantifier-alternation constraints by allowing one sort to have self-looping functions, under certain restrictions. To establish the symbolic model property for these fragments we construct a symbolic model for a formula from an arbitrary model. The construction and its correctness are proved in a generic fashion, which may be instantiated to other similarly restricted fragments.

Problem

• decidability
• first-order logic
• symbolic model property
• quantifier alternation
• fragments

Innovation

• symbolic structures
• symbolic model property
• decidability
• first-order logic fragments
• quantifier alternation