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. It was recently shown that $\mathsf{IS4}$ has the finite frame property and thus the finite model property. In this paper, we prove that $\mathsf{CS4}$ also enjoys 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 G\"odel--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 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 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 $\mathsf{S4I}$ 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 $\mathsf{S4I}$

翻译：逻辑系统$\mathsf{CS4}$和$\mathsf{IS4}$是模态逻辑$\mathsf{S4}$的两种主要直觉主义变体。这些逻辑是否具有有限模型性质一直是长期悬而未决的问题。近期研究表明$\mathsf{IS4}$具有有限框架性质从而具备有限模型性质。本文证明$\mathsf{CS4}$同样具有有限框架性质。此外，我们研究以下三种与$\mathsf{IS4}$密切相关的逻辑：通过在$\mathsf{IS4}$中添加Gödel–Dummett公理得到逻辑$\mathsf{GS4}$；该逻辑既是超直觉主义逻辑又是模糊逻辑，先前已被赋予实数值语义。我们为其提供另一种双关系语义，并证明关于该语义的强完备性。$\mathsf{GS4}$的扩展$\mathsf{GS4^c}$对应于在实数值语义上要求清晰的可达关系。我们为$\mathsf{GS4}$双关系语义附加额外合流条件后给出对应的双关系语义，并证明其强完备性。这两种逻辑在其实数值语义下均不具有有限模型性质，但我们证明它们对双关系语义具有有限框架性质。确立有限双关系框架性质可直接判定可判定性——此前这对这两种逻辑仍是开放问题。我们的证明给出了NEXPTIME上界。逻辑$\mathsf{S4I}$通过交换$\mathsf{IS4}$双关系语义中模态关系与直觉主义关系的角色得到。我们还证明了$\mathsf{S4I}$的有限框架性质，从而可获得其可判定性。