Implicational Completeness of Signed Resolution

1 Implicational completeness-a neglected topic Every serious computer scientist and logician knows that resolution is complete for rst-order clause logic. By this, of course, one means that the empty clause (representing contradiction) is derivable by resolution from every unsatissable set of clauses S. However, there is another { less well known { concept of completeness for clause logic, that is often referred to as \Lee's Theorem" (see, e.g., 8]): Char-tung Lee's dissertation 7] focused on an interesting observation that (in a corrected version and more adequate terminology) can be stated as follows: Theorem 1.1 (Lee) Let S be a set of clauses. For every non-tautological clause C that is logically implied by S there is clause D, derivable by resolution from S, such that D subsumes C. Observe that this theorem amounts to a strengthening of refutational completeness of resolution: If S is unsatissable then it implies every clause; but the only clause that subsumes every clause (including the empty clause) is the empty clause, which therefore must be derivable by resolution from S according to the theorem. At least from a logical point of view, Lee's \positive" completeness result is as interesting as refutational completeness. Nevertheless this classic result { which we prefer to call implicational completeness of resolution { is not even mentioned in most textbooks and survey articles on automated deduction. The main reason for this is probably the conception that implicational completeness, in contrast to refutational completeness, is of no practical signiicance. Moreover, it fails for all important reenements of Robinson's original resolution calculus. In addition, Lee's proof 7] is presented in an unsatisfactory manner (to say the least). A fourth reason for the widespread neglect of implicational completeness might be the fact that Lee (and others at that time) did not distinguish between implication and subsumption of clauses. However, nowadays, it is well known that the rst relation between clauses is undecidable 10], whereas sophisticated and eecient algorithms for testing the latter one are at the core of virtually all successful resolution theorem provers (see, e.g., 4]). With hindsight, this is decisive for the signiicance of Lee's Theorem. We will provide a new and independent proof of implicational completeness in a much more general setting, namely signed resolution. An additional motivation is that this result is needed for an interesting application: computing optimal rules for the handling of