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arXiv · 2508.03574

Analysis of logics with arithmetic

Michael Benedikt·Chia-Hsuan Lu·Tony Tan
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Abstract

We present new results on finite satisfiability of logics with counting and arithmetic. One result is a tight bound on the complexity of satisfiability of logics with so-called local Presburger quantifiers, which sum over neighbors of a node in a graph. A second contribution concerns computing a semilinear representation of the cardinalities associated with a formula in two variable logic extended with counting quantifiers. Such a representation allows you to get bounds not only on satisfiability for these logics, but for satisfiability in the presence of additional ``global cardinality constraints'': restrictions on cardinalities of unary formulas, expressed using arbitrary decidability logics over arithmetic. In the process, we provide simpler proofs of some key prior results on finite satisfiability and semi-linearity of the spectrum for these logics.

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Michael Benedikt, Chia-Hsuan Lu, Tony Tan. 2025-08-05. Analysis of logics with arithmetic. https://arxiv.org/abs/2508.03574

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