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11
Universal Profinite Domains
 Information and Computation
, 1987
"... . We introduce a bicartesian closed category of what we call profinite domains. Study of these domains is carried out through the use of an equivalent category of preorders in a manner similar to the information systems approach advocated by Dana Scott and others. A class of universal profinite dom ..."
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. We introduce a bicartesian closed category of what we call profinite domains. Study of these domains is carried out through the use of an equivalent category of preorders in a manner similar to the information systems approach advocated by Dana Scott and others. A class of universal profinite domains is defined and used to derive sufficient conditions for the profinite solution of domain equations involving continuous operators. As a special instance of this construction, a universal domain for the category SFP is demonstrated. Necessary conditions for the existence of solutions for domain equations over the profinites are also given and used to derive results about solutions of some equations. A new universal bounded complete domain is also demonstrated using an operator which has bounded complete domains as its fixed points. 1 Introduction. For our purposes a domain equation has the form X ¸ = F (X) where F is an operator on a class of semantic domains (typically, F is an endof...
Completeness of quasiuniform and syntopological spaces
 J. London Math. Soc
, 1994
"... In this paper we begin to develop the filter approach to (completeness of) quasiuniform spaces, proposed in [8, Section V]. It will be seen that this permits a more powerful and elegant account of completion to be given than was feasible using sequences or nets [8]. ..."
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In this paper we begin to develop the filter approach to (completeness of) quasiuniform spaces, proposed in [8, Section V]. It will be seen that this permits a more powerful and elegant account of completion to be given than was feasible using sequences or nets [8].
A stable programming language
 I&C
"... It is wellknown that stable models (as dIdomains, qualitative domains and coherence spaces) are not fully abstract for the languagePCF. This fact is related to the existence of stable parallel functions and of stable functions that are not monotone with respect to the extensional order, which cann ..."
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Cited by 5 (2 self)
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It is wellknown that stable models (as dIdomains, qualitative domains and coherence spaces) are not fully abstract for the languagePCF. This fact is related to the existence of stable parallel functions and of stable functions that are not monotone with respect to the extensional order, which cannot be defined by programs ofPCF. In this paper, a paradigmatic programming language namedStPCF is proposed, which extends the languagePCF with two additional operators. The operational description of the extended language is presented in an effective way, although the evaluation of one of the new operators cannot be formalized in a PCFlike rewrite system. SinceStPCF can define all finite cliques of coherence spaces the above gap with stable models is filled, consequently stable models are fully abstract for the extended language. 1
Computer theorem proving in math
, 2004
"... Abstract—We give an overview of issues surrounding computerverified theorem proving in the standard puremathematical context. This is based on my talk at the PQR conference (Brussels, June 2003). ..."
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Abstract—We give an overview of issues surrounding computerverified theorem proving in the standard puremathematical context. This is based on my talk at the PQR conference (Brussels, June 2003).
LOCATEDNESS AND OVERT SUBLOCALES
, 2009
"... Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in constructive locale theory. We show that the two notions are intimately connected. Bishop defines a ..."
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Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in constructive locale theory. We show that the two notions are intimately connected. Bishop defines a metric space to be compact if it is complete and totally bounded. A subset of a totally bounded set is again totally bounded iff it is located. So a closed subset of a Bishop compact set is Bishop compact iff it is located. We translate this result to formal topology. ‘Bishop compact ’ is translated as compact and overt. We propose a definition of located predicate on subspaces in formal topology. We call a sublocale located if it can be presented by a formal topology with a located predicate. We prove that a closed sublocale of a compact regular locale has a located predicate iff it is overt. Moreover, a Bishopclosed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt. Finally, we show by elementary methods that the points of the Vietoris locale of a compact regular locale are precisely its compact overt sublocales. We work constructively, predicatively and avoid the use of the axiom of countable choice. Consequently, all our results are valid in any predicative topos.
A Presentation Of The Initial LiftAlgebra
 Journal of Pure and Applied Algebra
, 1997
"... The object of study of the present paper may be considered as a model, in an elementary topos with a natural numbers object, of a nonclassical variation of the Peano arithmetic. The new feature consists in admitting, in addition to the constant (zero) s0 2 N and the unary operation (the success ..."
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The object of study of the present paper may be considered as a model, in an elementary topos with a natural numbers object, of a nonclassical variation of the Peano arithmetic. The new feature consists in admitting, in addition to the constant (zero) s0 2 N and the unary operation (the successor map) s1 : N ! N, arbitrary operations su : N u ! N of arities u `between 0 and 1'. That is, u is allowed to range over subsets of a singleton set.
Archive for Mathematical Logic manuscript No. (will be inserted by the editor) Topological Characterization of Scott Domains
"... The date of receipt and acceptance will be inserted by the editor Abstract First we introduce the notion of supercoherent topology which does not depend on any ordering. Then we show that a topology is supercoherent if and only if it is the Scott topology over a suitable algebraic dcpo. The main i ..."
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The date of receipt and acceptance will be inserted by the editor Abstract First we introduce the notion of supercoherent topology which does not depend on any ordering. Then we show that a topology is supercoherent if and only if it is the Scott topology over a suitable algebraic dcpo. The main ideas of the paper are a byproduct of the constructive approach to domain theory through information bases which we have proposed in a previous work, but the presentation here does not rely on that foundational framework. Key words Scott Domain – Algebraic dcpo – Formal Topology 1
A Cartesian Closed Category in MartinLöf’s intuitionistic type theory
, 2001
"... First, we briefly recall the main definitions of the theory of Information Bases and Translations. These mathematical structures are the basis to ..."
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First, we briefly recall the main definitions of the theory of Information Bases and Translations. These mathematical structures are the basis to