Fixed-parameter complexity of semantics for logic programs

A decision problem is called parameterized if its input is a pair of strings. One of these strings is referred to as a parameter. The following problem is an example of a parameterized decision problem with k serving as a parameter: given a prepositional logic program P and a nonnegative integer k, decide whether P has a stable model of size no more than k. Parameterized problems that are NP-complete often become solvable in polynomial time if the parameter is fixed. The problem to decide whether a program P has a stable model of size no more than k, where k is fixed and not a part of input, can be solved in time O(mnk), where m is the size of P and n is the number of atoms in P. Thus, this problem is in the class P. However, algorithms with the running time given by a polynomial of order k are not satisfactory even for relatively small values of k. The key question then is whether significantly better algorithms (with the degree of the polynomial not dependent on k) exist. To tackle it, we use the framework of fixed-parameter complexity, We establish the fixed-parameter complexity for several parameterized decision problems involving models, supported models, and stable models of logic programs. We also establish the fixed-parameter complexity for variants of these problems resulting from restricting attention to definite Horn programs and to purely negative programs. Most of the problems considered in the paper have high fixed-parameter complexity. Thus, it is unlikely that fixing bounds on models (supported models, stable models) will lead to fast algorithms to decide the existence of such models.