DELTA - A Bottom-up Preprocessor for Top-Down Theorem Provers - System Abstract

Top-down theorem provers with depth-rst search (e.g., PTTP [Sti88], METEOR [AL91], SETHEO [LSBB92]) have the general disadvantage that during the search the same goals have to be proven over and over again, thus causing a large amount of redundancy. Resolution-based bottom-up theorem provers (e.g., OTTER [McC90]), on the other hand, avoid this problem by performing backward and forward subsumption and by using elaborate storage and indexing techniques. Those provers, however, often lack the goal-orientedness of top-down provers. In order to combine the advantages of top-down and bottom-up theorem proving , we have developed the preprocessor DELTA. DELTA processes one part of the search space (the \bottom" part) in a preprocessing phase, using bottom-up techniques (see also [?]). It generates unit-clauses (e.g., by applying UR-resolution) which are added to the original formula. Then, this formula is processed by a top-down theorem prover in the usual way. During this top-down search, the additional unit clauses are used as generalized unit lemmata. Due to the structure of the search space and the combination of advantages of both approaches (subsumption in the preprocessing phase and goal-oriented search in the subsequent top-down search), a remarkable gain of eciency can be achieved in many cases. DELTA uses SETHEO [LSBB92] and its logic programming facilities for generating these unit clauses. In order to obtain a high eciency for the bottom-up phase, the unit clauses are generated level by level. This technique is similar to delta iteration as it is used in the eld of database research. Starting with the original formula, the following iteration step is performed (rst, we only consider Horn-formulae): we let SETHEO generate all new unit clauses which can be obtained by one UR-resolution step out of the current formula. This is accomplished by adding to the formula \most general queries" :p(X 1 ; : : : ; X n) for each predicate symbol p and variables X 1 ; : : : ; X n. For those queries, SETHEO searches for all solutions within a low bound bu (here, normally, a depth-bound (A-literal depth) of 2 is used, but it can be varied freely). For each obtained substitution , we generate the unit clause \p(X ; : : : ; X n)". Up to now SETHEO only uses locally effective subsumption techniques, based on SETHEO's built-in constraint mechanism. Although, in case of a depth bound of 2, anti-lemma constraints 2 simulate forward ? …