{"title":"Kleene代数模理论:具体kat的框架","authors":"Ryan Beckett, E. Campbell, M. Greenberg","doi":"10.1145/3519939.3523722","DOIUrl":null,"url":null,"abstract":"Kleene algebras with tests (KATs) offer sound, complete, and decidable equational reasoning about regularly structured programs. Interest in KATs has increased greatly since NetKAT demonstrated how well extensions of KATs with domain-specific primitives and extra axioms apply to computer networks. Unfortunately, extending a KAT to a new domain by adding custom primitives, proving its equational theory sound and complete, and coming up with an efficient implementation is still an expert’s task. Abstruse metatheory is holding back KAT’s potential. We offer a fast path to a “minimum viable model” of a KAT, formally or in code through our framework, Kleene algebra modulo theories (KMT). Given primitives and a notion of state, we can automatically derive a corresponding KAT’s semantics, prove its equational theory sound and complete with respect to a tracing semantics (programs are denoted as traces of states), and derive a normalization-based decision procedure for equivalence checking. Our framework is based on pushback, a generalization of weakest preconditions that specifies how predicates and actions interact. We offer several case studies, showing tracing variants of theories from the literature (bitvectors, NetKAT) along with novel compositional theories (products, temporal logic, and sets). We derive new results over unbounded state, reasoning about monotonically increasing, unbounded natural numbers. Our OCaml implementation closely matches the theory: users define and compose KATs with the module system.","PeriodicalId":140942,"journal":{"name":"Proceedings of the 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation","volume":"39 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2017-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Kleene algebra modulo theories: a framework for concrete KATs\",\"authors\":\"Ryan Beckett, E. Campbell, M. Greenberg\",\"doi\":\"10.1145/3519939.3523722\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Kleene algebras with tests (KATs) offer sound, complete, and decidable equational reasoning about regularly structured programs. Interest in KATs has increased greatly since NetKAT demonstrated how well extensions of KATs with domain-specific primitives and extra axioms apply to computer networks. Unfortunately, extending a KAT to a new domain by adding custom primitives, proving its equational theory sound and complete, and coming up with an efficient implementation is still an expert’s task. Abstruse metatheory is holding back KAT’s potential. We offer a fast path to a “minimum viable model” of a KAT, formally or in code through our framework, Kleene algebra modulo theories (KMT). Given primitives and a notion of state, we can automatically derive a corresponding KAT’s semantics, prove its equational theory sound and complete with respect to a tracing semantics (programs are denoted as traces of states), and derive a normalization-based decision procedure for equivalence checking. Our framework is based on pushback, a generalization of weakest preconditions that specifies how predicates and actions interact. We offer several case studies, showing tracing variants of theories from the literature (bitvectors, NetKAT) along with novel compositional theories (products, temporal logic, and sets). We derive new results over unbounded state, reasoning about monotonically increasing, unbounded natural numbers. Our OCaml implementation closely matches the theory: users define and compose KATs with the module system.\",\"PeriodicalId\":140942,\"journal\":{\"name\":\"Proceedings of the 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation\",\"volume\":\"39 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2017-07-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/3519939.3523722\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3519939.3523722","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Kleene algebra modulo theories: a framework for concrete KATs
Kleene algebras with tests (KATs) offer sound, complete, and decidable equational reasoning about regularly structured programs. Interest in KATs has increased greatly since NetKAT demonstrated how well extensions of KATs with domain-specific primitives and extra axioms apply to computer networks. Unfortunately, extending a KAT to a new domain by adding custom primitives, proving its equational theory sound and complete, and coming up with an efficient implementation is still an expert’s task. Abstruse metatheory is holding back KAT’s potential. We offer a fast path to a “minimum viable model” of a KAT, formally or in code through our framework, Kleene algebra modulo theories (KMT). Given primitives and a notion of state, we can automatically derive a corresponding KAT’s semantics, prove its equational theory sound and complete with respect to a tracing semantics (programs are denoted as traces of states), and derive a normalization-based decision procedure for equivalence checking. Our framework is based on pushback, a generalization of weakest preconditions that specifies how predicates and actions interact. We offer several case studies, showing tracing variants of theories from the literature (bitvectors, NetKAT) along with novel compositional theories (products, temporal logic, and sets). We derive new results over unbounded state, reasoning about monotonically increasing, unbounded natural numbers. Our OCaml implementation closely matches the theory: users define and compose KATs with the module system.