{"title":"Symmetric normalisation for intuitionistic logic","authors":"Nicolas Guenot, Lutz Straßburger","doi":"10.1145/2603088.2603160","DOIUrl":null,"url":null,"abstract":"We present two proof systems for implication-only intuitionistic logic in the calculus of structures. The first is a direct adaptation of the standard sequent calculus to the deep inference setting, and we describe a procedure for cut elimination, similar to the one from the sequent calculus, but using a non-local rewriting. The second system is the symmetric completion of the first, as normally given in deep inference for logics with a DeMorgan duality: all inference rules have duals, as cut is dual to the identity axiom. We prove a generalisation of cut elimination, that we call symmetric normalisation, where all rules dual to standard ones are permuted up in the derivation. The result is a decomposition theorem having cut elimination and interpolation as corollaries.","PeriodicalId":20649,"journal":{"name":"Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"919 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2014-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/2603088.2603160","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 7

Abstract

We present two proof systems for implication-only intuitionistic logic in the calculus of structures. The first is a direct adaptation of the standard sequent calculus to the deep inference setting, and we describe a procedure for cut elimination, similar to the one from the sequent calculus, but using a non-local rewriting. The second system is the symmetric completion of the first, as normally given in deep inference for logics with a DeMorgan duality: all inference rules have duals, as cut is dual to the identity axiom. We prove a generalisation of cut elimination, that we call symmetric normalisation, where all rules dual to standard ones are permuted up in the derivation. The result is a decomposition theorem having cut elimination and interpolation as corollaries.
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直觉逻辑的对称归一化
我们提出了结构演算中纯蕴涵直觉逻辑的两个证明系统。第一个是将标准序列演算直接应用于深度推理设置,我们描述了一个切消过程,类似于序列演算中的过程,但使用了非局部重写。第二个系统是第一个系统的对称完成,通常在具有DeMorgan对偶的逻辑的深度推理中给出:所有推理规则都有对偶,因为cut对恒等公理是对偶的。我们证明了切消的一个推广,我们称之为对称归一化,其中所有对偶于标准规则的规则都在推导中排列。其结果是一个分解定理,它以切消和插值作为推论。
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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