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13
Proof Normalization Modulo
, 1998
"... We consider a class of logical formalisms, in which firstorder logic is extended by identifying propositions modulo a given congruence. We particularly focus on the case where this congruence is induced by a confluent and terminating rewrite system over the propositions. This extension enhances the ..."
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Cited by 46 (17 self)
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We consider a class of logical formalisms, in which firstorder logic is extended by identifying propositions modulo a given congruence. We particularly focus on the case where this congruence is induced by a confluent and terminating rewrite system over the propositions. This extension enhances the power of firstorder logic and various formalisms, including higherorder logic, can be described in this framework. We conjecture that proof normalization and logical consistency always hold over this class of formalisms, provided some minimal conditions over the rewrite system are fulfilled. We prove this conjecture for some subcases, including higherorder logic. At last, we extend these results to classical sequent calculus.
Deciding Type Equivalence in a Language with Singleton Kinds
 In TwentySeventh ACM Symposium on Principles of Programming Languages
, 2000
"... Work on the TILT compiler for Standard ML led us to study a language with singleton kinds: S(A) is the kind of all types provably equivalent to the type A. Singletons are interesting because they provide a very general form of definitions for type variables, allow finegrained control of type comput ..."
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Cited by 39 (6 self)
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Work on the TILT compiler for Standard ML led us to study a language with singleton kinds: S(A) is the kind of all types provably equivalent to the type A. Singletons are interesting because they provide a very general form of definitions for type variables, allow finegrained control of type computations, and allow many equational constraints to be expressed within the type system.
Extension of MartinLöf's Type Theory with Record Types and Subtyping
, 1998
"... this paper, the implementation has been used to verify an abstract version of sorting by insertion in (Tasistro 1997). In this latter work, dependent record types are used to express speciøcations of abstract data types. The theory here developed is a direct successor of the calculus of substitution ..."
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Cited by 24 (3 self)
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this paper, the implementation has been used to verify an abstract version of sorting by insertion in (Tasistro 1997). In this latter work, dependent record types are used to express speciøcations of abstract data types. The theory here developed is a direct successor of the calculus of substitutions for type theory (MartinL#f 1992; Tasistro 1997) in the sense that record types can be seen as type constructions corresponding to contexts of variables ¯record objects becoming then the counterpart to substitutions. Several theories of records have been developed in the context of systems without dependent types, mainly with the motivation of providing foundations for concepts that appear in object oriented programming. Then, for instance, there is by now a standard way of encoding objects in the sense of object oriented programming as recursively deøned records. The general motivation mentioned departs from ours, which, as far as the theory of programming is concerned, is limited to that of providing basic means that allow the use of dependent types for expressing speciøcations of abstract data types and modules in a general way. The problem of formulating a type system for object oriented programming raises a number of questions that are simply not relevant for our purposes. As to dependent record types, they have been implemented in PVS (Owre et al. 1993), which is a theorem proving system based on classical higher order logic. The subtyping that record types induce is, however, not a part of this implementation. In the original type theory, it is possible to encode each particular instance of inclusion between types ff and fi by using a coercion function that injects the objects of type ff into the type fi. In (Barthe 1996; Bailey 1996; Sa#bi 1997) different mechanisms...
XML and Multilingual Document Authoring: Convergent Trends
 IN COLING
, 2000
"... Typical approaches to XML authoring view a XML document as a mixture o1' structure (tile tags) and surt'acc (text between the tags). We advocalc a radical appr(>ach where the surface disappears from lhc XML document altogether to be Dandled exclusively by rendering mech anisns. This move is based o ..."
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Cited by 20 (13 self)
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Typical approaches to XML authoring view a XML document as a mixture o1' structure (tile tags) and surt'acc (text between the tags). We advocalc a radical appr(>ach where the surface disappears from lhc XML document altogether to be Dandled exclusively by rendering mech anisns. This move is based on the view that lhc author's choices when authoring XML documents are best seen as languageneutral semantic decisions, that the strucltll 'O Cttll then be viewed as inletlingual content, tllct that the textual OUll)Ut should be derived lom this content by languagcspccilic realization mechanisms. thus assimilating XMI. authoring lo Multilingual l)ocuncnt Authoring. However; standard XMI. tools have imporlant limitations when used IBr such a purpose: (1) they al'e weak at propagttting semantic dependencies belwecn dil'lrcnt parts of the structure, and, (2) current XMI rendering tools arc illsuited for handling the grammatical combil atiol of lcxttal units. We present two related proposals for overcoming these limitations: one ((;1:) oriinatin in lhc tradition of mathematical proof cctitcrs and conslructivc type llleory, the other (l(J), a specialiT. atio off l)clinilc Clause G'allmlt's strongly inspired by GF.
Program Development in Constructive Type Theory
 Theoretical Computer Science
, 1992
"... We present the program development concept in a logical framework including constructive type theory and then show how to use such theories to derive programs from proofs of formal specifications. We are interested in two important facts that are the mechanization of the proof construction and the p ..."
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Cited by 4 (2 self)
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We present the program development concept in a logical framework including constructive type theory and then show how to use such theories to derive programs from proofs of formal specifications. We are interested in two important facts that are the mechanization of the proof construction and the possibility to express in the theory significiant concepts for programming (like inductively deøned types and general recursion). We give here a survey on some results and problems appearing in logical frameworks devoted to the programming with proofs approach.
Weak Transitivity in Coercive Subtyping
 TYPES FOR PROOFS AND PROGRAMS, VOLUME 2646 OF LNCS
, 2001
"... Coercive subtyping is a general approach to subtyping, inheritance and abbreviation in dependent type theories. A vital requirement for coercive subtyping is that of coherence which essentially says that coercions between any two types must be unique. Another important task for coercive subtyping is ..."
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Cited by 4 (4 self)
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Coercive subtyping is a general approach to subtyping, inheritance and abbreviation in dependent type theories. A vital requirement for coercive subtyping is that of coherence which essentially says that coercions between any two types must be unique. Another important task for coercive subtyping is to prove the admissibility or elimination of transitivity and substitution. In this paper, we propose and study the notion of Weak Transitivity, consider suitable subtyping rules for certain parameterised inductive types and prove its coherence and the admissibility of substitution and weak transitivity in the coercive subtyping framework.
Maximal and Partial Points in Formal Spaces
, 2002
"... this paper that if a formal topology S is setpresented, in the sense of Aczel, and has only maximal points, then the collection of its points Pt(S) is isomorphic to a set. This generalises a result by Curi (2001) on setrepresentability. The crucial result (Theorem 4.3) is that any suitably small su ..."
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Cited by 2 (0 self)
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this paper that if a formal topology S is setpresented, in the sense of Aczel, and has only maximal points, then the collection of its points Pt(S) is isomorphic to a set. This generalises a result by Curi (2001) on setrepresentability. The crucial result (Theorem 4.3) is that any suitably small subset of a point can be extended to a point picked from a prescribed small power set. This power set depends only on the data of the formal space. The proof involves what seems to be a new choice principle (Theorem 4.2). It is a generalisation of dependent choice from natural numbers to Wtypes and is provable in type theory
Implementation of Intuitionistic Type Theory and Realizability Theory
, 1995
"... Writing correct programming code is necessary in computer system development, where complete testing is not possible. Intuitionistic type theory leads to a mechanical generation of correct code by using specifications. The idea is that the specification of a program is its type, and the specificatio ..."
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Cited by 2 (0 self)
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Writing correct programming code is necessary in computer system development, where complete testing is not possible. Intuitionistic type theory leads to a mechanical generation of correct code by using specifications. The idea is that the specification of a program is its type, and the specification can be expressed by logical statements called wellformed formulas (wffs) and therefore proved by using mathematical axioms and inference rules of logic. Then, using the correspondences propositions are types are specifications and proofs are programs are values [16], a proof can be translated into a correct programming code. The fundamental idea of realizability theory is that a proof can be translated into not only correct, but also minimal programming code, which contains only computational values. Based on these theories, a realizability algorithm developed by John Hatcliff defines how the translation can be done. We analyzed Hatcliff's algorithm and implemented it in a system. System ...
Reflections On Formalism And Reductionism In Logic And Computer Science
"... This report contains a preprint (paper 1) and a reprint (paper 2). The first develops some epistemological views which were hinted in the second, in particular by stressing the need of a greater role of geometric insight and images in foundational studies and in approaches to cognition. The second p ..."
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This report contains a preprint (paper 1) and a reprint (paper 2). The first develops some epistemological views which were hinted in the second, in particular by stressing the need of a greater role of geometric insight and images in foundational studies and in approaches to cognition. The second paper is the "philosophical" part of a lecture in Type Theory, whose technical sections, omitted here, have been largely subsumed by subsequent publications (see references). The part reprinted below discusses more closely some historical remarks recalled in paper 1. 1. Reflections on formalism and reductionism in Logic and Computer Science (pp. 1  9)