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arXiv · 2605.05840

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

Neta Elad·Sharon Shoham
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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.

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Neta Elad, Sharon Shoham. 2026-05-07. Decidability Results for Fragments of First-Order Logic via a Symbolic Model Property. https://arxiv.org/abs/2605.05840

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