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24
A New Deconstructive Logic: Linear Logic
, 1995
"... The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different a ..."
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Cited by 102 (11 self)
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The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's , FD ([9, 11, 27, 31]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programmingwithproofs' ([22, 23]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for nonadditive proof nets, to be precise) using appropriate embeddings (socalled decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these `deconstructi...
Types, Abstraction, and Parametric Polymorphism, Part 2
, 1991
"... The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and P ..."
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Cited by 53 (1 self)
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The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and PLcategory models of the polymorphic typed lambda calculus. Treatments of Kripke relations and of complete relations on domains are included.
Inductively Defined Types in the Calculus of Constructions
 IN: PROCEEDINGS OF THE FIFTH CONFERENCE ON THE MATHEMATICAL FOUNDATIONS OF PROGRAMMING SEMANTICS. SPRINGER VERLAG LNCS
, 1989
"... We define the notion of an inductively defined type in the Calculus of Constructions and show how inductively defined types can be represented by closed types. We show that all primitive recursive functionals over these inductively defined types are also representable. This generalizes work by Böhm ..."
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Cited by 43 (2 self)
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We define the notion of an inductively defined type in the Calculus of Constructions and show how inductively defined types can be represented by closed types. We show that all primitive recursive functionals over these inductively defined types are also representable. This generalizes work by Böhm & Berarducci on synthesis of functions on term algebras in the secondorder polymorphiccalculus (F2). We give several applications of this generalization, including a representation of F2programs in F3, along with a definition of functions reify, reflect, and eval for F2 in F3. We also show how to define induction over inductively defined types and sketch some results that show that the extension of the Calculus of Construction by induction principles does not alter the set of functions in its computational fragment, F!. This is because a proof by induction can be realized by primitive recursion, which is already de nable in F!.
TypeBased Termination of Recursive Definitions
, 2002
"... This article The purpose of this paper is to introduce b, a simply typed calculus that supports typebased recursive definitions. Although heavily inspired from previous work by Giménez (Giménez 1998) and closely related to recent work by Amadio and Coupet (Amadio and CoupetGrimal 1998), the techn ..."
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Cited by 39 (3 self)
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This article The purpose of this paper is to introduce b, a simply typed calculus that supports typebased recursive definitions. Although heavily inspired from previous work by Giménez (Giménez 1998) and closely related to recent work by Amadio and Coupet (Amadio and CoupetGrimal 1998), the technical machinery behind our system puts a slightly different emphasis on the interpretation of types. More precisely, we formalize the notion of typebased termination using a restricted form of type dependency (a.k.a. indexed types), as popularized by (Xi and Pfenning 1998; Xi and Pfenning 1999). This leads to a simple and intuitive system which is robust under several extensions, such as mutually inductive datatypes and mutually recursive function definitions; however, such extensions are not treated in the paper
Deliverables: A Categorical Approach to Program Development in Type Theory
, 1992
"... This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack's ..."
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Cited by 24 (1 self)
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This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack's versatile LEGO implementation, which I use extensively to develop the mathematical constructions studied here. I systematically investigate Burstall's notion of deliverable, that is, a program paired with a proof of correctness. This approach separates the concerns of programming and logic, since I want a simple program extraction mechanism. The \Sigmatypes of the calculus enable us to achieve this. There are many similarities with the subset interpretation of MartinLof type theory. I show that deliverables have a rich categorical structure, so that correctness proofs may be decomposed in a principled way. The categorical combinators which I define in the system package up much logical bo...
The Impact of the Lambda Calculus in Logic and Computer Science
 Bulletin of Symbolic Logic
, 1997
"... One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the represent ..."
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Cited by 23 (0 self)
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One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand. Acknowledgement. The following persons provided help in various ways. Erik Barendsen, Jon Barwise, Johan van Benthem, Andreas Blass, Olivier Danvy, Wil Dekkers, Marko van Eekelen, Sol Feferman, Andrzej Filinski, Twan Laan, Jan Kuper, Pierre Lescanne, Hans Mooij, Robert Maron, Rinus Plasmeijer, Randy Pollack, Kristoffer Rose, Richard Shore, Rick Statman and Simon Thompson. Partial support came from the European HCM project Typed lambda calculus (CHRXCT920046), the Esprit Working Group Types (21900) and the Dutch NWO project WINST (612316607). 1. Introduction This paper is written to honor Church's gr...
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 18 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
On functors expressible in the polymorphic typed lambda calculus
 Logical Foundations of Functional Programming
, 1990
"... This is a preprint of a paper that has been submitted to Information and Computation. ..."
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Cited by 16 (1 self)
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This is a preprint of a paper that has been submitted to Information and Computation.
A General Storage Theorem for Integers in CallByName λCalculus
, 1993
"... The notion of storage operator introduced in [5, 6] appears to be an important tool in the study of data types in second order λcalculus. These operators are λterms which simulate callbyvalue in the callbyname strategy, and they can be used in order to modelize assignment instructions. The mai ..."
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Cited by 15 (4 self)
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The notion of storage operator introduced in [5, 6] appears to be an important tool in the study of data types in second order λcalculus. These operators are λterms which simulate callbyvalue in the callbyname strategy, and they can be used in order to modelize assignment instructions. The main result about storage operators is that there is a very simple second order type for them, using Gödel's "notnot translation" of classical into intuitionistic logic. We give here a new and simpler proof of a strengthened version of this theorem, which contains all previous results in intuitionistic and in classical logic ([6, 7]), and gives rise to new "storage theorems". Moreover, this result has a simple and intuitive meaning, in terms of realizability.
A Cube of Proof Systems for the Intuitionistic Predicate mu,nuLogic
 Dept. of Informatics, Univ. of Oslo
, 1997
"... This paper is an attempt at a systematizing study of the proof theory of the intuitionistic predicate ¯; logic (conventional intuitionistic predicate logic extended with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers). We identify eight pr ..."
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Cited by 6 (5 self)
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This paper is an attempt at a systematizing study of the proof theory of the intuitionistic predicate ¯; logic (conventional intuitionistic predicate logic extended with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers). We identify eight prooftheoretically interesting naturaldeduction calculi for this logic and propose a classification of these into a cube on the basis of the embeddibility relationships between these. 1 Introduction ¯,logics, i.e. logics with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers, have turned out to be a useful formalism in a number of computer science areas. The classical 1storder predicate ¯,logic can been used as a logic of (nondeterministic) imperative programs and as a database query language. It is also one of the relation description languages studied in descriptive complexity theory (finite model theory) (for a survey on this hi...