Some Issues Concerning Fixed-Points in Computational Logic: Quasi-Metrics, Multivalued Mappings and the Knaster-Tarski Theorem
Many questions concerning the semantics of disjunctive databases and of logic programming systems depend on the fixed points of various multivalued mappings and operations determined by the database or program. We discuss known versions for multivalued mappings of the Knaster-Tarski theorem and of the Banach contraction mapping theorem and formulate a version of the classical fixed point theorem (sometimes attributed to Kleene) which is new. All these results have applications to the semantics of disjunctive logic programs, and we will describe a class of programs to which the new theorem can be applied. We also show that a unification of the latter two theorems is possible, using quasi-metrics, which parallels the well-known unification of Rutten and Smyth in the case of conventional programming language semantics.
& Seda, A. K.
(1999). Some Issues Concerning Fixed-Points in Computational Logic: Quasi-Metrics, Multivalued Mappings and the Knaster-Tarski Theorem. Topology Proceedings, 24, 223-250.