The Limits of Horn Logic Programs

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.Comment: 11 pages, added new results. Welcome any comments to [email protected]

Investigation of design and execution alternatives for the committed choice non-deterministic logic languages

The general area of developing, applying and studying new and parallel models of computation is motivated by a need to overcome the limits of current Von Neumann based architectures. A key area of research in understanding how new technology can be applied to Al problem solving is through using logic languages. Logic programming languages provide a procedural interpretation for sentences of first order logic, mainly using a class of sentence called Horn clauses. Horn clauses are open to a wide variety of parallel evaluation models, giving possible speed-ups and alternative parallel models of execution. The research in this thesis is concerned with investigating one class of parallel logic language known as Committed Choice Non-Deterministic languages. The investigation considers the inherent parallel behaviour of Al programs implemented in the CCND languages and the effect of various alternatives open to language implementors and designers. This is achieved by considering how various Al programming techniques map to alternative language designs and the behaviour of these Al programs on alternative implementations of these languages. The aim of this work is to investigate how Al programming techniques are affected (qualitatively and quantitatively) by particular language features. The qualitative evaluation is a consideration of how Al programs can be mapped to the various CCND languages. The applications considered are general search algorithms (which focuses on the committed choice nature of the languages); chart parsing (which focuses on the differences between safe and unsafe languages); and meta-level inference (which focuses on the difference between deep and flat languages). The quantitative evaluation considers the inherent parallel behaviour of the resulting programs and the effect of possible implementation alternatives on this inherent behaviour. To carry out this quantitative evaluation we have implemented a system which improves on the current interpreter based evaluation systems. The new system has an improved model of execution and allows severa

An Algebraic Preservation Theorem for Aleph-Zero Categorical Quantified Constraint Satisfaction

We prove an algebraic preservation theorem for positive Horn definability in aleph-zero categorical structures. In particular, we define and study a construction which we call the periodic power of a structure, and define a periomorphism of a structure to be a homomorphism from the periodic power of the structure to the structure itself. Our preservation theorem states that, over an aleph-zero categorical structure, a relation is positive Horn definable if and only if it is preserved by all periomorphisms of the structure. We give applications of this theorem, including a new proof of the known complexity classification of quantified constraint satisfaction on equality templates