Constructive S4 modal logics with the finite birelational frame property

Abstract

The logics \(\mathsf {CS4} \) and \(\mathsf {IS4} \) are the two leading intuitionistic variants of the modal logic \(\mathsf {S4} \) . Whether the finite model property holds for each of these logics have been long-standing open problems. In this paper, we resolve the first of these problems, proving that \(\mathsf {CS4} \) has the finite frame property. Additionally, we investigate the following three logics closely related to \(\mathsf {IS4} \) . The logic \(\mathsf {GS4} \) is obtained by adding the Gödel–Dummett axiom to \(\mathsf {IS4} \) ; it is both a superintuitionistic and a fuzzy logic and has previously been given a real-valued semantics. We provide an alternative birelational semantics and prove strong completeness with respect to this semantics. The extension \(\mathsf {GS4}^c \) of \(\mathsf {GS4} \) corresponds to requiring a crisp accessibility relation on the real-valued semantics. We give \(\mathsf {GS4}^c \) a birelational semantics corresponding to an extra confluence condition on the \(\mathsf {GS4} \) birelational semantics and prove strong completeness. Neither of these two logics have the finite model property with respect to their real-valued semantics, but we prove that they have the finite frame property with respect to their birelational semantics. Establishing the finite birelational frame property immediately establishes decidability, which was previously open for these two logics. Our proofs yield coNEXPTIME upper bounds for \(\mathsf {CS4} \) , \(\mathsf {GS4} \) , and \(\mathsf {GS4}^c \) . The logic \(\mathsf {S4I} \) is obtained from \(\mathsf {IS4} \) by reversing the roles of the modal and intuitionistic relations in the birelational semantics. We also prove the finite frame property, and thereby decidability, for \(\mathsf {S4I} \) , although our proof does not yield an elementary complexity bound. In summary, we prove that \(\mathsf {CS4} \) , \(\mathsf {GS4} \) , \(\mathsf {GS4}^c \) , and \(\mathsf {S4I} \) all have the finite birelational frame property, with the first three having a coNEXPTIME validity problem.

The authors' abstract, as published at the source. Journal of the ACM, 2026 · DOI ↗