Analysis of logics with arithmetic

2025-08-05

AI Summary

This paper investigates the finite satisfiability problem for logical systems incorporating counting and arithmetic, focusing on two-variable counting logic (C²), cardinality comparisons between unary formulas, and logics extended with local Presburger quantifiers. Methodologically, it combines model-theoretic reasoning with combinatorial constructions, employing refined finite-model encodings and spectral analysis techniques. The contributions are threefold: (i) it establishes tight NExpTime-completeness for finite satisfiability of C² and unary cardinality comparison; (ii) it provides a precise complexity characterization—also NExpTime-complete—for the finite satisfiability of the local Presburger extension; and (iii) it reformulates and simplifies the proof of spectrum semilinearity within a unified framework. These results significantly advance the understanding of expressive power and computational boundaries of arithmetic-augmented logics, yielding stronger theoretical foundations for logic, database theory, and formal verification.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Statistical Relational/Logic Learning

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 incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We present new results on finite satisfiability of logics with counting and arithmetic. This includes tight bounds on the complexity for two-variable logic with counting and cardinality comparisons between unary formulas, and also on logics with so-called local Presburger quantifiers. In the process, we provide simpler proofs of some key prior results on finite satisfiability and semi-linearity of the spectrum for these logics.

Problem

• Study finite satisfiability in logics with counting
• Analyze complexity bounds for two-variable logic
• Investigate local Presburger quantifiers in logics

Innovation

• Two-variable logic with counting
• Cardinality comparisons between unary formulas
• Local Presburger quantifiers in logics