{"title":"Normalization in the simply typed -calculus","authors":"Péter Battyányi, Karim Nour","doi":"10.1017/s096012952200041x","DOIUrl":null,"url":null,"abstract":"In this paper, in connection with the program of extending the Curry–Howard isomorphism to classical logic, we study the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline2.png\" /><jats:tex-math> $\\lambda \\mu$ </jats:tex-math></jats:alternatives></jats:inline-formula>-calculus of Parigot emphasizing the difference between the original version of Parigot and the version of de Groote in terms of normalization properties. In order to talk about a satisfactory representation of the integers, besides the usual <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline3.png\" /><jats:tex-math> $\\beta$ </jats:tex-math></jats:alternatives></jats:inline-formula>-, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline4.png\" /><jats:tex-math> $\\mu$ </jats:tex-math></jats:alternatives></jats:inline-formula>-, and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline5.png\" /><jats:tex-math> $\\mu '$ </jats:tex-math></jats:alternatives></jats:inline-formula>-reductions, we consider the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline6.png\" /><jats:tex-math> $\\lambda \\mu$ </jats:tex-math></jats:alternatives></jats:inline-formula>-calculus augmented with the reduction rules <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline7.png\" /><jats:tex-math> $\\rho$ </jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline8.png\" /><jats:tex-math> $\\theta$ </jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline9.png\" /><jats:tex-math> $\\varepsilon$ </jats:tex-math></jats:alternatives></jats:inline-formula>. We show that we need all of these rules for this purpose. Then we prove that, with the syntax of Parigot, the calculus enjoys the strong normalization property even when we add the rules <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline10.png\" /><jats:tex-math> $\\rho$ </jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline11.png\" /><jats:tex-math> $\\theta$ </jats:tex-math></jats:alternatives></jats:inline-formula>, and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline12.png\" /><jats:tex-math> $\\epsilon$ </jats:tex-math></jats:alternatives></jats:inline-formula>, while the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline13.png\" /><jats:tex-math> $\\lambda \\mu$ </jats:tex-math></jats:alternatives></jats:inline-formula>-calculus presented with the more flexible de Groote-style syntax, in contrast, has only the weak normalization property. In particular, we present a normalization algorithm for the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline14.png\" /><jats:tex-math> $\\beta \\mu \\mu '\\rho \\theta \\varepsilon$ </jats:tex-math></jats:alternatives></jats:inline-formula>-reduction in the de Groote-style calculus.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"39 11","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2023-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Structures in Computer Science","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1017/s096012952200041x","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: 0
Abstract
In this paper, in connection with the program of extending the Curry–Howard isomorphism to classical logic, we study the $\lambda \mu$ -calculus of Parigot emphasizing the difference between the original version of Parigot and the version of de Groote in terms of normalization properties. In order to talk about a satisfactory representation of the integers, besides the usual $\beta$ -, $\mu$ -, and $\mu '$ -reductions, we consider the $\lambda \mu$ -calculus augmented with the reduction rules $\rho$ , $\theta$ and $\varepsilon$ . We show that we need all of these rules for this purpose. Then we prove that, with the syntax of Parigot, the calculus enjoys the strong normalization property even when we add the rules $\rho$ , $\theta$ , and $\epsilon$ , while the $\lambda \mu$ -calculus presented with the more flexible de Groote-style syntax, in contrast, has only the weak normalization property. In particular, we present a normalization algorithm for the $\beta \mu \mu '\rho \theta \varepsilon$ -reduction in the de Groote-style calculus.
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.