Computing large and small stable models

  • Authors:

  • Mirosł/aw Truszczyń/ski

  • Affiliations:

  • Department of Computer Science, University of Kentucky, Lexington, KY 40506-0046, USA/ (e-mail: mirek@cs.uky.edu)

  • Venue:
  • Theory and Practice of Logic Programming
  • Year:
  • 2002

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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(mc), where m is the size of the program and 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].