A fixed point theorem in partial quasi-metric spaces and an application to Software Engineering

Abstract

Scott (1970) [11] introduced qualitative fixed point techniques as a suitable mathematical tool for program verification. Inspired by the fact that the Scott mathematical tools do not include metric implements, Matthews (1994) [8] introduced the concept of partial metric space with the aim of reconciling the Scott fixed point techniques with metric spaces and proved a fixed point theorem for self-mappings in partial metric spaces providing, thus, quantitative techniques useful, in the spirit of Scott, in denotational semantics. Schellekens (1995) [10] showed that the original Scott ideas can be also applied to asymptotic complexity analysis of algorithms via quantitative fixed point techniques for self-mappings in quasi-metric spaces. Later on Cerdà-Uguet et al. (2012) [3] showed that, contrarily to the case of Matthews partial metric spaces, partial quasi-metrics are useful for modeling the algorithmic complexity by means of quantitative fixed point techniques that preserve the Scott ideas and Schellekens techniques concurrently.In the present paper, we focus our efforts on developing a quantitative fixed point technique, based on partial quasi-metric spaces, that allows us to provide a suitable mathematical tool for program verification and complexity analysis simultaneously and, in addition, preserves the Scott ideas and the essence of Matthews and Schellekens techniques. Moreover, we show its applicability to discuss the running time of computing an algorithm using a recursive denotational specification and the meaning of such an specification.