A Hierarchy of Tractable Subsets for Computing Stable Models

TL;DR

This paper introduces a hierarchy of knowledge base classes that allows efficient computation of stable models, with each class having bounded complexity and the ability to identify the minimal class for any given base.

Contribution

The paper defines a hierarchy of knowledge base classes, enabling polynomial-time identification and bounded stable models, advancing the understanding of tractable subsets for stable model computation.

Findings

Omega_1 includes all stratified knowledge bases.

Knowledge bases in Omega_k have at most k stable models.

Every knowledge base belongs to some class in the hierarchy.

Abstract

Finding the stable models of a knowledge base is a significant computational problem in artificial intelligence. This task is at the computational heart of truth maintenance systems, autoepistemic logic, and default logic. Unfortunately, it is NP-hard. In this paper we present a hierarchy of classes of knowledge bases, Omega_1,Omega_2,..., with the following properties: first, Omega_1 is the class of all stratified knowledge bases; second, if a knowledge base Pi is in Omega_k, then Pi has at most k stable models, and all of them may be found in time O(lnk), where l is the length of the knowledge base and n the number of atoms in Pi; third, for an arbitrary knowledge base Pi, we can find the minimum k such that Pi belongs to Omega_k in time polynomial in the size of Pi; and, last, where K is the class of all knowledge bases, it is the case that union{i=1 to infty} Omega_i = K, that is, every knowledge base belongs to some class in the hierarchy.