We study the algorithmic properties of first-order monomodal logics of frames $\langle {\textrm{I}\!\textrm{N}}, \leqslant \rangle $, $\langle {\textrm{I}\!\textrm{N}}, < \rangle $, $\langle \mathbb {Q}, \leqslant \rangle $, $\langle \mathbb {Q}, < \rangle $, $\langle {\textrm{I}\!\textrm{R}}, \leqslant \rangle $, $\langle {\textrm{I}\!\textrm{R}}, < \rangle $, as well as some related logics, in languages with restrictions on the number of individual variables as well as the number and arity of predicate letters. We show that the logics of frames based on $ {\textrm{I}\!\textrm{N}}$ are $\varPi ^1_1$-hard—thus, not recursively enumerable—in languages with two individual variables, one monadic predicate letter and one proposition letter. We also show that the logics of frames based on $\mathbb {Q}$ and ${\textrm{I}\!\textrm{R}}$ are $\varSigma ^0_1$-hard in languages with the same restrictions. Similar results are obtained for a number of related logics.

中文翻译：

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受限语言中线性 Kripke 框架的一阶模态逻辑的算法性质

我们研究帧 $\langle {\textrm{I}\!\textrm{N}}、\leqslant \rangle $、$\langle {\textrm{I}\!\textrm 的一阶单峰逻辑的算法性质{N}}, < \rangle $, $\langle \mathbb {Q}, \leqslant \rangle $, $\langle \mathbb {Q}, < \rangle $, $\langle {\textrm{I}\!\textrm{R}}, \leqslant \rangle $, $\langle {\textrm{I}\!\textrm{R}}, < \rangle $，以及一些相关的逻辑，在语言中对单个变量的数量以及谓词字母的数量和数量有限制。我们证明了基于 $ {\textrm{I}\!\textrm{N}}$ 的帧的逻辑是 $\varPi ^1_1$-hard - 因此，不能递归枚举 - 在具有两个单独变量的语言中，一个单子谓词和一个命题字母。我们还展示了基于 $\mathbb {Q}$ 和 ${\textrm{I}\! \textrm{R}}$ 在具有相同限制的语言中是 $\varSigma ^0_1$-hard。对于许多相关逻辑，获得了类似的结果。