arXiv:cs/0002001v1 [cs.LO] 3 Feb 2000 Computing large and small stable models 1 Miroslaw Truszczy´ nski University of Kentucky Lexington, KY 40506-0046, USA mirek@cs.uky.edu Abstract In this paper, we focus on the problem of existence and computing of small and large stable models. We show that for every fixed integer k, there is a linear-time algorithm to decide the problem LSM (large stable models problem): does a logic program P have a stable model of size at least |P |− k. In contrast, we show that the problem SSM (small stable models problem) to decide whether a logic program P has a stable model of size at most k is much harder. We present two algorithms for this problem but their running time is given by polynomials of order depending on k. We show that the problem SSM is fixed- parameter intractable by demonstrating that it is W [2]-hard. This result implies that it is unlikely, an algorithm exists to compute stable models of size at most k that would run in time O(n c ), where c is a constant independent of k. We also provide an upper bound on the fixed-parameter complexity of the problem SSM by showing that it belongs to the class W [3]. 1 Introduction The stable model semantics by Gelfond and Lifschitz [10] is one of the two most widely studied semantics for normal logic programs, the other one being the well-founded seman- tics by Van Gelder, Ross and Schlipf [17]. Among 2-valued semantics, the stable model semantics is commonly regarded as the one providing the correct meaning to the negation operator in logic programming. It coincides with the least model semantics on the class of Horn programs, and with the well-founded semantics and the perfect model semantics on the class of stratified programs [1]. In addition, the stable model semantics is closely related to the notion of a default extension by Reiter [12, 4]. Logic programming with sta- ble model semantics has applications in knowledge representation, planning and reasoning about action. It was also recently proposed as a computational paradigm especially well suited for solving combinatorial optimization and constraint satisfaction problems [14, 15]. The problem with the stable model semantics is that, even in the propositional case, reasoning with logic programs under the stable model semantics is computationally hard. It is well-known that deciding whether a finite propositional logic program has a stable model is NP-complete [13]. Consequently, it is not at all clear that logic programming with the stable model semantics can serve as a practical computational tool. 1 This is a full version of an extended abstract presented at the International Conference on Logic Pro- gramming, ICLP-99 and included in the proceedings published by MIT Press. 1