Graphical Conditions for the Existence, Unicity and Number of Regular Models
This research addresses a problem relevant to the field of artificial intelligence, specifically for those working with logic programs and Boolean networks, providing a deeper understanding of regular models.
The authors tackled the problem of analyzing the existence, unicity, and number of regular models for normal logic programs, and found necessary and sufficient conditions for their existence and unicity, as well as upper bounds for their number. The results generalize existing findings and provide new insights into Boolean network theory.
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.