Completeness of the Connection Graph Proof Procedure for Unit-Refutable Clause Sets

In this paper it is shown that R. Kowalski's connection graph proof procedure /Ko75/ terminates with the empty clause for every unit-refutable clause set, provided an exhaustive search strategy is employed. This result holds for unrestricted tautology deletion, whereas subsumption requires certain precautions. The results are shown for an improved version of the connection graph resolution rule which generatesfewer links than the original one. The new inference rule not only leads to a smaller search space but it also permits a more efficient implementation. The proofs are based on refutation trees as in /HR78/ and are applied immediately at the general level. Hence the unsolved problems resulting from the classical lifting techniques in the context of connection graphs are avoided. Finally a counterexample is presented at the propositional level which shows that unrestricted deletion cf tautologies destroys completeness for non-unit-refutable clause sets.