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Explicit Provability And Constructive Semantics
 Bulletin of Symbolic Logic
, 2001
"... In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is noth ..."
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Cited by 117 (22 self)
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In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a BrouwerHeytingKolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #calculus.
An Algorithm for TypeChecking Dependent Types
 Science of Computer Programming
, 1996
"... We present a simple typechecker for a language with dependent types and let expressions, with a simple proof of correctness. Introduction Type Theory provides an interesting approach to the problem of (interactive) proofchecking. Instead of introducing, like in LCF [10], an abstract data type of t ..."
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Cited by 46 (4 self)
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We present a simple typechecker for a language with dependent types and let expressions, with a simple proof of correctness. Introduction Type Theory provides an interesting approach to the problem of (interactive) proofchecking. Instead of introducing, like in LCF [10], an abstract data type of theorems, it uses the proofsasprograms analogy and reduces the problem of proof checking to the problem of typechecking in a programming language with dependent types [5]. This approach presents several advantages, well described in [11,5], among those being the possibility of independent proof verification and of a uniform treatment for naming constants and theorems. It is crucial however for this approach to proofchecking to have a simple and reliable typechecking algorithm. Since the core part of such languages, like the ones described in [5,7], seems very simple, there may be some hope for such a short and simple typechecker for dependent types. Indeed, de Bruijn sketches such an al...
The Barendregt Cube with Definitions and Generalised Reduction
, 1997
"... In this paper, we propose to extend the Barendregt Cube by generalising reduction and by adding definition mechanisms. We show that this extension satisfies all the original properties of the Cube including Church Rosser, Subject Reduction and Strong Normalisation. Keywords: Generalised Reduction, ..."
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Cited by 39 (18 self)
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In this paper, we propose to extend the Barendregt Cube by generalising reduction and by adding definition mechanisms. We show that this extension satisfies all the original properties of the Cube including Church Rosser, Subject Reduction and Strong Normalisation. Keywords: Generalised Reduction, Definitions, Barendregt Cube, Church Rosser, Subject Reduction, Strong Normalisation. Contents 1 Introduction 3 1.1 Why generalised reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Why definition mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 The item notation for definitions and generalised reduction . . . . . . . . . . 4 2 The item notation 7 3 The ordinary typing relation and its properties 10 3.1 The typing relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Properties of the ordinary typing relation . . . . . . . . . . . . . . . . . . . . 13 4 Generalising reduction in the Cube 15 4.1 The generalised...
Strong Normalisation in the πCalculus
, 2001
"... We introduce a typed πcalculus where strong normalisation is ensured by typability. Strong normalisation is a useful property in many computational contexts, including distributed systems. In spite of its simplicity, our type discipline captures a wide class of converging namepassing interactive b ..."
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Cited by 32 (17 self)
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We introduce a typed πcalculus where strong normalisation is ensured by typability. Strong normalisation is a useful property in many computational contexts, including distributed systems. In spite of its simplicity, our type discipline captures a wide class of converging namepassing interactive behaviour. The proof of strong normalisability combines methods from typed lcalculi and linear logic with processtheoretic reasoning. It is adaptable to systems involving state and other extensions. Strong normalisation is shown to have significant consequences, including finite axiomatisation of weak bisimilarity, a fully abstract embedding of the simplytyped lcalculus with products and sums and basic liveness in interaction.
The Calculus of Algebraic Constructions
 In Proc. of the 10th Int. Conf. on Rewriting Techniques and Applications, LNCS 1631
, 1999
"... Abstract. In a previous work, we proved that an important part of the Calculus of Inductive Constructions (CIC), the basis of the Coq proof assistant, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by hi ..."
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Cited by 28 (10 self)
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Abstract. In a previous work, we proved that an important part of the Calculus of Inductive Constructions (CIC), the basis of the Coq proof assistant, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higherorder rewrite rules. In this paper, we prove that almost all CIC can be seen as a CAC, and that it can be further extended with nonstrictly positive types and inductiverecursive types together with nonfree constructors and patternmatching on defined symbols. 1.
Embedding pure type systems in the lambdaPicalculus modulo
 TLCA
, 2007
"... The lambdaPicalculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambdaPicalculus modulo. We show in this paper that this simple extension is surprisingly expressive and, in particular, that all fu ..."
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Cited by 21 (6 self)
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The lambdaPicalculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambdaPicalculus modulo. We show in this paper that this simple extension is surprisingly expressive and, in particular, that all functional Pure Type Systems, such as the system F, or the Calculus of Constructions, can be embedded in it. And, moreover, that this embedding is conservative under termination hypothesis.
Inductive types in the calculus of algebraic constructions
 FUNDAMENTA INFORMATICAE 65(12) (2005) 61–86 JOURNAL VERSION OF TLCA’03
, 2005
"... In a previous work, we proved that almost all of the Calculus of Inductive Constructions (CIC), the basis of the proof assistant Coq, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higherorder rewrit ..."
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Cited by 17 (4 self)
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In a previous work, we proved that almost all of the Calculus of Inductive Constructions (CIC), the basis of the proof assistant Coq, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higherorder rewrite rules. In this paper, we prove that CIC as a whole can be seen as a CAC, and that it can be extended with nonstrictly positive types and inductiverecursive types together with nonfree constructors and patternmatching on defined symbols.
Secondorder unification and type inference for Churchstyle polymorphism
 In Conference Record of POPL 98: The 25TH ACM SIGPLANSIGACT Symposium on Principles of Programming Languages
, 1998
"... We present a proof of the undecidability of type inference for the Churchstyle system F  an abstraction of polymorphism. A natural reduction from the secondorder unification problem to type inference leads to strong restriction on instances  arguments of variables cannot contain variables. T ..."
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Cited by 14 (0 self)
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We present a proof of the undecidability of type inference for the Churchstyle system F  an abstraction of polymorphism. A natural reduction from the secondorder unification problem to type inference leads to strong restriction on instances  arguments of variables cannot contain variables. This requires another proof of the undecidability of the secondorder unification since known results use variables in arguments of other variables. Moreover, our proof uses elementary techniques, which is important from the methodological point of view, because Goldfarb's proof [Gol81] highly relies on the undecidability of the tenth Hilbert's problem. 1 1 Introduction The Churchstyle system F was independently introduced by Girard [Gir72] and Reynolds [Rey74] as an extension of the simplytyped calculus a type system introduced of H. B. Curry [Cur69]. As usual for type systems, the decidability of so called sequent decision problems was considered. A sequent decision problem in some ty...
A unified approach to Type Theory through a refined λcalculus
, 1994
"... In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. ..."
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Cited by 14 (13 self)
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In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic.
Parametricity and dependent types
, 2010
"... Reynolds ’ abstraction theorem shows how a typing judgement in System F can be translated into a relational statement (in second order predicate logic) about inhabitants of the type. We obtain a similar result for a single lambda calculus (a pure type system), in which terms, types and their relatio ..."
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Cited by 13 (2 self)
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Reynolds ’ abstraction theorem shows how a typing judgement in System F can be translated into a relational statement (in second order predicate logic) about inhabitants of the type. We obtain a similar result for a single lambda calculus (a pure type system), in which terms, types and their relations are expressed. Working within a single system dispenses with the need for an interpretation layer, allowing for an unusually simple presentation. While the unification puts some constraints on the type system (which we spell out), the result applies to many interesting cases, including dependentlytyped ones. Categories and Subject Descriptors F.3.3 [Logics and Meanings