Extremal problems in logic programming and stable model computation

TL;DR

This paper investigates the maximum number of stable models in various classes of logic programs, characterizes the programs that attain these maxima, and discusses implications for algorithm design.

Contribution

It establishes maximum stable model counts for different logic program classes and introduces algorithms inspired by Davis-Putnam for computing all stable models.

Findings

Maximum stable models for all logic programs with n clauses

Algorithms with worst-case search space of O(3^{n/3})

Characterization of programs attaining maximum stable models

Abstract

We study the following problem: given a class of logic programs C, determine the maximum number of stable models of a program from C. We establish the maximum for the class of all logic programs with at most n clauses, and for the class of all logic programs of size at most n. We also characterize the programs for which the maxima are attained. We obtain similar results for the class of all disjunctive logic programs with at most n clauses, each of length at most m, and for the class of all disjunctive logic programs of size at most n. Our results on logic programs have direct implication for the design of algorithms to compute stable models. Several such algorithms, similar in spirit to the Davis-Putnam procedure, are described in the paper. Our results imply that there is an algorithm that finds all stable models of a program with n clauses after considering the search space of size O(3^{n/3}) in the worst case. Our results also provide some insights into the question of representability of families of sets as families of stable models of logic programs.