The Limits of Horn Logic Programs

TL;DR

This paper investigates the behavior of the least Herbrand models of sequences of Horn logic programs, identifying conditions under which the models converge and exploring the stability of fixed points using dynamical systems concepts.

Contribution

It establishes a syntactic condition ensuring the convergence of least Herbrand models of Horn logic program sequences and links this condition to stability analysis.

Findings

The least Herbrand model of the limit equals the limit of models under the condition.

Finite Horn programs satisfying the condition have recursive least Herbrand models.

The condition guarantees stability of fixed points in Horn logic programs.

Abstract

Given a sequence $\{\Pi_n\}$ of Horn logic programs, the limit $\Pi$ of $\{\Pi_n\}$ is the set of the clauses such that every clause in $\Pi$ belongs to almost every $\Pi_n$ and every clause in infinitely many $\Pi_n$'s belongs to $\Pi$ also. The limit program $\Pi$ is still Horn but may be infinite. In this paper, we consider if the least Herbrand model of the limit of a given Horn logic program sequence $\{\Pi_n\}$ equals the limit of the least Herbrand models of each logic program $\Pi_n$. It is proved that this property is not true in general but holds if Horn logic programs satisfy an assumption which can be syntactically checked and be satisfied by a class of Horn logic programs. Thus, under this assumption we can approach the least Herbrand model of the limit $\Pi$ by the sequence of the least Herbrand models of each finite program $\Pi_n$. We also prove that if a finite Horn logic program satisfies this assumption, then the least Herbrand model of this program is recursive. Finally, by use of the concept of stability from dynamical systems, we prove that this assumption is exactly a sufficient condition to guarantee the stability of fixed points for Horn logic programs.