Weighted Graphs: A Tool for Studying the Halting Problem and Time Complexity in Term Rewriting Systems and Logic Programming

This study is based on the halting and complexity problems for a simple class of logic programs in PROLOG-like languages. Any Prolog program can be expressed in the form of an overlap of some simpler programs whose structures are basic and can be studied formally. The simplest recursive rules are studied here and the weighted graph is introduced to characterise their behaviour. This new syntactic object, the weighted graph, generalises the directed graph. Unfoldings of directed graphs generate infinite regular trees that I generalise by weighting the arrows and putting periods on the variables. The weights along a branch are added during unfolding and the result (modulo of the period) indexes variables. Hence, their interpretations are non-regular trees because of the infinity of variables. This paper presents some of the formal properties of these graphs, finite and infinite interpretation and unification. Although they have a consistency apart from all possible applications, weighted graphs characterise the behaviour of recursive rules in the form L:-R. They express themost general fixpoint of these rules and range across a finite sequence of recursive rewritings. Within global rewriting systems and logic programming, the halting problem and the existence of solutions are proved to be decidable for this simple recursive rule with linear goals and facts, and the complexity is shown to be at most linear. Although these problems are undecidable for slightly more complex schemes, it is hoped that from the weighted graphs of each recursive sub-structure of a Prolog program, the whole behaviour of the program will be understandable. Then, the weighted graphs would be the nucleus of an efficient and methodological logic programming, which could be called Structured Logic Programming.