Aspects of Coherence in Dependence Logic

2026-05-29

AI Summary

This study investigates the decidability and computational complexity of formula satisfiability in dependence logic, with a focus on how k-coherence conditions can mitigate the difficulty of model checking. By leveraging team semantics, second-order logic, and complexity theory, the paper establishes that first-order rewritability of quantifier-free dependence logic (DQF) is equivalent to k-coherence. It provides the first precise complexity characterization: coherence checking for DQF lies in the co-recursively enumerable (co-r.e.) class, whereas coherence for full dependence logic is highly undecidable. In the propositional setting, the problem is shown to be EXP^NP-complete. These results highlight the pivotal role of coherence as a balancing factor between expressive power and computational cost in logical systems.

Abstract

Dependence logic extends first-order logic with dependence atoms asserting that the value of a variable is determined by the values of certain other variables. The semantics of dependence logic has a second-order character and involves sets of assignments, called teams, instead of individual assignments as in the classical Tarski semantics. Since the model-checking problem is known to be NP-complete even for quantifier-free dependence logic (DQF) formulas, researchers have pursued conditions on formulas that make this problem tractable. In 2010, Jarmo Kontinen introduced the notion of k-coherence for dependence logic formulas, where k is a positive integer. This notion asserts that if the formula is satisfied in a structure by all k-element subteams of a given team, then the given team itself satisfies the formula. It has been proved that k-coherent DQF-formulas have a tame model-checking problem, because such formulas admit a first-order rewriting. In this paper, we investigate the structural and algorithmic aspects of coherence. We show that if a DQF-formula is first-order ewritable, then it is k-coherent for some positive integer k. Thus, for DQF-formulas, coherence is equivalent to first-order rewritability. Furthermore, we show that an analogous result holds for universally quantified dependence logic formulas under a stronger notion of coherence. After this, we focus on the complexity of deciding if a given dependence logic formula is k-coherent. We establish that this decision problem is highly undecidable for arbitrary dependence logic formulas, while for DQF-formulas this problem is co-recursively enumerable. Furthermore, we pinpoint the computational complexity of the coherence problem for propositional dependence logic formulas by showing that this problem is complete for the second level of the exponential hierarchy.

Problem

• Dependence Logic
• Coherence
• Model-checking
• First-order Rewritability
• Computational Complexity

Innovation

• k-coherence
• first-order rewritability
• dependence logic
• model-checking complexity
• undecidability