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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 139 (25 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.
DomainFree Pure Type Systems
 JOURNAL OF FUNCTIONAL PROGRAMMING
, 1993
"... Pure type systems make use of domainfull λabstractions λx:D.M . We present a variant of pure type systems, which we call domainfree pure type systems, with domainfree λabstractions λx.M. Domainfree pure type systems have a number of advantages over both pure type systems and socalled type ass ..."
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Cited by 40 (8 self)
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Pure type systems make use of domainfull λabstractions λx:D.M . We present a variant of pure type systems, which we call domainfree pure type systems, with domainfree λabstractions λx.M. Domainfree pure type systems have a number of advantages over both pure type systems and socalled type assignment systems (they also have some disadvantages) and have been used in theoretical developments as well as in implementations of proofassistants. We study the basic properties of domainfree pure type systems, establish their formal relationship with pure type systems and type assignment systems, and give a number of applications of these correspondences.
CPS Translations and Applications: The Cube and Beyond
 Higher Order and Symbolic Computation
, 1996
"... Continuation passing style (CPS) translations of typed calculi have numerous applications. However, the range of these applications has been conned by the fact that CPS translations are known for nondependent type systems only, thus excluding wellknown systems like the calculus of constructions ( ..."
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Cited by 11 (2 self)
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Continuation passing style (CPS) translations of typed calculi have numerous applications. However, the range of these applications has been conned by the fact that CPS translations are known for nondependent type systems only, thus excluding wellknown systems like the calculus of constructions (CC) and the logical frameworks (LF). This paper presents techniques for CPS translating systems with dependent types, with an emphasis on pure typetheoretical applications.
Explicit substitutions for the lambda Deltacalculus
"... . The \Deltacalculus is a calculus with a controllike operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce \Deltaexp, an explicit substitution calculus for \Delta, and study its properties. In particular, we show that \Deltaexp preserves ..."
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. The \Deltacalculus is a calculus with a controllike operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce \Deltaexp, an explicit substitution calculus for \Delta, and study its properties. In particular, we show that \Deltaexp preserves strong normalisation, which provides us with the first example moreover a very natural one indeed of explicit substitution calculus which is not structurepreserving and has the preservation of strong normalisation property. One particular application of this result is to prove that the simply typed version of \Deltaexp is strongly normalising. In addition, we show that Plotkin's callbyname continuationpassing style translation may be extended to \Deltaexp and that the extended translation preserves typing. This seems to be the first study of CPS translations for calculi of explicit substitutions. 1 Introduction Explicit substitutions were introduced by Abadi, Cardelli, Curien and L...
Explicit substitutions for the calculus
"... Abstract. The calculus is a calculus with a controllike operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce exp, an explicit substitution calculus for , and study its properties. In particular, we show that exp preserves strong normalisa ..."
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Abstract. The calculus is a calculus with a controllike operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce exp, an explicit substitution calculus for , and study its properties. In particular, we show that exp preserves strong normalisation, which provides us with the rst example {moreover a very natural one indeed{ of explicit substitution calculus which is not structurepreserving and has the preservation of strong normalisation property. One particular application of this result is to prove that the simply typed version of exp is strongly normalising. In addition, we show that Plotkin's callbyname continuationpassing style translation may be extended to exp and that the extended translation preserves typing. This seems to be the rst study of CPS translations for calculi of explicit substitutions. 1
Classical Fω, orthogonality and symmetric candidates
"... We present a version of system Fω, called F Cω, in which the layer of type constructors is essentially the traditional one of Fω, whereas provability of types is classical. The proofterm calculus accounting for the classical reasoning is a variant of Barbanera and Berardi’s symmetric λcalculus. We ..."
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We present a version of system Fω, called F Cω, in which the layer of type constructors is essentially the traditional one of Fω, whereas provability of types is classical. The proofterm calculus accounting for the classical reasoning is a variant of Barbanera and Berardi’s symmetric λcalculus. We prove that the whole calculus is strongly normalising. For the layer of type constructors, we use Tait and Girard’s reducibility method combined with orthogonality techniques. For the (classical) layer of terms, we use Barbanera and Berardi’s method based on a symmetric notion of reducibility candidate. We prove that orthogonality does not capture the fixpoint construction of symmetric candidates. We establish the consistency of F Cω, and relate the calculus to the traditional system Fω, also when the latter is extended with axioms for classical logic. 1
www.elsevier.com/locate/apal Classical Fω, orthogonality and symmetric candidates
, 2008
"... We present a version of system Fω, called Fcω, in which the layer of type constructors is essentially the traditional one of Fω, whereas provability of types is classical. The proofterm calculus accounting for the classical reasoning is a variant of Barbanera and Berardi’s symmetric λcalculus. We ..."
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We present a version of system Fω, called Fcω, in which the layer of type constructors is essentially the traditional one of Fω, whereas provability of types is classical. The proofterm calculus accounting for the classical reasoning is a variant of Barbanera and Berardi’s symmetric λcalculus. We prove that the whole calculus is strongly normalising. For the layer of type constructors, we use Tait and Girard’s reducibility method combined with orthogonality techniques. For the (classical) layer of terms, we use Barbanera and Berardi’s method based on a symmetric notion of reducibility candidate. We prove that orthogonality does not capture the fixpoint construction of symmetric candidates. We establish the consistency of Fcω, and relate the calculus to the traditional system Fω, also when the latter is extended with axioms for classical logic.
The Relevance of ProofIrrelevance: A MetaTheoretical Study of Generalised Calculi of Constructions
"... . We propose a general technique, inspired from proofirrelevance, to prove strong normalisation and consistency for extensions of the Calculus of Constructions. 1 Introduction The Calculus of Constructions (CC) [12] is a powerful typed calculus which may be used both as a programming language an ..."
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. We propose a general technique, inspired from proofirrelevance, to prove strong normalisation and consistency for extensions of the Calculus of Constructions. 1 Introduction The Calculus of Constructions (CC) [12] is a powerful typed calculus which may be used both as a programming language and as a logical framework. However, CC is minimal in the sense that the generalised function space \Pi x : A: B is its only type constructor. The minimality of CC imposes strict limits to its applicability and has given rise to a spate of proposals to include new term/type constructors: algebraic types [3, 7], fixpoints [1], control operators [6] and inductive types [21] to mention only the examples considered in this papersome proposals are actually concerned with the more general setting of Pure Type Systems [4]. While most of these calculi, which we call Generalised Calculi of Constructions (GCCs), are known to enjoy metatheoretical properties similar to CC itself, there are no genera...
Explicit Substitutions for the λΔcalculus
"... The λΔcalculus is a λcalculus with a controllike operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce λΔexp, an explicit substitution calculus for λΔ, and study its properties. In particular, we show that λΔexp preserves strong normalisa ..."
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The λΔcalculus is a λcalculus with a controllike operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce λΔexp, an explicit substitution calculus for λΔ, and study its properties. In particular, we show that λΔexp preserves strong normalisation, which provides us with the first example  moreover a very natural one indeed  of explicit substitution calculus which is not structurepreserving and has the preservation of strong normalisation property. One particular application of this result is to prove that the simply typed version of λΔexp is strongly normalising. In addition, we show that Plotkin's callbyname continuationpassing style translation may be extended to λΔexp and that the extended translation preserves typing. This seems to be the first study of CPS translations for calculi of explicit substitutions.