We present a fixed point theorem for a class of (potentially) non-monotonic functions over specially structured complete lattices. The theorem has as a special case the Knaster-Tarski fixed point theorem when restricted to the case of monotonic functions and Kleene's theorem when the functions are additionally continuous. From the practical side, the theorem has direct applications in the semantics of negation in logic programming. In particular, it leads to a more direct and elegant proof of the least fixed point result of [Rondogiannis and W.W.Wadge, ACM TOCL 6(2): 441-467 (2005)]. Moreover, the theorem appears to have potential for possible applications outside the logic programming domain.
@cite_7: The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and many-valued logic, lattice theory, game theory, and topology. One source of this richness is the inherent non-monotonicity of its negation, something that does not have close parallels with the machinery of other programming paradigms. Nonetheless, much of the work on logic programming semantics seems to exist side by side with similar work done for imperative and functional programming, with relatively minimal contact between communities. In this paper we summarize one variety of approaches to the semantics of logic programs: that based on fixpoint theory. We do not attempt to cover much beyond this single area, which is already remarkably fruitful. We hope readers will see parallels with, and the divergences from the better known fixpoint treatments developed for other programming methodologies.
The results reported in this paper are connected to previous work on the development of an abstract fixed point theory for non-monotonic operators. Pioneering in this respect is the work of Fitting @cite_7 who used the abstract framework of lattices and operators on lattices in order to characterize all major semantic approaches of logic programming. Despite its abstract nature, Fitting's work is centered around the theory of logic programming.