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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 G\"odel--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 GS4ᶜ 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
Balbiani et al. (Thu,) studied this question.