{"title":"带域的隐含克莱因代数和部分正确性的子结构逻辑","authors":"Igor Sedlár","doi":"10.1017/s0960129524000045","DOIUrl":null,"url":null,"abstract":"<p>We show that Kozen and Tiuryn’s substructural logic of partial correctness <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301160033202-0894:S0960129524000045:S0960129524000045_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathsf{S}$</span></span></img></span></span> embeds into the equational theory of Kleene algebra with domain, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301160033202-0894:S0960129524000045:S0960129524000045_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathsf{KAD}$</span></span></img></span></span>. We provide an implicational formulation of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301160033202-0894:S0960129524000045:S0960129524000045_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathsf{KAD}$</span></span></img></span></span> which sets <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301160033202-0894:S0960129524000045:S0960129524000045_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathsf{S}$</span></span></img></span></span> in the context of implicational extensions of Kleene algebra.</p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Implicational Kleene algebra with domain and the substructural logic of partial correctness\",\"authors\":\"Igor Sedlár\",\"doi\":\"10.1017/s0960129524000045\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We show that Kozen and Tiuryn’s substructural logic of partial correctness <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301160033202-0894:S0960129524000045:S0960129524000045_inline1.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathsf{S}$</span></span></img></span></span> embeds into the equational theory of Kleene algebra with domain, <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301160033202-0894:S0960129524000045:S0960129524000045_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathsf{KAD}$</span></span></img></span></span>. We provide an implicational formulation of <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301160033202-0894:S0960129524000045:S0960129524000045_inline3.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathsf{KAD}$</span></span></img></span></span> which sets <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301160033202-0894:S0960129524000045:S0960129524000045_inline4.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathsf{S}$</span></span></img></span></span> in the context of implicational extensions of Kleene algebra.</p>\",\"PeriodicalId\":49855,\"journal\":{\"name\":\"Mathematical Structures in Computer Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-03-04\",\"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/s0960129524000045\",\"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/s0960129524000045","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}
Implicational Kleene algebra with domain and the substructural logic of partial correctness
We show that Kozen and Tiuryn’s substructural logic of partial correctness $\mathsf{S}$ embeds into the equational theory of Kleene algebra with domain, $\mathsf{KAD}$. We provide an implicational formulation of $\mathsf{KAD}$ which sets $\mathsf{S}$ in the context of implicational extensions of Kleene algebra.
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.
$\mathsf{S}$ embeds into the equational theory of Kleene algebra with domain, 
$\mathsf{KAD}$. We provide an implicational formulation of 
$\mathsf{KAD}$ which sets 
$\mathsf{S}$ in the context of implicational extensions of Kleene algebra.
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