We study the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate modal logics of finite Kripke frames on the logics’ algorithmic properties. A finite frame is a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that most predicate modal logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, |$\varPi ^0_1$|-hard—in languages with three individual variables and a single monadic predicate letter. This applies to the logics of finite frames of the predicate extensions of the sublogics of propositional modal logics |$\textbf{GL}$|⁠, |$\textbf{Grz}$| and |$\textbf{KTB}$|—among them, |$\textbf{K}$|⁠, |$\textbf{T}$|⁠, |$\textbf{D}$|⁠, |$\textbf{KB}$|⁠, |$\textbf{K4}$| and |$\textbf{S4}$|⁠.

中文翻译：

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受限语言中有限Kripke框架的一阶模态逻辑的算法性质

我们研究了在有限Kripke框架的一阶谓词模态逻辑的语言中，限制单个变量的数量以及谓词字母的数量和种类对​​逻辑算法特性的影响。有限框架是具有有限可能世界的框架。我们认为的语言没有常量，函数符号或等号。我们证明了有限Kripke框架的自然类的大多数谓词模态逻辑不是递归可枚举的-更确切地说，| $ \ varPi ^ 0_1 $ | -hard-在具有三个单独变量和一个单子谓词字母的语言中。这适用于命题模态逻辑| $ \ textbf {GL} $ |⁠，| $ \ textbf { Grz} $ |的子逻辑的谓词扩展的有限框架的逻辑。和| $ \ textbf {KTB} $ | 在其中，| $ \ textbf {K} $ |⁠，| $ \ textbf {T} $ |⁠，| $ \ textbf {D} $ |⁠，| $ \ textbf {KB} $ |⁠，| $ \ textbf {K4} $ | 和| $ \ textbf {S4} $ |⁠。