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Pure Type Systems without Explicit Contexts 没有显式上下文的纯类型系统
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2010-09-11 DOI: 10.4204/EPTCS.34.6
H. Geuvers, R. Krebbers, J. McKinna, F. Wiedijk
{"title":"Pure Type Systems without Explicit Contexts","authors":"H. Geuvers, R. Krebbers, J. McKinna, F. Wiedijk","doi":"10.4204/EPTCS.34.6","DOIUrl":"https://doi.org/10.4204/EPTCS.34.6","url":null,"abstract":"We present an approach to type theory in which the typing judgments do not have explicit contexts. Instead of judgments of shape G‘ A : B, our systems just have judgments of shape A : B. A key feature is that we distinguish free and bound variables even in pseudo-terms. Specifically we give the rules of the ‘Pure Type System’ class of type theories in this style. We prove that the typing judgments of these systems correspond in a natural way with those of Pure Type Systems as traditionally formulated. I.e., our systems have exactly the same well-typed terms as traditional presentations of type theory. Our system can be seen as a type theory in which all type judgments share an identical, infinite, typing context that has infinitely many variables for each possible type. For this reason we call our system G¥. This name means to suggest that our type judgment A : B should be read as G¥‘ A : B, with a fixed infinite type context called G¥.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"197 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115010719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 19
Pattern Unification for the Lambda Calculus with Linear and Affine Types 线性和仿射λ演算的模式统一
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2010-09-11 DOI: 10.4204/EPTCS.34.9
Anders Schack-Nielsen, C. Schürmann
{"title":"Pattern Unification for the Lambda Calculus with Linear and Affine Types","authors":"Anders Schack-Nielsen, C. Schürmann","doi":"10.4204/EPTCS.34.9","DOIUrl":"https://doi.org/10.4204/EPTCS.34.9","url":null,"abstract":"We define the pattern fragment for higher-order unification problems in linear and affine type theory and give a deterministic unification algorithm that computes most general unifiers.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132259283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
Explicit Substitutions for Contextual Type Theory 语境类型理论的显式替代
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2010-09-01 DOI: 10.4204/EPTCS.34.3
Andreas Abel, B. Pientka
{"title":"Explicit Substitutions for Contextual Type Theory","authors":"Andreas Abel, B. Pientka","doi":"10.4204/EPTCS.34.3","DOIUrl":"https://doi.org/10.4204/EPTCS.34.3","url":null,"abstract":"In this paper, we present an explicit substitution calculus which distinguishes between ordinary bound variables and meta-variables. Its typing discipline is derived from contextual modal type theory. We first present a dependently typed lambda calculus with explicit substitutions for ordinary variables and explicit meta-substitutions for meta-variables. We then present a weak head normalization procedure which performs both substitutions lazily and in a single pass thereby combining substitution walks for the two different classes of variables. Finally, we describe a bidirectional type checking algorithm which uses weak head normalization and prove soundness.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127394228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
Representing Isabelle in LF 代表伊莎贝尔在LF
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2010-09-01 DOI: 10.4204/EPTCS.34.8
Florian Rabe
{"title":"Representing Isabelle in LF","authors":"Florian Rabe","doi":"10.4204/EPTCS.34.8","DOIUrl":"https://doi.org/10.4204/EPTCS.34.8","url":null,"abstract":"LF has been designed and successfully used as a meta-logical framework to represent and reason about object logics. Here we design a representation of the Isabelle logical framework in LF using the recently introduced module system for LF. The major novelty of our approach is that we can naturally represent the advanced Isabelle features of type classes and locales. \u0000Our representation of type classes relies on a feature so far lacking in the LF module system: morphism variables and abstraction over them. While conservative over the present system in terms of expressivity, this feature is needed for a representation of type classes that preserves the modular structure. Therefore, we also design the necessary extension of the LF module system.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127582638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Theory support for weak higher order abstract syntax in Isabelle/HOL Isabelle/HOL中弱高阶抽象语法的理论支持
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2009-08-02 DOI: 10.1145/1577824.1577827
E. Gunter, Christopher J. Osborn, A. Popescu
{"title":"Theory support for weak higher order abstract syntax in Isabelle/HOL","authors":"E. Gunter, Christopher J. Osborn, A. Popescu","doi":"10.1145/1577824.1577827","DOIUrl":"https://doi.org/10.1145/1577824.1577827","url":null,"abstract":"We describe the theoretical underpinnings to support the construction of an extension to the Isabelle/HOL theorem prover to support the creation of datatypes for weak higher-order abstract syntax, and give an example of its application. This theoretical basis is centered around the concept of variable types (i.e. types whose elements are variables), and the concept of two terms in a given type having the \"same structure\" up to a given set of substitutions (the difference set) of one variable for another as allowed by that set. We provide an axiomatization of types for which the notion of having the same structure is well-behaved with the axiomatic class of same_struct_thy. We show that being a same_struct_thy is preserved by products, sums and certain function spaces.\u0000 Within a same_struct_thy, not all terms necessarily have the same structure as anything, including themselves. Those terms having the same structure as themselves relative to the empty difference set are said to be proper. A proper function from variables to terms corresponds to an abstraction of a variable in a term and also corresponds to substitution of variables for that variable in the term. Proper functions form the basis for a formalization of weak higher-order abstract syntax.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"23 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131923577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 11
A practical module system for LF 一个实用的LF模块系统
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2009-08-02 DOI: 10.1145/1577824.1577831
Florian Rabe, C. Schürmann
{"title":"A practical module system for LF","authors":"Florian Rabe, C. Schürmann","doi":"10.1145/1577824.1577831","DOIUrl":"https://doi.org/10.1145/1577824.1577831","url":null,"abstract":"Module systems for proof assistants provide administrative support for large developments when mechanizing the meta-theory of programming languages and logics. We describe a module system for the logical framework LF that is based on two main primitives: signatures and signature morphisms. Signatures are defined as collections of constant declarations, and signature morphisms as homo-morphism in between them. Our design is semantically transparent in the sense that it is always possible to elaborate modules into the module free version of LF. We have implemented our design as part of the Twelf system and rewritten parts of the Twelf example library to take advantage of the module system.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133282184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 41
Formalizing a strong normalization proof for Moggi's computational metalanguage: a case study in Isabelle/HOL-nominal Moggi计算元语言的强规范化证明的形式化:Isabelle/HOL-nominal案例研究
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2009-08-02 DOI: 10.1145/1577824.1577834
C. Doczkal, Jan Schwinghammer
{"title":"Formalizing a strong normalization proof for Moggi's computational metalanguage: a case study in Isabelle/HOL-nominal","authors":"C. Doczkal, Jan Schwinghammer","doi":"10.1145/1577824.1577834","DOIUrl":"https://doi.org/10.1145/1577824.1577834","url":null,"abstract":"Lindley and Stark have given an elegant proof of strong normalization for various lambda calculi whose type systems preclude a direct inductive definition of Girard-Tait style logical relations, such as the simply typed lambda calculus with sum types or Moggi's calculus with monadic computation types. The key construction in their proof is a notion of relational TT-lifting, which is expressed with the help of stacks of evaluation contexts. We describe a formalization of Lindley and Stark's strong normalization proof for Moggi's computational metalanguage in Isabelle/HOL, using the nominal package.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130069143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
The calculus of nominal inductive constructions: an intensional approach to encoding name-bindings
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2009-08-02 DOI: 10.1145/1577824.1577836
Edwin M. Westbrook, Aaron Stump, Evan Austin
{"title":"The calculus of nominal inductive constructions: an intensional approach to encoding name-bindings","authors":"Edwin M. Westbrook, Aaron Stump, Evan Austin","doi":"10.1145/1577824.1577836","DOIUrl":"https://doi.org/10.1145/1577824.1577836","url":null,"abstract":"Although name-bindings are ubiquitous in computer science, they are well-known to be cumbersome to encode and reason about in logic and type theory. There are many proposed solutions to this problem in the literature, but most of these proposals, however, have been extensional, meaning they are defined in terms of other concepts in the theory. This makes it difficult to apply these proposals in intensional theories like the Calculus of Inductive Constructions, or CIC.\u0000 In this paper, we introduce an approach to encoding name-bindings that is intensional, as it attempts to capture the meaning of a name-binding in itself. This approach combines in a straightforward manner with CIC to form the Calculus of Nominal Inductive Constructions, or CNIC. CNIC supports induction over data containing bindings, comparing of names for equality, and associating meta-language types with names in a fashion similar to HOAS, features which have been shown difficult to support in practice.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124957332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 14
Universal algebra over lambda-terms and nominal terms: the connection in logic between nominal techniques and higher-order variables 泛代数上的λ项和名义项:名义技术和高阶变量之间的逻辑联系
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2009-08-02 DOI: 10.1145/1577824.1577835
M. Gabbay, D. Mulligan
{"title":"Universal algebra over lambda-terms and nominal terms: the connection in logic between nominal techniques and higher-order variables","authors":"M. Gabbay, D. Mulligan","doi":"10.1145/1577824.1577835","DOIUrl":"https://doi.org/10.1145/1577824.1577835","url":null,"abstract":"This paper develops the correspondence between equality reasoning with axioms using λ-terms syntax, and reasoning using nominal terms syntax. Both syntaxes involve name-abstraction: λ-terms represent functional abstraction; nominal terms represent atomsabstraction in nominal sets.\u0000 It is not evident how to relate the two syntaxes because their intended denotations are so different. We use universal algebra, the logic of equational reasoning, a logical foundation based on an equality judgement form which is spartan but which is sufficiently expressive to encode mathematics in theory and practice.\u0000 We investigate how syntax, algebraic theories, and derivability relate across λ-theories (algebra over λ-terms) and nominal algebra theories.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116944444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
Higher-order abstract syntax in classical higher-order logic 经典高阶逻辑中的高阶抽象语法
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2009-08-02 DOI: 10.1145/1577824.1577826
Douglas J. Howe
{"title":"Higher-order abstract syntax in classical higher-order logic","authors":"Douglas J. Howe","doi":"10.1145/1577824.1577826","DOIUrl":"https://doi.org/10.1145/1577824.1577826","url":null,"abstract":"Higher-Order Abstract Syntax, or HOAS, is a technique for using a higher-order logic as a metalanguage for an object language with binding operators. It avoids formalizing syntactic details related to variable binding. This paper gives an extension to classical higher-order logic that supports HOAS. The logic we work with is the core of the logics employed in the widely used systems HOL and Isabelle/HOL. The extension adds recursive types, and a new type constructor for parametric functions. Using these additions, we can solve, for example, the archetypal recursive type equation for a HOAS representation of the syntax of the untyped lambda-calculus: T = (T x T) + (T ↪ T), where the function type is the new parametric one. The usual HOAS induction principles can be derived. The bulk of the technical development in the paper is a semantics of the new logic, extending the usual set-theoretic semantics of classical higher-order logic.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124341114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
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