简单类型微积分的归一化

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2023-01-12 DOI:10.1017/s096012952200041x
Péter Battyányi, Karim Nour
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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":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2023-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":null,\"pages\":null},\"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}","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

摘要

本文结合将Curry-Howard同构扩展到经典逻辑的方案,研究Parigot的$\lambda \mu$ -演算,强调Parigot的原始版本与de Groote版本在规格化性质上的区别。为了讨论整数的令人满意的表示,除了通常的$\beta$ -, $\mu$ -和$\mu '$ -约简之外,我们还考虑了$\lambda \mu$ -演算与约简规则$\rho$, $\theta$和$\varepsilon$的扩充。我们证明了我们需要所有这些规则来达到这个目的。然后我们证明了在Parigot语法下,即使添加了$\rho$、$\theta$和$\epsilon$规则,微积分仍然具有强的规格化性质,而采用更灵活的de groote风格语法的$\lambda \mu$ -微积分却只有弱的规格化性质。特别地,我们提出了一种归一化算法,用于de Groote-style演算中的$\beta \mu \mu '\rho \theta \varepsilon$ -约简。
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Normalization in the simply typed -calculus
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.
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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.
期刊最新文献
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