Automatic Termination Analysis of Programs Containing Arithmetic Predicates

For logic programs with arithmetic predicates, showing termination is not easy, since the usual order for the integers is not well-founded. A new method, easily incorporated in the TermiLog system for automatic termination analysis, is presented for showing termination in this case. The method consists of the following steps: First, a nite abstract domain for representing the range of integers is deduced automatically. Based on this abstraction, abstract interpretation is applied to the program. The result is a nite number of atoms abstracting answers to queries which are used to extend the technique of query-mapping pairs. For each query-mapping pair that is potentially non-terminating, a bounded (integer-valued) termination function is guessed. If traversing the pair decreases the value of the termination function, then termination is established. Simple functions often su ce for each query-mapping pair, and that gives our approach an edge over the classical approach of using a single termination function for all loops, which must inevitably be more complicated and harder to guess automatically. It is worth noting that the termination of McCarthy's 91 function can be shown automatically using our method. In summary, the proposed approach is based on combining a nite abstraction of the integers with the technique of the query-mapping pairs, and is essentially capable of dividing a termination proof into several cases, such that a simple termination function su ces for each case. Consequently, the whole process of proving c 1999 Published by Elsevier Science B. V.