Constructive S4 modal logics with the finite birelational frame property

The logics 𝖢𝖲4 and 𝖨𝖲4 are the two leading intuitionistic variants of the modal logic 𝖲4. Whether the finite model property holds for each of these logics have been long-standing open problems. It was recently shown that 𝖨𝖲4 has the finite frame property and thus the finite model property. In this paper, we prove that 𝖢𝖲4 also enjoys the finite frame property. Additionally, we investigate the following three logics closely related to 𝖨𝖲4. The logic 𝖦𝖲4 is obtained by adding the Gödel–Dummett axiom to 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 𝖦𝖲4^𝖼 of 𝖦𝖲4 corresponds to requiring a crisp accessibility relation on the real-valued semantics. We give a birelational semantics corresponding to an extra confluence condition on the 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 for their birelational semantics. Establishing the finite birelational frame property immediately establishes decidability, which was previously open for these two logics. Our proofs yield NEXPTIME upper bounds. The logic 𝖲4𝖨 is obtained from 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 𝖲4𝖨