A linear logic framework for multimodal logics

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2022-10-01 DOI:10.1017/S0960129522000366
Bruno Xavier, C. Olarte, Elaine Pimentel
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引用次数: 1

Abstract

Abstract One of the most fundamental properties of a proof system is analyticity, expressing the fact that a proof of a given formula F only uses subformulas of F. In sequent calculus, this property is usually proved by showing that the $\mathsf{cut}$ rule is admissible, i.e., the introduction of the auxiliary lemma H in the reasoning “if H follows from G and F follows from H, then F follows from G” can be eliminated. The proof of cut admissibility is usually a tedious, error-prone process through several proof transformations, thus requiring the assistance of (semi-)automatic procedures. In a previous work by Miller and Pimentel, linear logic ( $\mathsf{LL}$ ) was used as a logical framework for establishing sufficient conditions for cut admissibility of object logical systems (OL). The OL’s inference rules are specified as an $\mathsf{LL}$ theory and an easy-to-verify criterion sufficed to establish the cut-admissibility theorem for the OL at hand. However, there are many logical systems that cannot be adequately encoded in $\mathsf{LL}$ , the most symptomatic cases being sequent systems for modal logics. In this paper, we use a linear-nested sequent ( $\mathsf{LNS}$ ) presentation of $\mathsf{MMLL}$ (a variant of LL with subexponentials), and show that it is possible to establish a cut-admissibility criterion for $\mathsf{LNS}$ systems for (classical or substructural) multimodal logics. We show that the same approach is suitable for handling the $\mathsf{LNS}$ system for intuitionistic logic.
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多模态逻辑的线性逻辑框架
证明系统的一个最基本的性质是解析性,它表示一个给定公式F的证明只使用F的子公式。在序演学中,通常通过证明$\mathsf{cut}$规则是可接受的来证明这一性质,即在“如果H从G引出,F从H引出,那么F从G引出”推理中引入辅助引理H是可以消除的。切割可采性的证明通常是一个冗长且容易出错的过程,需要经过多次证明转换,因此需要(半)自动化程序的帮助。在Miller和Pimentel之前的工作中,线性逻辑($\mathsf{LL}$)被用作建立对象逻辑系统(OL)切容许的充分条件的逻辑框架。OL的推理规则被指定为$\mathsf{LL}$理论和一个易于验证的准则,足以建立手头OL的可割性定理。然而,有许多逻辑系统不能在$\mathsf{LL}$中充分编码,最典型的情况是模态逻辑的顺序系统。在本文中,我们使用$\mathsf{MMLL}$的线性嵌套序列($\mathsf{LNS}$)表示,并证明了对于(经典或子结构)多模态逻辑,$\mathsf{LNS}$系统可以建立一个切容许准则。我们证明了同样的方法也适用于处理直觉逻辑的$\mathsf{LNS}$系统。
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
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