Generic Models of Logic Programs

The notion of minima] Herbrand models can be generalized to goal-generic models A, which have the property that P (= V ( i i A • • • A L n ) iff A |= V(Li A • •• A Ln ) , where P is a program and + L i , . . . , L n is a goal. In the first part of this paper we characterize a large class of goal-generic models for definite Horn programs. These are minimal models which satisfy Clark's Equality Theory and which contain an infinite number of 'unnamed' elements. Predictably, the situation deteriorates when negation is allowed. There are simple stratified programs like Q = {p(0)+-; q(0)<—<p(x); r(0)«—'q(O) } with no perfect model A mch that, for all normal goals «—Li,... ,Ln, PERF(Q) |= V(Li A • • • A L n ) iff A \= V(Li A • • • A Ln). However, there do exist two models A, B such that PERF(Q) f= V(Li A • • • A Ln) iff {A, B} \= V(Li A • • • A £„) , suggesting the conjecture that all stratified programs P have finite goal-generic models sets in PERF(P). In the second part of this paper we show that this conjecture is false. The proof depends on a characterization of model domain elements by way of Cauchy term sequences, a finitary treatment of infinite terms. A model A realizes a Cauchy term sequence T = {U)ieN if there is some element in the domain of A which is an instance of every term in T, and A omits T otherwise. Basic results from model theory show that the models of Clark's Equality Theory display a rich variety in their realization and omission of Cauchy term sequences. We give a program P which can determine if a model rrjiir^ any given computable Cauchy sequence, and show that P is a counterexample to the conjecture.