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The type theoretic interpretation of constructive set theory: inductive definitions
 Logic, Methodology and Philosophy of Science VII
, 1986
"... Abstract. We present a generalisation of the typetheoretic interpretation of constructive set theory into MartinLöf type theory. The original interpretation treated logic in MartinLöf type theory via the propositionsastypes interpretation. The generalisation involves replacing MartinLöf t ..."
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Abstract. We present a generalisation of the typetheoretic interpretation of constructive set theory into MartinLöf type theory. The original interpretation treated logic in MartinLöf type theory via the propositionsastypes interpretation. The generalisation involves replacing MartinLöf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in type theories allows us to study reinterpretations of logic, such as the doublenegation translation.
A uniform presentation of suplattices, quantales and frames by means of in preordered sets, pretopologies and formal topologies , Preprint no. 19 of Dipartimento di Matematica P. e Appl. , Universita di Padova
, 1993
"... We introduce the notion of innitary preorder and use it to obtain a predicative presentation of suplattices by generators and relations. The method is uniform in that it extends in a modular way to obtain a presentation of quantales, as \suplattices on monoids", by using the notion of pretopo ..."
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Cited by 21 (5 self)
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We introduce the notion of innitary preorder and use it to obtain a predicative presentation of suplattices by generators and relations. The method is uniform in that it extends in a modular way to obtain a presentation of quantales, as \suplattices on monoids", by using the notion of pretopology. Our presentation is then applied to frames, the link with Johnstone's presentation of frames is spelled out, and his theorem on freely generated frames becomes a special case of our results on quantales. The main motivation of this paper is to contribute to the development of formal topology. That is why all our denitions and proofs can be expressed within an intuitionistic and predicative foundation, like constructive type theory.
Programming interfaces and basic topology
 Annals of Pure and Applied Logic
, 2005
"... A pattern of interaction that arises again and again in programming, is a “handshake”, in which two agents exchange data. The exchange is thought of as provision of a service. Each interaction is initiated by a specific agent —the client or Angel, and concluded by the other —the server or Demon. We ..."
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Cited by 9 (2 self)
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A pattern of interaction that arises again and again in programming, is a “handshake”, in which two agents exchange data. The exchange is thought of as provision of a service. Each interaction is initiated by a specific agent —the client or Angel, and concluded by the other —the server or Demon. We present a category in which the objects —called interaction structures in the paper — serve as descriptions of services provided across such handshaken interfaces. The morphisms —called (general) simulations— model components that provide one such service, relying on another. The morphisms are relations between the underlying sets of the interaction structures. The proof that a relation is a simulation can serve (in principle) as an executable program, whose specification is that it provides the service described by its domain, given an implementation of the service described by its codomain.
Presenting dcpos and dcpo algebras
 Proceedings of the 24th Conference on the Mathematical Foundations of Programming Semantics (MFPS XXIV), Electronic Notes in Theoretical Computer Science
"... Dcpos can be presented by a preorder of generators and inequational relations expressed as covers. Algebraic operations on the generators (possibly with their results being ideals of generators) can be extended to the dcpo presented, provided the covers are “stable ” for the operations. The resultin ..."
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Cited by 9 (1 self)
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Dcpos can be presented by a preorder of generators and inequational relations expressed as covers. Algebraic operations on the generators (possibly with their results being ideals of generators) can be extended to the dcpo presented, provided the covers are “stable ” for the operations. The resulting dcpo algebra has a natural universal characterization and satisfies all the inequational laws satisfied by the generating algebra. Applications include known “coverage theorems ” from locale theory. 1
Recursion on the partial continuous functionals
 Logic Colloquium ’05
, 2006
"... We describe a constructive theory of computable functionals, based on the partial continuous functionals as their intendend domain. Such a task had long ago been started by Dana Scott [28], under the wellknown abbreviation ..."
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Cited by 7 (5 self)
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We describe a constructive theory of computable functionals, based on the partial continuous functionals as their intendend domain. Such a task had long ago been started by Dana Scott [28], under the wellknown abbreviation
Localic completion of generalized metric spaces I
, 2005
"... Abstract. Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero selfdistance law and the triangle inequality. We describe a complet ..."
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Cited by 7 (0 self)
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Abstract. Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero selfdistance law and the triangle inequality. We describe a completion of gms’s by Cauchy filters of formal balls. In terms of Lawvere’s approach using categories enriched over [0, ∞], the Cauchy filters are equivalent to flat left modules. The completion generalizes the usual one for metric spaces. For quasimetrics it is equivalent to the Yoneda completion in its netwise form due to Künzi and Schellekens and thereby gives a new and explicit characterization of the points of the Yoneda completion. Nonexpansive functions between gms’s lift to continuous maps between the completions. Various examples and constructions are given, including finite products. The completion is easily adapted to produce a locale, and that part of the work is constructively valid. The exposition illustrates the use of geometric logic to enable pointbased reasoning for locales. 1.
On some peculiar aspects of the constructive theory of pointfree spaces
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Spatiality for formal topologies
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2006
"... We define what it means for a formal topology to be spatial, and investigate properties related to spatiality both in general and in examples. ..."
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Cited by 5 (2 self)
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We define what it means for a formal topology to be spatial, and investigate properties related to spatiality both in general and in examples.