Finite-model, decidability, and complexity results for broader closure logics

Determine whether finite-model property and decidability results can be extended from closure logics without modal inference rules to broader classes of closure logics, and determine the complexity of validity and satisfiability for logics characterized by closure models, accounting for the computational cost of the closure operation when finite generators produce infinite closures.

Background

The paper establishes finite-model constructions and primitive-recursive decidability only for logics whose proof systems contain no modal inference rules. Its closure semantics permits a finite generator G(w) to determine an infinite set cl(G(w)), so the computational behavior of the closure operation may affect both decision procedures and complexity bounds. The paper therefore identifies unresolved questions about extending these metatheoretic results to broader classes of modal logics and about analyzing the complexity of validity and satisfiability in this setting.

References

Questions concerning the finite model property, decidability, and computational complexity also remain open for broader classes of logics in closure semantics. In the present paper, finite model constructions and primitive-recursive decidability were obtained only for logics without modal inference rules, and it is natural to ask how far these results can be extended. A further problem is determining the complexity of validity and satisfiability in logics characterized by closure models.

— Closure Semantics for Weak Modal Logics: Completeness and Representations  (2610.01433 - Nishimura, 1 Oct 2026) in Section 6, “Prospects”