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24
Propositions as [Types]
, 2001
"... Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevanc ..."
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Cited by 38 (3 self)
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Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics using regular categories. We show that dependent type theory with the unit type, strong extensional equality types, strong dependent sums, and bracket types is the internal type theory of regular categories, in the same way that the usual dependent type theory with dependent sums and products is the internal type theory of locally cartesian closed categories. We also show how to interpret rstorder logic in type theory with brackets, and we make use of the translation to compare type theory with logic. Specically, we show that the propositionsastypes interpretation is complete with respect to a certain fragment of intuitionistic rstorder logic. As a consequence, a modied doublenegation translation into type theory (without bracket types) is complete for all of classical rstorder logic.
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.
A minimalist twolevel foundation for constructive mathematics
, 2008
"... We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other lev ..."
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Cited by 19 (7 self)
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We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model. This twolevel theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the “proofsasprograms” paradigm and acts as a programming language.
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 11 (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.
A modular typechecking algorithm for type theory with singleton types and proof irrelevance
 IN TLCA’09, VOLUME 5608 OF LNCS
, 2009
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Formal Topologies on the Set of FirstOrder Formulae
 Journal of Symbolic Logic
, 1998
"... this paper that the question has a simple negative answer. This raised further natural questions on what can be said about the points of these two topologies; we give some answers. The observation that topological models for firstorder theories can expressed in the framework of locales appears, for ..."
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Cited by 5 (2 self)
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this paper that the question has a simple negative answer. This raised further natural questions on what can be said about the points of these two topologies; we give some answers. The observation that topological models for firstorder theories can expressed in the framework of locales appears, for instance, in Fourman and Grayson [6], where the analogy between points of a locale and models of a theory is emphasised; the identification of formal points with Henkin sets, gives a precise form to this analogy. We replace the use of locales by formal topology, which can be expressed in a predicative framework such as MartinLof's type theory. Prooftheoretic issues are also considered by Dragalin [4], who presents a topological completeness proof using only finitary inductive definitions. Palmgren and Moerdijk [10] is also concerned with constructions of models: using sheaf semantics, they obtain a stronger conservativity result than the one in [3]. We will first investigate the difference between the DedekindMacNeille cover and the inductive cover. It easy to see that \Delta DM is stronger than \Delta I , that is, OE \Delta I U implies OE \Delta DM U , but the converse does not hold in general. The notion of point is not primitive in formal topology and therefore it is natural to require that a formal topology has some notion of positivity defined on the basic neighbourhoods; that a neighbourhood is positive then corresponds to, in ordinary point based topology, that it is inhabited by some point. We will show several negative results on positivity, both for the inductive topology and the DedekindMacNeille topology. The points of an inductive topology correspond to Henkin sets, but the DedekindMacNeille topology has, in general, no points. Our reasoning is constructi...
Steps Towards a Dynamic Constructivism
"... this paper is necessarily an overview. I hope in the future to find the energy to write up a complete version, which will include all the details, arguments, connections and references omitted here. Also, if my exposition is somewhat ideological and my language nontechnical, it is because my goal is ..."
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Cited by 5 (1 self)
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this paper is necessarily an overview. I hope in the future to find the energy to write up a complete version, which will include all the details, arguments, connections and references omitted here. Also, if my exposition is somewhat ideological and my language nontechnical, it is because my goal is to take the initial steps towards a new conception.