Oscar Valero

A quantitative computational model for complete partial metric spaces via formal balls

Given a partial metric space (X, p), we use (BX, ⊑dp) to denote the poset of formal balls of the associated quasi-metric space (X, dp). We obtain characterisations of complete partial metric spaces and sup-separable complete partial metric spaces in terms of domain-theoretic properties of (BX, ⊑dp). In particular, we prove that a partial metric space (X, p) is complete if and only if the poset (BX, ⊑dp) is a domain. Furthermore, for any complete partial metric space (X, p), we construct a Smyth complete quasi-metric q on BX that extends the quasi-metric dp such that both the Scott topology and the partial order ⊑dp are induced by q. This is done using the partial quasi-metric concept recently introduced and discussed by H. P. Kunzi, H. Pajoohesh and M. P. Schellekens (Kunzi et al. 2006). Our approach, which is inspired by methods due to A. Edalat and R. Heckmann (Edalat and Heckmann 1998), generalises to partial metric spaces the constructions given by R. Heckmann (Heckmann 1999) and J. J. M. M. Rutten (Rutten 1998) for metric spaces.

Denotational semantics for programming languages, balanced quasi-metrics and fixed points

Abstract A new mathematical model is introduced for the study of the domain of words. We do it by means of the introduction of a suitable balanced quasi-metric on the set of all words over an alphabet. It will be shown that this construction has better quasi-metric and topological properties than several classical constructions. We also prove a fixed point theorem which allows us to develop an application for the study of probabilistic divide and conquer algorithms.

Domain theoretic characterisations of quasi-metric completeness in terms of formal balls†

We characterise those quasi-metric spaces (X, d) whose poset BX of formal balls satisfies the condition (*) \begin{linenomath}\begin{equation} \text{for every } (x,r),(y,s)\in \mathbf{B}X,\ (x,r)\ll (y,s)\Leftrightarrow d(x,y)X, d) is Smyth-complete if and only if BX is a dcpo satisfying condition (*). We also give characterisations in terms of formal balls for sequentially Yoneda complete quasi-metric spaces and for Yoneda complete T1 quasi-metric spaces. Finally, we discuss several properties of the Heckmann quasi-metric on the formal balls of any quasi-metric space.

On the structure of the space of complexity partial functions

Dual complexity spaces were introduced by Romaguera and Schellekens in order to obtain a robust mathematical model for the complexity analysis of algorithms and programs. This model is based on the notions of a cone and of a quasi-metric space. Later on, the structure of the dual complexity spaces was modified with the purpose of giving quantitative measures of the improvements in the complexity of programs. This new complexity structure was presented as an ordered cone endowed with an invariant extended quasi-metric. Here we construct a general dual complexity space by using (complexity) partial functions. This new complexity structure is a pointed ordered cone endowed with a subinvariant bicomplete extended quasi-metric as complexity distance. We apply this approach to modelling certain processes that arise, in a natural way, in symbolic computation.

The Baire Partial Quasi-Metric Space: A Mathematical Tool for Asymptotic Complexity Analysis in Computer Science

In 1994, S.G. Matthews introduced the notion of partial metric space in order to obtain a suitable mathematical tool for program verification (Ann. N.Y. Acad. Sci. 728:183–197, 1994). He gave an application of this new structure to parallel computing by means of a partial metric version of the celebrated Banach fixed point theorem (Theor. Comput. Sci. 151:195–205, 1995). Later on, M.P. Schellekens introduced the theory of complexity (quasi-metric) spaces as a part of the development of a topological foundation for the asymptotic complexity analysis of programs and algorithms (Electron. Notes Theor. Comput. Sci. 1:211–232, 1995). The applicability of this theory to the asymptotic complexity analysis of Divide and Conquer algorithms was also illustrated by Schellekens. In particular, he gave a new proof, based on the use of the aforenamed Banach fixed point theorem, of the well-known fact that Mergesort algorithm has optimal asymptotic average running time of computing. In this paper, motivated by the utility of partial metrics in Computer Science, we discuss whether the Matthews fixed point theorem is a suitable tool to analyze the asymptotic complexity of algorithms in the spirit of Schellekens. Specifically, we show that a slight modification of the well-known Baire partial metric on the set of all words over an alphabet constitutes an appropriate tool to carry out the asymptotic complexity analysis of algorithms via fixed point methods without the need for assuming the convergence condition inherent to the definition of the complexity space in the Schellekens framework. Finally, in order to illustrate and to validate the developed theory we apply our results to analyze the asymptotic complexity of Quicksort, Mergesort and Largesort algorithms. Concretely we retrieve through our new approach the well-known facts that the running time of computing of Quicksort (worst case behaviour), Mergesort and Largesort (average case behaviour) are in the complexity classes

The complexity space of partial functions: a connection between complexity analysis and denotational semantics

\mathcal{O}(n^{2})

Asymptotic Complexity of Algorithms via the Nonsymmetric Hausdorff Distance

,

Complete partial metric spaces have partially metrizable computational models

\mathcal{O}(n\log_{2}(n))

A quasi-metric computational model from modular functions on monoids

and

Approximating SP-orders through total preorders: incomparability and transitivity through permutations

\mathcal{O}(2(n-1)-\log_{2}(n))