How Complicated is the Set of Stable Models of a Recursive Logic Program?

In Logic Programming, ”negation as failure” has become an important area of research. ”Negation as failure” is ubiquitous in deductive databases ([Minker, 1987]), in truth maintenance systems ([Doyle, 1979]), and in default logic ([Reiter, 1980]). It is a naturally occurring “non-monotonic logic”. The original (breadth-first) PROLOG is a (monotonic) classical logic of pure Horn clauses. There, every Horn program possesses a least Herbrand model. Correspondingly, an important tool for understanding the semantics of PROLOG programs allowing “negation as failure” has been stable models ([Gelfond and Lifschitz, 1988]). But, unlike the situation for pure Horn clause logic, a program here may have no least stable model in the Herbrand universe. In fact, a program may have no stable model or many different stable models. Exactly how complicated can the set of all stable models be? In this paper we determine this for programs with a recursive set of axioms. See [Rogers, 1967] for all unexplained terminology. We show that, up to a 1:1 recursive Department of Computer Science, University Kentucky, Lexington, KY 40506–0027. Work partially supported by NSF grant RII-8610671 and Kentucky EPSCoR program and ARO contract DAAL03-89-K0124. Mathematical Sciences Institute, Cornell University, Ithaca, NY 14853. Work partially supported by NSF grant DMS-8902797 and ARO contract DAAG629-85-C-0018. Department of Mathematics, University of California at San Diego, La Jolla, CA 92903. Work partially supported by NSF grant DMS-9006413.