Consistency of Clark's completion and existence of stable models

1 Abstract The most general notion of canonical model for a logic program with negation is the one of stable model 9]. In 7] the stable models of a logic program are characterized by the well-supported Herbrand models of the program, and a new xed point semantics that formalizes the bottom-up truth maintenance procedure of 4] is based on that characterization. Here we focus our attention on the abstract notion of well-supportedness in order to derive suucient conditions for the existence of stable models. We show that if a logic program is positive-order-consistent (i.e. there is no innnite decreasing chain w.r.t. the positive dependencies in the atom dependency graph of) then the Herbrand models of comp(() coincide with the stable models of. From this result and the ones of 10] 17] 2] on the consistency of Clark's completion, we obtain suf-cient conditions for the existence of stable models for positive-order-consistent programs. Then we show that a negative cycle free program can have no stable model if it is not positive-order-consistent, but that order-consistency alone 17] 2] (this condition generalizes call-consistency 10] 18], it is independent of positive-order-consistency) is suucient to ensure the existence of a stable model. The last result is based on a characterization of stable models due to 5] and on a generalization of the result of 17] on the consistency of Clark's completion to innnite logic programs. A logic program is a nite set of rules of the form L 1 A is an atom, called the conclusion and denoted by concl(R), and the L i 's are literals (i.e. atoms or negated atoms), called the premises and denoted by prem(R). We denote the subset of positive atoms in prem(R) by pos(R), and