Title:

Studies in the completeness and efficiency of theoremproving by resolution

Inference systems Τ and search strategies E for T are distinguished from proof procedures β = (T,E) The completeness of procedures is studied by studying separately the completeness of inference systems and of search strategies. Completeness proofs for resolution systems are obtained by the construction of semantic trees. These systems include minimal αrestricted binary resolution, minimal αrestricted Mclash resolution and maximal pseudoclash resolution. Certain refinements of hyperresolution systems with equality axioms are shown to be complete and equivalent to refinements of the pararmodulation method for dealing with equality. The completeness and efficiency of search strategies for theoremproving problems is studied in sufficient generality to include the case of search strategies for pathsearch problems in graphs. The notion of theoremproving problem is defined abstractly so as to be dual to that of and" or tree. Special attention is given to resolution problems and to search strategies which generate simpler before more complex proofs. For efficiency, a proof procedure (T,E) requires an efficient search strategy E as well as an inference system T which admits both simple proofs and relatively few redundant and irrelevant derivations. The theory of efficient proof procedures outlined here is applied to proving the increased efficiency of the usual method for deleting tautologies and subsumed clauses. Counterexamples are exhibited for both the completeness and efficiency of alternative methods for deleting subsumed clauses. The efficiency of resolution procedures is improved by replacing the single operation of resolving a clash by the two operations of generating factors of clauses and of resolving a clash of factors. Several factoring methods are investigated for completeness. Of these the mfactoring method is shown to be always more efficient than the WosRobinson method.
