{"title":"A linear logic framework for multimodal logics","authors":"Bruno Xavier, C. Olarte, Elaine Pimentel","doi":"10.1017/S0960129522000366","DOIUrl":null,"url":null,"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 \n$\\mathsf{cut}$\n 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 ( \n$\\mathsf{LL}$\n ) 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 \n$\\mathsf{LL}$\n 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 \n$\\mathsf{LL}$\n , the most symptomatic cases being sequent systems for modal logics. In this paper, we use a linear-nested sequent ( \n$\\mathsf{LNS}$\n ) presentation of \n$\\mathsf{MMLL}$\n (a variant of LL with subexponentials), and show that it is possible to establish a cut-admissibility criterion for \n$\\mathsf{LNS}$\n systems for (classical or substructural) multimodal logics. We show that the same approach is suitable for handling the \n$\\mathsf{LNS}$\n system for intuitionistic logic.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"1176 - 1204"},"PeriodicalIF":0.4000,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Structures in Computer Science","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1017/S0960129522000366","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
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.
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.