A Topological View of the Kowalski - Van Emden Theorem

1 Introduction. In 8] we showed how several issues of importance in logic programming can prootably be viewed by casting them as questions concerned with topologies on spaces of interpretations and topological continuity of operators associated with logic programs. In doing this, we were building on earlier work of Batarekh and Subrahmanian 1, 2, 3] and extending and generalising their ideas. For reasons of space, it was convenient then to restrict ourselves to consideration of just one topology, the query topology as deened in 2], and the generalisation of it we called the atomic topology Q as deened in 7, 8]. However , in addition to the query topology, Batarekh and Subrahmanian 2] have discussed a topology which they call the positive query topology and have established equality between it and the well known Scott topology. Once again, their treatment is subject to a restriction they call the \matching condition" which they impose on all preinter-pretations they admit of the underlying rst order language in question. Unfortunately, this restricts them, eeectively, to dealing only with the Herbrand preinterpretation. Our purpose in this note is to consider a generalisation of the positive query topology, called the positive atomic topology, which we denote by Q +. In doing this we proceed to \remove" the existential quantiier in a (positive) query just as was done in 8]. In this process, no restriction whatever is placed on the preinterpretations we consider and it turns out that Q + coincides with the positive query topology ii the matching condition is satissed. Moreover, it is easy to see that Q + can be regarded as a product of two-point spaces each of which is endowed with the Scott topology and that, consequently, Q + coincides with the Scott topology in complete generality. We go on to show how Q + can be taken as a foundation for the declarative semantics of deenite logic programs by deriving the Kowalski-van Emden theorem as a consequence of our main results. That it should be possible to derive such results is no surprise given the role of the Scott topology in computing. Nevertheless, the approach we adopt has some mathematical interest for two reasons. First, it motivates the introduction of the greatest limit of a net of interpretations. Secondly, it brings out the topological interpretation