A General Framework for Automatic Termination Analysis of Logic Programs

This paper describes a general framework for automatic termi- nation analysis of logic programs, where we understand by "termination" the finiteness of the LD-tree constructed for the program and a given query. A general property of mappings from a certain subset of the branches of an infinite LD-tree into a finite set is proved. From this result several ter- mination theorems are derived, by using different finite sets. The first two are formulated for the predicate dependency and atom dependency graphs. Then a general result for the case of the query-mapping pairs relevant to a program is proved (cf. (29,21)). The correctness of the TermiLog system described in (22) follows from it. In this system it is not possible to prove termination for programs involving arithmetic predicates, since the usual or- der for the integers is not well-founded. A new method, which can be easily incorporated in TermiLog or similar systems, is presented, which makes it possible to prove termination for programs involving arithmetic predicates. It is based on combining a finite abstraction of the integers with the tech- nique of the query-mapping pairs, and is essentially capable of dividing a termination proof into several cases, such that a simple termination function suffices for each case. Finally several possible extensions are outlined.