Multivalued Mappings, Fixed-Point Theorems and Disjunctive Databases

Abstract

In this paper, we discuss the semantics of disjunctive programs and databases and show how multivalued mappings and their fixed points arise naturally within this context. A number of fixed-point theorems for multivalued mappings are considered, some of which are already known and some of which are new. The notion of a normal derivative of a disjunctive program is introduced. Normal derivatives are normal logic programs which are determined by the disjunctive program. Thus, the well-known single-step operator associated with a normal derivative is single-valued, and its fixed points can be found by well-established means. It is shown how fixed points of the multivalued mapping determined by a disjunctive program relate to the fixed points of the single-step operators coming from its normal derivatives. This procedure has potential for simplifying the construction of models of disjunctive databases, and this point is discussed.Most of the results for multivalued mappings rest on corresponding, known results concerning fixed points of single-valued mappings. Since the latter results are frequently referred to, they have been collected together for convenience in a survey which should be of independent interest as well as being preparatory for the later results. Finally, a number of problems and issues raised by this work are discussed. * The first named author acknowledges financial support under grant SC/98/621 from Enterprise Ireland.3rd Irish Workshop on Formal Methods (IWFM-99).

Review Article - A study of some fixed point theorems for various types of maps

“…Husain and Latif (1991) proved some fixed point theorems for multi-valued contractive-type and nonexpansive type maps on complete metric spaces and on certain closed bounded convex subsets of Banach spaces. Hitzler and Seda (1999) discussed the semantics of disjunctive programs and databases and show how multivalued mappings and their fixed points arise naturally within this context. A number of fixed-point theorems for multivalued mappings are considered, some of which are already known and some of which are new.…”

Fixed point theorem in ultrametric space

“…More general programs, involving negation in clauses lead to the ultrametric space of maps from the Herbrand base with values 0, 1; in this space the values of the distance are the subsets of the Herbrand base. The fixed point of the immediate consequence operator gives conditions for the existence of models for the program, see Priess-Crampe and Ribenboim [9,13] and Hitzler and Seda [4,5].…”