A connection between computer science and fuzzy theory: midpoints and running time of computing

Following the mathematical formalism introduced by M. Schellekens [Elec- tronic Notes in Theoret. Comput. Sci. 1 (1995), 211-232] in order to give a common foundation for Denotational Semantics and Complexity Analysis, we obtain an application of the theory of midpoints for asymmetric distances de ned between fuzzy sets to the complexity analysis of algorithms and pro- grams. In particular we show that the average running time for the algorithm known as Largetwo is exactly a midpoint between the best and the worst case running time of computingPeer Reviewe

Uniqueness of fixpoints of single-step operators determined by Belnap's four-valued logic

Recently, Hitzler and Seda showed how a domain-theoretic proof can be given of the fact that, for a locally hierarchical program, the single-step operator TP , de�ned in two-valued logic, has a unique �xed point. Their approach employed a construction which turned a ScottErshov domain into a generalized ultrametric space. Finally, a �xed-point theorem of PriessCrampe and Ribenboim was applied to TP to establish the result. In this paper, we extend these methods and results to the corresponding well-known single-step operators �P and P determined by P and de�ned, respectively, in three-valued and four-valued logic

Metric 1-spaces

A generalization of metric space is presented which is shown to admit a theory strongly related to that of ordinary metric spaces. To avoid the topological effects related to dropping any of the axioms of metric space, first a new, and equivalent, axiomatization of metric space is given which is then generalized from a fresh point of view. Naturally arising examples from metric geometry are presented

Fixpoint semantics for logic programming a survey

AbstractThe variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and many-valued logic, lattice theory, game theory, and topology. One source of this richness is the inherent non-monotonicity of its negation, something that does not have close parallels with the machinery of other programming paradigms. Nonetheless, much of the work on logic programming semantics seems to exist side by side with similar work done for imperative and functional programming, with relatively minimal contact between communities. In this paper we summarize one variety of approaches to the semantics of logic programs: that based on fixpoint theory. We do not attempt to cover much beyond this single area, which is already remarkably fruitful. We hope readers will see parallels with, and the divergences from the better known fixpoint treatments developed for other programming methodologies

Hyperconvex hulls in catergories of quasi-metric spaces

Includes bibliographical references.Isbell showed that every metric space has an injective hull, that is, every metric space has a “minimal” hyperconvex metric superspace. Dress then showed that the hyperconvex hull is a tight extension. In analogy to Isbell’s theory Kemajou et al. proved that each T₀-quasi-metric space X has a q-hyperconvex hull QX , which is joincompact if X is joincompact. They called a T₀-quasi-metric space q-hyperconvex if and only if it is injective in the category of T₀-quasi-metric spaces and non-expansive maps. Agyingi et al. generalized results due to Dress on tight extensions of metric spaces to the category of T₀-quasi-metric spaces and non-expansive maps. In this dissertation, we shall study tight extensions (called uq-tight extensions in the following) in the categories of T₀-quasi-metric spaces and T₀-ultra-quasimetric spaces. We show in particular that most of the results stay the same as we move from T₀-quasi-metric spaces to T₀-ultra-quasi-metric spaces. We shall show that these extensions are maximal among the uq-tight extensions of the space in question. In the second part of the dissertation we shall study the q-hyperconvex hull by viewing it as a space of minimal function pairs. We will also consider supseparability of the space of minimal function pairs. Furthermore we study a special subcollection of bicomplete supseparable quasi-metric spaces: bicomplete supseparable ultra-quasi-metric spaces. We will show the existence and uniqueness (up to isometry) of a Urysohn Γ-ultra-quasi-metric space, for an arbitrary countable set Γ of non-negative real numbers including 0