Inference of termination conditions

Numerical computations form an essential part of almost any real-world program. Clearly, in order for a termination analyser to be of practical use it should contain a mechanism for inferring termination of such computations. However, this topic attracted less attention of the research community. In works of Der-showitz et al. 5] and Ruggieri 9] such termination analysis techniques were presented. In 8] a termination inference technique for constraints logic programming was suggested. The study of termination for constraint logic programs 9] provide the necessary theoretical results and imply the equivalence of a variant of acceptability 1] and termination. Unfortunately, this work does not provide a methodology for actual proving termination or inferring conditions that imply it. Alternatively, Mesnard 8] provides techniques for inferring termination conditions, but does not consider inherently non well-founded CLP-domains. In this paper we present a termination inference technique for numerical loops based on the well-known constraints based approach 4], further extending 3], and on the adornments technique 6, 7]. We restrict our interest only to integer loops, since termination of real number computations is often implementation dependent (see 5]). Example 1. This example illustrates two possible sources of termination. First, if no rule is applicable for some query (e.g. ??p(?1)), and second, if some rules are applicable, but the execution terminates(e.g. ?? p(3) or ?? p(7)). We distinguish between these cases and infer termination conditions separately. A condition ensuring termination in the rst case is deened syntactically, based on the constraints appearing in rules bodies. One of the major diiculties while analysing termination in the second case is deening correct level-mappings. On the one hand, level-mappings should map atoms to natural numbers. On the other hand, level-mappings should reeect the possibly negative numeric values of integer arguments. This contradiction is solved by step-by-step inference of bounded integer arguments and reening the deenition of the level-mapping.