Abstract interpretation: A kind of magic

Abstract

Magic sets and, more recently, magic templates are used in the field of deductive databases to facilitate efficient bottom up evaluation of database queries. Roughly speaking a top down computation is simulated by first transforming the program and then executing the new program bottom up. In this paper we give a new and very simple proof that this approach is equivalent to the collecting interpretation of the abstract interpretation framework of C. Mellish. As a side-effect we also prove that "bottom up" abstract interpretation based on the magic templates transformation is equally powerful as this particular abstract interpretation framework, but less powerful than other (more precise) abstract interpretation frameworks.