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32
Power domains and iterated function systems
 Information and Computation
, 1996
"... We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domaintheoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniquene ..."
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Cited by 37 (11 self)
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We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domaintheoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniqueness of the attractor of a weakly hyperbolic IFS and the invariant measure of a weakly hyperbolic IFS with probabilities, extending the classic results of Hutchinson for hyperbolic IFSs in this more general setting. We also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure, extending the corresponding algorithms for hyperbolic IFSs. We then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent IFS and obtain an algorithm to generate the invariant distribution on the digitised screen. The generalised Riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution. For hyperbolic recurrent IFSs and Lipschitz maps, one can estimate the integral up to any threshold of accuracy.] 1996 Academic Press, Inc. 1.
The regularlocallycompact coreflection of stably locally compact locale
 Journal of Pure and Applied Algebra
, 2001
"... The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally comp ..."
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Cited by 21 (8 self)
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The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally compact locales and perfect maps,
Multi Lingual Sequent Calculus and Coherent Spaces
 Fundamenta Informaticae
, 1997
"... We study a Gentzen style sequent calculus where the formulas on the left and right of the turnstile need not necessarily come from the same logical system. Such a sequent can be seen as a consequence between different domains of reasoning. We discuss the ingredients needed to set up the logic ge ..."
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Cited by 18 (8 self)
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We study a Gentzen style sequent calculus where the formulas on the left and right of the turnstile need not necessarily come from the same logical system. Such a sequent can be seen as a consequence between different domains of reasoning. We discuss the ingredients needed to set up the logic generalized in this fashion.
Computational Content of Classical Logic
 SEMANTICS AND LOGICS OF COMPUTATION
, 1996
"... This course is an introduction to the research trying to connect the proof theory of classical logic and computer science. We omit important and standard topics, among them the connection between the computational interpretation of classical logic and the programming operator callcc. Instead, here ..."
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Cited by 16 (0 self)
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This course is an introduction to the research trying to connect the proof theory of classical logic and computer science. We omit important and standard topics, among them the connection between the computational interpretation of classical logic and the programming operator callcc. Instead, here we put the emphasis on actual mathematical examples. We analyse the following questions: what can be the meaning of a noneoeective proof of an existential statement, a statement that claims the existence of a nite object that satises a decidable property? Is it clear that a noneoeective proof has a meaning at all? Can we always say that this proof contains implicitly, if not explicitly, some eoeective witness? Is this witness unique? By putting the emphasis on actual mathematical examples, we follow Gentzen who founded natural deduction by analysing concrete mathematical examples, like Euclid's proof of the innity of prime numbers. We
C ∗ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED KTHEORY
, 810
"... Abstract. We define the filtrated Ktheory of a C ∗algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated Ktheory. For finite spaces with totally ordered lattice of open subsets, this spect ..."
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Cited by 14 (0 self)
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Abstract. We define the filtrated Ktheory of a C ∗algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated Ktheory. For finite spaces with totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated Ktheory is not yet a complete invariant. We describe two C ∗algebras over a space X with four points that have isomorphic filtrated Ktheory without being KK(X)equivalent. For this space X, we enrich filtrated Ktheory by another Ktheory functor to a complete invariant up to KK(X)equivalence that satisfies a Universal Coefficient Theorem. 1.
Tropological systems are points of quantales
 J. Pure Appl. Algebra
, 2002
"... We address two areas in which quantales have been used. One is of a topological nature, whereby quantales or involutive quantales are seen as generalized noncommutative spaces, and its main purpose so far has been to investigate the spectrum of noncommutative C*algebras. The other sees quantales as ..."
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Cited by 13 (7 self)
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We address two areas in which quantales have been used. One is of a topological nature, whereby quantales or involutive quantales are seen as generalized noncommutative spaces, and its main purpose so far has been to investigate the spectrum of noncommutative C*algebras. The other sees quantales as algebras of abstract experiments on physical or computational systems, and has been applied to the study of the semantics of concurrent systems. We investigate connections between the two areas, in particular showing that concurrent systems, in the form of either settheoretic or localic tropological systems, can be identified with points of quantales by means of a suitable adjunction, which indeed holds for a much larger class of socalled “tropological models”. We show that in the case of tropological models in factor quantales, which still generalize tropological systems, the identification of models and (generalized) points preserves all the information needed for describing the observable behaviour of systems. We also define a notion of morphism of models that generalizes previous definitions of morphism of systems, and show that morphisms, too, can be defined in terms of either side of the adjunction, in fact giving us isomorphisms of categories. The relation between completeness notions for tropological systems and spatiality for quantales is also addressed, and a preliminary partial preservation result is obtained.
A Categorical Outlook on Relational Modalities and Simulations
, 2002
"... We characterise bicategories of spans, relations and partial maps universally in terms of factorisations involving maps . We apply this characterisation to show that the standard modalities 2 and arise canonically as the extension of a predicate logic from functions to (abstract) relations . ..."
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Cited by 8 (2 self)
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We characterise bicategories of spans, relations and partial maps universally in terms of factorisations involving maps . We apply this characterisation to show that the standard modalities 2 and arise canonically as the extension of a predicate logic from functions to (abstract) relations .
Localic suplattices and tropological systems
 THEORETICAL COMPUTER SCIENCE
, 2003
"... The approach to process semantics using quantales and modules is topologized by considering tropological systems whose sets of states are replaced by locales and which satisfy a suitable stability axiom. A corresponding notion of localic suplattice (algebra for the lower powerlocale monad) is descri ..."
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Cited by 7 (3 self)
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The approach to process semantics using quantales and modules is topologized by considering tropological systems whose sets of states are replaced by locales and which satisfy a suitable stability axiom. A corresponding notion of localic suplattice (algebra for the lower powerlocale monad) is described, and it is shown that there are contravariant functors from suplattices to localic suplatices and, for each quantale Q, from left Qmodules to localic right Qmodules. A proof technique for third completeness due to Abramsky and Vickers is reset constructively, and an example of application to failures semantics is given.
Uniform Ideals and Strictness Analysis
 In Proc. 18th Int'l Coll. on Automata, Languages and Programming (ICALP
, 1991
"... We propose a notion of uniform ideal (certain Scottclosed sets) to characterise strictness properties. This enables us to explain why Hughes' and Wadler's H projection for lazy list strictness analysis is not in general expressible as an abstract interpretation property of the standard se ..."
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Cited by 6 (2 self)
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We propose a notion of uniform ideal (certain Scottclosed sets) to characterise strictness properties. This enables us to explain why Hughes' and Wadler's H projection for lazy list strictness analysis is not in general expressible as an abstract interpretation property of the standard semantics. We give circumstances when it is so expressible. Doing so casts light on Burn's HB projection and his question of its relationship to H. Uniform ideals are a generalisation of the sets of values corresponding to types in (simple) polymorphic type systems. Wadler's doublylifted abstract domain constructor for lazy lists can be seen as a special case which only uses certain uniform ideals. The conuence of strictness and type theory furthers Kuo and Mishra's notion of \strictness types". Summary of results We characterise strictness properties as uniform ideals. This enables us to give abstract interpretation properties to show that a function on list(t 1 +t 2 ) is Hstrict (Wadler an...