Graphical Conditions for the Existence, Unicity and Number of Regular Models

TL;DR

This paper establishes graphical conditions on dependency graphs to analyze the existence, uniqueness, and quantity of regular models in finite ground normal logic programs, linking logic programming with Boolean network theory.

Contribution

It introduces new graphical criteria for regular model analysis, generalizes previous results, and connects logic programming with Boolean network theory.

Findings

Necessary condition for non-trivial regular models

Sufficient condition for unique regular models

Upper bounds on the number of regular models

Abstract

The regular models of a normal logic program are a particular type of partial (i.e. 3-valued) models which correspond to stable partial models with minimal undefinedness. In this paper, we explore graphical conditions on the dependency graph of a finite ground normal logic program to analyze the existence, unicity and number of regular models for the program. We show three main results: 1) a necessary condition for the existence of non-trivial (i.e. non-2-valued) regular models, 2) a sufficient condition for the unicity of regular models, and 3) two upper bounds for the number of regular models based on positive feedback vertex sets. The first two conditions generalize the finite cases of the two existing results obtained by You and Yuan (1994) for normal logic programs with well-founded stratification. The third result is also new to the best of our knowledge. Key to our proofs is a connection that we establish between finite ground normal logic programs and Boolean network theory.