{"title":"计算高维类型理论","authors":"C. Angiuli, R. Harper, Todd Wilson","doi":"10.1145/3009837.3009861","DOIUrl":null,"url":null,"abstract":"Formal constructive type theory has proved to be an effective language for mechanized proof. By avoiding non-constructive principles, such as the law of the excluded middle, type theory admits sharper proofs and broader interpretations of results. From a computer science perspective, interest in type theory arises from its applications to programming languages. Standard constructive type theories used in mechanization admit computational interpretations based on meta-mathematical normalization theorems. These proofs are notoriously brittle; any change to the theory potentially invalidates its computational meaning. As a case in point, Voevodsky's univalence axiom raises questions about the computational meaning of proofs. We consider the question: Can higher-dimensional type theory be construed as a programming language? We answer this question affirmatively by providing a direct, deterministic operational interpretation for a representative higher-dimensional dependent type theory with higher inductive types and an instance of univalence. Rather than being a formal type theory defined by rules, it is instead a computational type theory in the sense of Martin-Löf's meaning explanations and of the NuPRL semantics. The definition of the type theory starts with programs; types are specifications of program behavior. The main result is a canonicity theorem stating that closed programs of boolean type evaluate to true or false.","PeriodicalId":20657,"journal":{"name":"Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages","volume":"7 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"45","resultStr":"{\"title\":\"Computational higher-dimensional type theory\",\"authors\":\"C. Angiuli, R. Harper, Todd Wilson\",\"doi\":\"10.1145/3009837.3009861\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Formal constructive type theory has proved to be an effective language for mechanized proof. By avoiding non-constructive principles, such as the law of the excluded middle, type theory admits sharper proofs and broader interpretations of results. From a computer science perspective, interest in type theory arises from its applications to programming languages. Standard constructive type theories used in mechanization admit computational interpretations based on meta-mathematical normalization theorems. These proofs are notoriously brittle; any change to the theory potentially invalidates its computational meaning. As a case in point, Voevodsky's univalence axiom raises questions about the computational meaning of proofs. We consider the question: Can higher-dimensional type theory be construed as a programming language? We answer this question affirmatively by providing a direct, deterministic operational interpretation for a representative higher-dimensional dependent type theory with higher inductive types and an instance of univalence. Rather than being a formal type theory defined by rules, it is instead a computational type theory in the sense of Martin-Löf's meaning explanations and of the NuPRL semantics. The definition of the type theory starts with programs; types are specifications of program behavior. The main result is a canonicity theorem stating that closed programs of boolean type evaluate to true or false.\",\"PeriodicalId\":20657,\"journal\":{\"name\":\"Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages\",\"volume\":\"7 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2017-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"45\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/3009837.3009861\",\"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 44th ACM SIGPLAN Symposium on Principles of Programming Languages","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3009837.3009861","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Formal constructive type theory has proved to be an effective language for mechanized proof. By avoiding non-constructive principles, such as the law of the excluded middle, type theory admits sharper proofs and broader interpretations of results. From a computer science perspective, interest in type theory arises from its applications to programming languages. Standard constructive type theories used in mechanization admit computational interpretations based on meta-mathematical normalization theorems. These proofs are notoriously brittle; any change to the theory potentially invalidates its computational meaning. As a case in point, Voevodsky's univalence axiom raises questions about the computational meaning of proofs. We consider the question: Can higher-dimensional type theory be construed as a programming language? We answer this question affirmatively by providing a direct, deterministic operational interpretation for a representative higher-dimensional dependent type theory with higher inductive types and an instance of univalence. Rather than being a formal type theory defined by rules, it is instead a computational type theory in the sense of Martin-Löf's meaning explanations and of the NuPRL semantics. The definition of the type theory starts with programs; types are specifications of program behavior. The main result is a canonicity theorem stating that closed programs of boolean type evaluate to true or false.