The application of goal - oriented heuristics for proving equational theorems via the unfailing Knuth - Bendix completion procedure

In this report we present a case study of employing goal-oriented heuristics when proving equational theorems with the (unfailing) Knuth-Bendix completion procedure. The theorems are taken from the domain of lattice ordered groups. It will be demonstrated that goal-oriented (heuristic) criteria for selecting the next critical pair can in many cases significantly reduce the search effort and hence increase performance of the proving system considerably. The heuristic, goal-oriented criteria are on the one hand based on so-called “measures” measuring occurrences and nesting of function symbols, and on the other hand based on matching subterms. We also deal with the property of goal-oriented heuristics to be particularly helpful in certain stages of a proof. This fact can be addressed by using them in a framework for distributed (equational) theorem proving, namely the “teamwork-method”. 0. Introduction The completion-procedure initially proposed by D.E. Knuth and P.B. Bendix (the KB-procedure [KB70]) together with further extensions and improvements (the unfailing KB-procedure (UKB-procedure) [BDP89]) has also proved to be an important tool for proving theorems in equational theories. The major drawback of its usefulness for proving resides in what it was originally designed for, namely for deriving a complete (i.e. convergent) set of rules from a given set of (equational) axioms to yield a decision procedure for the respective equational theory. Since there are in this case hardly any hints to what kind of rules resp. equations might be needed, the strategies (heuristics) employed in the completion process are entirely forward oriented. This way of proceeding does not make sense in case the UKB-procedure is used for proving if no convergent set of rules can be generated. Under these conditions, the theorem to be proved (the goal) can give valuable clues how the rules and equations the UKB-procedure should generate may look like. Forward oriented strategies completely ignore this kind of information and always exhibit the same behaviour regardless of the given goal. This is neither satisfactory nor acceptable because goal-oriented strategies can considerably reduce the search-effort by pruning the most of the time enormous search space and thus substantially increase efficiency. Therefore 1. This work was supported by the Deutsche Forschungsgemeinschaft (DFG).