Complexity function and complexity of validity of modal and superintuitionistic propositional logics

IF 0.7 4区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Journal of Logic and Computation Pub Date : 2023-01-03 DOI:10.1093/logcom/exac085

Mikhail Rybakov, Dmitry Shkatov

引用次数: 0

Abstract

Abstract We consider the relationship between the algorithmic properties of the validity problem for a modal or superintuitionistic propositional logic and the size of the smallest Kripke countermodels for non-theorems of the logic. We establish the existence, for every degree of unsolvability, of a propositional logic whose validity problem belongs to the degree and whose every non-theorem is refuted on a Kripke frame that validates the logic and has the size linear in the length of the non-theorem. Such logics are obtained among the normal extensions of the propositional modal logics $\textbf {KTB}$, $\textbf {GL}$ and $\textbf {Grz}$ as well as in the lattice of superintuitionistic propositional logics. This shows that the computational complexity of a modal or superintuitionistic propositional logic is, in general, not related to the size of the countermodels for its non-theorems.

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模态与超直觉命题逻辑的复杂性函数与有效性复杂性

摘要研究了模态或超直觉命题逻辑有效性问题的算法性质与该逻辑非定理的最小Kripke反模型的大小之间的关系。对于每一个不可解度，我们建立了一个命题逻辑的存在性，它的有效性问题属于该度，它的每一个非定理都在一个验证逻辑的Kripke框架上被驳斥，该框架的大小与非定理的长度呈线性关系。在命题模态逻辑$\textbf {KTB}$、$\textbf {GL}$和$\textbf {Grz}$的正规扩展中以及超直觉命题逻辑的格中都得到了这样的逻辑。这表明，模态或超直觉命题逻辑的计算复杂性通常与其非定理的反模型的大小无关。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Logic and Computation 工程技术-计算机：理论方法

CiteScore

1.90

自引率

14.30%

发文量

审稿时长

6-12 weeks

期刊介绍： Logic has found application in virtually all aspects of Information Technology, from software engineering and hardware to programming and artificial intelligence. Indeed, logic, artificial intelligence and theoretical computing are influencing each other to the extent that a new interdisciplinary area of Logic and Computation is emerging. The Journal of Logic and Computation aims to promote the growth of logic and computing, including, among others, the following areas of interest: Logical Systems, such as classical and non-classical logic, constructive logic, categorical logic, modal logic, type theory, feasible maths.... Logical issues in logic programming, knowledge-based systems and automated reasoning; logical issues in knowledge representation, such as non-monotonic reasoning and systems of knowledge and belief; logics and semantics of programming; specification and verification of programs and systems; applications of logic in hardware and VLSI, natural language, concurrent computation, planning, and databases. The bulk of the content is technical scientific papers, although letters, reviews, and discussions, as well as relevant conference reviews, are included.