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12
Using Typed Lambda Calculus to Implement Formal Systems on a Machine
 Journal of Automated Reasoning
, 1992
"... this paper and the LF. In particular the idea of having an operator T : Prop ! Type appears already in De Bruijn's earlier work, as does the idea of having several judgements. The paper [24] describes the basic features of the LF. In this paper we are going to provide a broader illustration of its a ..."
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Cited by 83 (14 self)
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this paper and the LF. In particular the idea of having an operator T : Prop ! Type appears already in De Bruijn's earlier work, as does the idea of having several judgements. The paper [24] describes the basic features of the LF. In this paper we are going to provide a broader illustration of its applicability and discuss to what extent it is successful. The analysis (of the formal presentation) of a system carried out through encoding often illuminates the system itself. This paper will also deal with this phenomenon.
Dialogue Pragmatics and Context Specification
 In Abduction, Belief and Context in Dialogue; studies in computational
, 2000
"... Introduction Pragmatics is commo,nly understood to be concerned with studying the relations between linguistic phenomena and properties of the context of use. The understanding of these relations is important in many areas of theoretical and applied research, from grammatical analysis to sociolingu ..."
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Cited by 44 (14 self)
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Introduction Pragmatics is commo,nly understood to be concerned with studying the relations between linguistic phenomena and properties of the context of use. The understanding of these relations is important in many areas of theoretical and applied research, from grammatical analysis to sociolinguistic field studies. One area where the importance of these relations has become particularly clear is the design of language understanding systems. Such systems are extremely limited, brittle, and unpractical if they do not have powerful ways to make use of contextual information in computing the meanings of utterances. The question of how this can be achieved in an effective and principled way forms one of the major obstacles in building such systems. Computational pragmatics, the study of how contextual information can be effectively brought to bear in language understanding and production processes, hopes to contribute to removing this obstacle. One way in which contextual infor
A short and flexible proof of Strong Normalization for the Calculus of Constructions
, 1994
"... this paper can still go through (with slightly more technical effort) in case one can distinguish cases according to whether a specific subterm is a type or kind in a fixed context. The other property of type systems that is really actually required for the constructions in this paper to go through ..."
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Cited by 15 (0 self)
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this paper can still go through (with slightly more technical effort) in case one can distinguish cases according to whether a specific subterm is a type or kind in a fixed context. The other property of type systems that is really actually required for the constructions in this paper to go through is a slight strengthening of the Stripping property (also called Generation). This property says, for example, that if \Gamma ` v:T:M : U has a derivation D, then one can find a subderivation of
Induction principles formalized in the Calculus of Constructions
 Programming of Future Generation Computers. Elsevier Science
, 1988
"... The Calculus of Constructions is a higherorder formalism for writing constructive proofs in a natural deduction style, inspired from work of de Bruijn [2, 3], Girard [12], MartinLöf [14] and Scott [18]. The calculus and its syntactic theory were presented in Coquand’s thesis [7], and an implementa ..."
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Cited by 10 (4 self)
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The Calculus of Constructions is a higherorder formalism for writing constructive proofs in a natural deduction style, inspired from work of de Bruijn [2, 3], Girard [12], MartinLöf [14] and Scott [18]. The calculus and its syntactic theory were presented in Coquand’s thesis [7], and an implementation by the author was used to mechanically verify a substantial number of proofs demonstrating the power of expression of the formalism [9]. The Calculus of Constructions is proposed as a foundation for the design of programming environments where programs are developed consistently with formal specifications. The current paper shows how to define inductive concepts in the calculus. A very general induction schema is obtained by postulating all elements of the type of interest to belong to the standard interpretation associated with a predicate map. This is similar to the treatment of D. Park [16], but the power of expression of the formalism permits a very direct treatment, in a language that is formalized enough to be actually implemented on computer. Special instances of the induction schema specialize to Nœtherian induction and Structural induction over any algebraic type. Computational Induction is treated in an axiomatization of Domain Theory in Constructions. It is argued that the resulting principle is more powerful than LCF’s [13], since the restriction on admissibility is expressible in the object language. Notations We assume the reader is familiar with the Calculus of Constructions, as presented in [7, 9, 10, 11]. More precisely, we shall use in the present paper the extended system defined in Section 11 of [8]. The notation [x: A]B stands for the algorithm with formal parameter x of type A and body B, whereas (x: A)B stands for the product of types B indexed by x ranging over A. Thus square brackets are used for λabstraction, whereas parentheses stand for product formation. The atom P rop is the type of logical propositions. The atom T ype stands for the first level in the predicative hierarchy of types (and thus we have P rop: T ype). We abbreviate (x: A)B into A → B whenever x does not occur in B. When B: P rop, we think of (x: A)B as the universally quantified proposition ∀x: A·B. When x does not occur in B and A: P rop,
On the Definition of the Etalong Normal Form in Type Systems of the Cube
 Informal Proceedings of the Workshop on Types for Proofs and Programs
, 1993
"... The smallest transitive relation ! on welltyped normal terms such that if t is a strict subterm of u then t ! u and if T is the normal form of the type of t and the term t is not a sort then T ! t is wellfounded in the type systems of the cube. Thus every term admits a jlong normal form. Introdu ..."
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Cited by 7 (0 self)
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The smallest transitive relation ! on welltyped normal terms such that if t is a strict subterm of u then t ! u and if T is the normal form of the type of t and the term t is not a sort then T ! t is wellfounded in the type systems of the cube. Thus every term admits a jlong normal form. Introduction In this paper we prove that the smallest transitive relation ! on welltyped normal terms such that ffl if t is a strict subterm of u then t ! u, ffl if T is the normal form of the type of t and the term t is not a sort then T ! t is wellfounded in the type systems of the cube [1]. This result is proved using the notion of marked terms introduced by de Vrijer [6]. A motivation for this theorem is to define the jlong form of a normal term in these type systems. In simply typed calculus, to define the jlong form of a normal term we first define the jlong form of a variable x of type P 1 ! ::: ! P n ! P (P atomic) as the term [y 1 : P 1 ]:::[y n : P n ](x y 0 1 ::: y 0 n ) w...
The Calculus of Constructions and Higher Order Logic
 In preparation
, 1992
"... The Calculus of Constructions (CC) ([Coquand 1985]) is a typed lambda calculus for higher order intuitionistic logic: proofs of the higher order logic are interpreted as lambda terms and formulas as types. It is also the union of Girard's system F! ([Girard 1972]), a higher order typed lambda calcul ..."
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Cited by 6 (0 self)
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The Calculus of Constructions (CC) ([Coquand 1985]) is a typed lambda calculus for higher order intuitionistic logic: proofs of the higher order logic are interpreted as lambda terms and formulas as types. It is also the union of Girard's system F! ([Girard 1972]), a higher order typed lambda calculus, and a first order dependent typed lambda calculus in the style of de Bruijn's Automath ([de Bruijn 1980]) or MartinLof's intuitionistic theory of types ([MartinLof 1984]). Using the impredicative coding of data types in F! , the Calculus of Constructions thus becomes a higher order language for the typing of functional programs. We shall introduce and try to explain CC by exploiting especially the first point of view, by introducing a typed lambda calculus that faithfully represent higher order predicate logic (so for this system the CurryHoward `formulasastypes isomorphism' is really an isomorphism.) Then we discuss some propositions that are provable in CC but not in the higher or...
A games semantics for reductive logic and proofsearch
 GaLoP 2005: Games for Logic and Programming Languages
, 2005
"... Abstract. Theorem proving, or algorithmic proofsearch, is an essential enabling technology throughout the computational sciences. We explain the mathematical basis of proofsearch as the combination of reductive logic together with a control régime. Then we present a games semantics for reductive l ..."
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Cited by 3 (0 self)
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Abstract. Theorem proving, or algorithmic proofsearch, is an essential enabling technology throughout the computational sciences. We explain the mathematical basis of proofsearch as the combination of reductive logic together with a control régime. Then we present a games semantics for reductive logic and show how it may be used to model two important examples of control, namely backtracking and uniform proof. 1 Introduction to reductive logic and proofsearch Theorem proving, or algorithmic proofsearch, is an essential enabling technology throughout the computational sciences. We explain the mathematical basis of proofsearch as the combination of reductive logic together with a control régime. Then we present a games semantics for reductive logic and show how it may be used to model two important
The DenK architecture: a pragmatic approach to user interfaces
, 1996
"... In this paper we present the basic principles underlying the DenKsystem, a generic cooperative interface combining linguistic and visual interaction. The system integrates results from fundamental research in knowledge representation, communication, natural language semantics and pragmatics, and ..."
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In this paper we present the basic principles underlying the DenKsystem, a generic cooperative interface combining linguistic and visual interaction. The system integrates results from fundamental research in knowledge representation, communication, natural language semantics and pragmatics, and objectoriented animation. Our design incorporates a cooperative and knowledgeable electronic assistant that communicates with a user in natural language, and an application domain, which is presented visually. The assistant, that we call the cooperator, has an information state that is represented in a rich form of Type Theory, a formalism that enables us to model the inherent cognitive dynamics of a dialogue participant. Pragmatic issues in manmachine interaction, concerning the use of natural language and knowledge in cooperative communication, are central to our approach. Keywords: multimodal interaction, knowledge representation, natural language semantics, pragmatics, type t...
Preuve de correction de la compilation de MiniML en code CAM dans le système d'aide à la démonstration COQ
, 1995
"... Machine). Notre objectif a 'et'e de m'ecaniser une preuve pr'esent'ee dans l'article de J. Despeyroux [9] et 'ecrite `a l'aide du langage Typol. Nous utilisons des s'emantiques naturelles pour mod'eliser l"evaluation de nos langages. Nous ne sommes parvenus que partiellement `a m'ecaniser cette preu ..."
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Machine). Notre objectif a 'et'e de m'ecaniser une preuve pr'esent'ee dans l'article de J. Despeyroux [9] et 'ecrite `a l'aide du langage Typol. Nous utilisons des s'emantiques naturelles pour mod'eliser l"evaluation de nos langages. Nous ne sommes parvenus que partiellement `a m'ecaniser cette preuve de correction. En effet, les sp'ecifications naturelles des langages source et cible contiennent des termes rationnels difficiles `a axiomatiser dans l"etat actuel du syst`eme. Nous proposons un d'ecoupage de la preuve isolant cette difficult'e. (Abstract: pto) Samuel.Boutin@inria.fr Unit'e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T'el'ephone : (33 1) 39 63 55 11  T'el'ecopie : (33 1) 39 63 53 30 Proving Correctness of the Translation from MiniML to the CAM with the Coq Proof Development System Abstract: In this report we show how we proved correctness of the translation from a small applicative language with rec...
Informal Proceedings Of The 1993 Workshop On Types For Proofs And Programs, Nijmegen
, 1993
"... Clauses : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 375 iii Foreword This document is the preliminary proceedings of the workshop of the Esprit Basic Research Project 6453 "Types for Proofs and Programs" held at the University of Nijmegen, the Netherlands, from ..."
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Clauses : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 375 iii Foreword This document is the preliminary proceedings of the workshop of the Esprit Basic Research Project 6453 "Types for Proofs and Programs" held at the University of Nijmegen, the Netherlands, from May 24th until May 28th 1993. The workshop was organised by Henk Barendregt and Herman Geuvers. Local arrangements were made by Marielle van der Zandt, Erik Barendsen, Herman Geuvers and Mark Ruys. These proceedings have been collected from L a T E X sources, using email. It contains 22 papers from the 35 talks that were presented at the workshop. Very useful support in solving the L a T E X puzzles was provided by Erik Barendsen. This document can be obtained by anonymous ftp from the University of Nijmegen: Type ftp ftp.cs.kun.nl anonymous (as login) [your email address] (as password) cd /pub/csi/CompMath/Types bin get NijmegenTypes.ps.Z bye iv Workshop Programme Types for Proofs an...