Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Van-Giang Trinh, Belaid Benhamou, Sylvain Soliman, François Fages

TL;DR

The authors address how graphical properties of a finite ground normal logic program's dependency graph influence the existence, uniqueness, and multiplicity of its regular models. They build a bridge to Boolean networks via trap-space semantics, showing that regular models correspond to $ ext{≤}_s$-minimal stable trap spaces and that the BN encoding preserves key graph-theoretic structure (e.g., $ ext{tg}_{sp}(P)= ext{sstg}(f)$). They prove three main results: (i) the presence of negative cycles is necessary for non-trivial regular models, (ii) absence of positive cycles suffices for the unicity of regular models, and (iii) two bounds on the number of regular models based on a positive feedback vertex set $U^{+}$, namely $3^{|U^{+}|}$ in general and $2^{|U^{+}|}$ for tight programs. This framework generalizes prior results for well-founded stratification programs and provides new tools for analyzing logic programs through the lens of Boolean-network dynamics, with potential extensions to Datalog and argumentation frameworks.

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.

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

TL;DR

The authors address how graphical properties of a finite ground normal logic program's dependency graph influence the existence, uniqueness, and multiplicity of its regular models. They build a bridge to Boolean networks via trap-space semantics, showing that regular models correspond to -minimal stable trap spaces and that the BN encoding preserves key graph-theoretic structure (e.g., ). They prove three main results: (i) the presence of negative cycles is necessary for non-trivial regular models, (ii) absence of positive cycles suffices for the unicity of regular models, and (iii) two bounds on the number of regular models based on a positive feedback vertex set , namely in general and for tight programs. This framework generalizes prior results for well-founded stratification programs and provides new tools for analyzing logic programs through the lens of Boolean-network dynamics, with potential extensions to Datalog and argumentation frameworks.

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.

Paper Structure

This paper contains 15 sections, 22 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normal logic programs
  4. Stable and supported partial models
  5. Dependency and transition graphs
  6. Stable and supported trap spaces
  7. Boolean networks
  8. Finite ground normal logic programs and Boolean networks
  9. Graphical analysis results
  10. Preparations
  11. Unicity of regular and stable models
  12. Existence of non-trivial regular models
  13. Unicity of regular models
  14. Upper bound for number of regular models
  15. Conclusion and perspectives

Key Result

Theorem 1

Let $P$ be a finite ground program and $f$ be its encoded BN. Then $\text{ig}({f}) \subseteq \text{dg}({P})$.

Figures (2)

  • Figure 1: (a) $\text{dg}({P_1})$, (b) $\text{tg}_{st}({P_1})$, and (c) $\text{tg}_{sp}({P_1})$.
  • Figure 2: (a) $\text{ig}({f_1})$, (b) $\text{sstg}({f_1})$, and (c) $\text{astg}({f_1})$.

Theorems & Definitions (37)

  • Definition 1
  • Example 1
  • Example 2
  • Definition 2
  • Theorem 1
  • proof
  • Lemma 2: derived from Theorem 4.5 of DBLP:conf/birthday/InoueS12
  • Theorem 3
  • proof
  • Theorem 4: DBLP:conf/birthday/InoueS12
  • ...and 27 more