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 domain-theoretic 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 everywh...

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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,

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.

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 non-eoeective 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

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 set-theoretic 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 so-called “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.

We propose a notion of uniform ideal (certain Scott-closed 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 doubly-lifted 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 H-strict (Wadler an...

In this paper we give a review of the Plotkin powerdomain construction over algebraic cpo's. Algebraic cpo's are cpo's that are completely determined by their collection of finite elements. We show how one can build an algebraic cpo out of any pre-ordered set using the method of chain completion. We apply this method to define the powerdomain over an algebraic cpo. We then show how to interpret the powerdomain as a subset of the powerset of the base domain together with a suit- able ordering relation. This ordering relation is the Plotkin order and it extends the Egli-Milner order. We give a necessary and sufficient condition on the base domain to ensure that the Plotkin order and the Egli-Milner order coincide. We also show how one can construct continuous functions between powerdomains out of functions between the underlying base domains. It follows that the powerdomain construction is a continuous functor on the category of algebraic cpo's.

Unlike a uniformity, a quasi-uniformity is not determined by its quasiuniform covers. However, a classical construction, due to Fletcher, which assigns a transitive quasi-uniformity to each family of interior-preserving open covers, allows to describe all transitive quasi-uniformities on the topological spaces in terms of those families of covers. In this paper we develop a pointfree generalization of this, which solves a problem posed by G. C. L. Brümmer, together with various examples and applications that illustrate its remarkable usefulness. By this construction, many kinds of interiorpreserving open covers (e.g. locally finite, open spectrum, well-monotone) induce compatible quasi-uniformities on an arbitrary frame.

Abstract. We present a flexible approach for extracting hierarchical classifications from data, which employs the logic of affirmative assertions. The basic observation is that each set of rules induced by the data canonically determines a classificational hierarchy. We give a characterization of how the chosen rule type affects the structure of the induced hierarchy. Moreover, we show how our approach is related to Formal Concept Analysis. The framework is then applied to the induction of hierarchical classifications from an amino acid database. Based on this example, the pros and cons of several types of hierarchies are discussed with respect to criteria such as compactness of representation, suitability for inference tasks, and intelligibility for the human user. 1