Fixed-parameter complexity of semantics for logic programs

TL;DR

This paper investigates the fixed-parameter complexity of decision problems related to the existence of small stable models in logic programs, showing most are computationally hard and unlikely to admit efficient algorithms.

Contribution

It establishes the fixed-parameter complexity classifications for various problems involving models of logic programs, including restricted classes like Horn and negative programs.

Findings

Most problems have high fixed-parameter complexity.

Fast algorithms for small model bounds are unlikely.

Complexity results hold for restricted program classes.

Abstract

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 problem: given a propositional logic program P and a non-negative integer k, decide whether P has a stable model of size no more than k, is an example of a parameterized decision problem with k serving as a parameter. 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(mn^k), 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 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.