Resolution Theorem Proving

Lifting is one of the most important general techniques for accelerating dii-cult computations. Lifting was used by Alan Robinson in the early 1960's to accelerate rst order inference Robinson, 1965]. A complete inference procedure had been done by Martin Davis and Hillary Putnam a few years prior to Robinson's work Davis and Putnam, 1960]. Unfortunately, it was a ground procedure and suuered from a large search space generated by unguided selection of gound instances of quantiied formulas. Robinson designed a ground procedure for which lifting was particularly simple and demonstrated empirically that lifting greatly improves the performance of this procedure and that the lifted version is far superior to the Davis-Putnam procedure for general purpose theorem proving. Robinson's lifted inference process became known as resolution. Because of the tremendous advantage of a lifted procedure over a ground procedure, resolution immediately became the central paradigm of automated rst order inference. Over the years resolution has been greatly reened. The basic procedure is deened below and several reenements are described. Although lifting is a good technique for accelerating diicult computations, lifting alone has not brought computers near human-level competence in mathematical reasoning. It is still instructive, however, to study the classical resolution theorem proving techniques. The study of resolution theorem proving has resulted in the development of a large variety of general purpose search heuris-tics known as \strategies". In order to using lifting one must rst construct a ground procedure. The rst step in constructing the ground procedure is to convert inference problems to a simple normal form.