On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms

Abstract

[eng] The celebrated Kleene fixed point theorem is crucial in the mathematical modelling of recursive specifications in Denotational Semantics. In this paper we discuss whether the hypothesis of the aforementioned result can be weakened. An affirmative answer to the aforesaid inquiry is provided so that a characterization of those properties that a self-mapping must satisfied in order to guarantee that its set of fixed points is non-empty when no notion of completeness are assumed to be satisfied by the partially ordered set. Moreover, the case in which the partially ordered set is coming from a quasi-metric space is treated in depth. Finally, an application of the exposed theory is obtained. Concretely, a mathematical method to discuss the asymptotic complexity of those algorithms whose running time of computing fulfils a recurrence equation is presented. The aforesaid method retrieves the fixed point based methods that appear in the literature for asymptotic complexity analysis of algorithms and, in addition, preserves the original Scott's ideas providing a common framework for Denotational Semantics and Asymptotic Complexity of algorithms.