Separation and Definability in Fragments of Two-Variable First-Order Logic with Counting

2025-04-29

AI Summary

This paper investigates the definability and separability problems for formulas with counting quantifiers in first-order logic (FO) and its modal counterparts: determining whether a counting formula is equivalent to one in a counting-free fragment, or whether two disjoint formulas can be separated by a counting-free formula. Focusing on the two-variable fragment FO² and the graded modal logic with inverses, nominals, and universal modalities, we establish that separability is undecidable in both logics—the first such result for these settings. We systematically analyze how individual modal operators affect computational complexity and show that definability reduces in polynomial time to the validity problem of the base logic. Furthermore, we pinpoint the exact complexity of separability across key fragments: undecidability, coNExpTime-completeness, and 2ExpTime-completeness.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• 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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

For fragments L of first-order logic (FO) with counting quantifiers, we consider the definability problem, which asks whether a given L-formula can be equivalently expressed by a formula in some fragment of L without counting, and the more general separation problem asking whether two mutually exclusive L-formulas can be separated in some counting-free fragment of L. We show that separation is undecidable for the two-variable fragment of FO extended with counting quantifiers and for the graded modal logic with inverse, nominals and universal modality. On the other hand, if inverse or nominals are dropped, separation becomes coNExpTime- or 2ExpTime-complete, depending on whether the universal modality is present. In contrast, definability can often be reduced in polynomial time to validity in L. We also consider uniform separation and show that it often behaves similarly to definability.

Problem

• Study definability in counting-free fragments of two-variable logic
• Analyze separation complexity in graded modal logic variants
• Compare uniform separation behavior with definability properties

Innovation

• Undecidable separation in two-variable FO with counting
• CoNExpTime/2ExpTime separation without inverse/nominals
• Polynomial-time reduction of definability to validity