Quasi-Metrics and the Semantics of Logic Programs

Quasi-metrics have been used in several places in the literature on domain theory and the formal semantics of programming languages. In this paper, we consider the role of quasi-metrics in the xed point semantics of logic programs, examining in detail a quite general process by which xed points of immediate consequence operators can be found. This work takes as its starting point: (i) Fitting's recent application of the Banach contraction mapping theorem in logic programming; (ii) a theorem of Rutten which generalizes both the contraction mapping theorem and the Knaster-Tarski theorem; (iii) Smyth's work on totally bounded spaces and compact ordered spaces as domains of computation. Our results therefore are theoretical and to be viewed as a contribution to the mathematical foundations of computer science.