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A Concurrent Logical Framework: The Propositional Fragment
, 2003
"... We present the propositional fragment CLF0 of the Concurrent Logical Framework (CLF). CLF extends the Linear Logical Framework to allow the natural representation of concurrent computations in an object language. The underlying type theory uses monadic types to segregate values from computations ..."
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Cited by 31 (3 self)
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We present the propositional fragment CLF0 of the Concurrent Logical Framework (CLF). CLF extends the Linear Logical Framework to allow the natural representation of concurrent computations in an object language. The underlying type theory uses monadic types to segregate values from computations. This separation leads to a tractable notion of definitional equality that identifies computations di#ering only in the order of execution of independent steps. From a logical point of view our type theory can be seen as a novel combination of lax logic and dual intuitionistic linear logic. An encoding of a small Petri net exemplifies the representation methodology, which can be summarized as "concurrent computations as monadic expressions ".
Pure type systems with judgemental equality
 Journal Functional Programming
"... In a typing system, there are two approaches that may be taken to the notion of equality. One can use some external relation of convertibility defined on the terms of the grammar, such as βconvertibility or βηconvertibility; or one can introduce a judgement form for equality into the rules of the ..."
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Cited by 19 (0 self)
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In a typing system, there are two approaches that may be taken to the notion of equality. One can use some external relation of convertibility defined on the terms of the grammar, such as βconvertibility or βηconvertibility; or one can introduce a judgement form for equality into the rules of the typing system itself. For quite some time, it has been an open problem whether the two systems produced by these two choices are equivalent. This problem is essentially the problem of proving that the Subject Reduction property holds in the system with judgemental equality. In this paper, we shall prove that the equivalence holds for all functional Pure Type Systems (PTSs). The proof essentially consists of proving the ChurchRosser Theorem for a typed version of parallel onestep reduction. This method should generalise easily to many typing systems which satisfy the Uniqueness of Types property. 1
A Modular Hierarchy of Logical Frameworks
, 2004
"... Abstract. We present a method for defining logical frameworks as a collection of features which are defined and behave independently of one another. Each feature is a set of grammar clauses and rules of deduction such that the result of adding the feature to a framework is a conservative extension o ..."
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Cited by 13 (4 self)
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Abstract. We present a method for defining logical frameworks as a collection of features which are defined and behave independently of one another. Each feature is a set of grammar clauses and rules of deduction such that the result of adding the feature to a framework is a conservative extension of the framework itself. We show how several existing logical frameworks can be so built, and how several much weaker frameworks defined in this manner are adequate for expressing a wide variety of object logics. 1
Object Languages in a TypeTheoretic MetaFramework
 Workshop of Proof Transformation and Presentation and Proof Complexities (PTP'01
, 2001
"... . This paper concerns techniques for providing a convenient syntax for object languages implemented via a typetheoretic Logical Framework, and reports on work in progress. We first motivate the need for a typetheoretic logical framework. Firstly, we take the logical framework seriously as a me ..."
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. This paper concerns techniques for providing a convenient syntax for object languages implemented via a typetheoretic Logical Framework, and reports on work in progress. We first motivate the need for a typetheoretic logical framework. Firstly, we take the logical framework seriously as a metalanguage for implementing object languages (including object type theories). Another reason is the goal of building domainspecific reasoning tools which are implemented using type theory technology but do not require great expertise in type theory to use productively. We then present several examples of bidirectional translations between an encoding in the framework language and a more convenient syntax. The paper ends by discussing several techniques for implementing the translations and properties that we may require for the translation. Coercive subtyping is shown to help in the translation. 1
Under consideration for publication in J. Functional Programming 1 Pure Type Systems with Judgemental Equality
"... In a typing system, there are two approaches that may be taken to the notion of equality. One can use some external relation of convertibility defined on the terms of the grammar, such as βconvertibility or βηconvertibility; or one can introduce a judgement form for equality into the rules of the ..."
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In a typing system, there are two approaches that may be taken to the notion of equality. One can use some external relation of convertibility defined on the terms of the grammar, such as βconvertibility or βηconvertibility; or one can introduce a judgement form for equality into the rules of the typing system itself. For quite some time, it has been an open problem whether the two systems produced by these two choices are equivalent. This problem is essentially the problem of proving that the Subject Reduction property holds in the system with judgemental equality. In this paper, we shall prove that the equivalence holds for all functional Pure Type Systems (PTSs). The proof essentially consists of proving the ChurchRosser Theorem for a typed version of parallel onestep reduction. This method should generalise easily to many typing systems which satisfy the Uniqueness of Types property. 1
Coercive Subtyping in LambdaFree Logical Frameworks
"... Abstract. Coercive subtyping is a powerful approach to subtyping in dependent type theories, but its theoretical properties are often difficult to prove. Lambdafree logical frameworks such as TF have shown themselves to be a powerful tool for investigating the theory of logical frameworks, thanks t ..."
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Abstract. Coercive subtyping is a powerful approach to subtyping in dependent type theories, but its theoretical properties are often difficult to prove. Lambdafree logical frameworks such as TF have shown themselves to be a powerful tool for investigating the theory of logical frameworks, thanks to the close correspondance between a lambdafree frame and a traditional framework such as LF. We show how a type theory with coercive subtyping may be defined within TF. An operation of typecasting plays the role that coercive application plays in LF. We show that the resulting systems in TF and LF are equivalent, and how several results may be proven more easily in TF and then lifted to LF.
λTypes on the λCalculus with Abbreviations
, 2007
"... In this paper the author presents λδ, a λtyped λcalculus with a single λ binder and abbreviations. This calculus pursues the reuse of the term constructions both at the level of types and at the level of contexts as the main goal. Up to conversion λδ shares with Church λ → the subset of typable te ..."
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In this paper the author presents λδ, a λtyped λcalculus with a single λ binder and abbreviations. This calculus pursues the reuse of the term constructions both at the level of types and at the level of contexts as the main goal. Up to conversion λδ shares with Church λ → the subset of typable terms but in the “propositions as types ” perspective it can encode the implicative fragment of predicative logic without quantifiers because dependent types are allowed. λδ enjoys the properties of Church λ → (mainly subject conversion, strong normalization and decidability of type inference) and, in addition, it satisfies the correctness of types and the uniqueness of types up to conversion. We stress that λδ differs from the Automathrelated λcalculi in that they do not provide for an abbreviation construction at the level of terms. Moreover, unlike many λcalculi, λδ features a type hierarchy with an infinite number of levels both above and below any reference point.
The Formal System λδ
, 2008
"... The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction wh ..."
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The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction while it provides for an abbreviation mechanism at the level of terms. λδ enjoys some important desirable properties such as the confluence of reduction, the correctness of types, the uniqueness of types up to conversion, the subject reduction of the type assignment, the strong normalization of the typed terms and, as a corollary, the decidability of type inference problem.