Tableau-Based Theorem Proving

20 Reiner HahnleTheorem 4.3 (Completeness). If the clause set S is unsatisable, then there isaclause tableau proof for S.Proof. Herbrand's Theorem 2.2 provides a nite, unsatisable set S of groundinstances of S. By Theorem 4.2 there is a closed ground clause tableau T for S and,by Theorem 4.1, there is a closed clause tableau for S.As announced already, the next goal is to nd complete restrictions of clausetableaux. It is sucient toprovide a suitable instance of Theorem 4.2, whenevera ground X-tableau proof T lifts to a rst-order X-tableau proof T . It is usuallysucient tocheck that the proof of Theorem 4.1 can be used unaltered.From the point of proof search, restricting the tableau calculus means to excludecertain choices in select clause and select pair and to x select branch in some way.4.3. ConnectionsConnection conditions have been pioneered by Davydov [17], Bibel [11, 12], andAndrews [1].4.3.1. Strong ConnectionsA major drawback of the tableau calculus is that the extension rule 4.1.(ii) is appliedcompletely unguided which can clutter up tableaux with many nodes that do notcontribute to a proof.Example 4.2. Consider the two clause tableaux for S = fp(x) _ q(x); r(x) _s(x); :p(a); :q(a); :r(b); :s(b)g displayed in Figure 6. The tableau on the rightconstitutes a minimal proof, while the second extension step in the tableau on theleft is completely unrelated to the initial step.true:p(a)true:p(a)r(x 1 )s(x 1 )p(x 1 )q(x 1 ):r(b)x 1 =bp(x 2 ).q(x 2 ).x 1 =a:q(a)idFigure 6. Redundant nodes in a tableau.Denition 4.3 ([36]). A connection tableau is a clause tableau in which every nonleafnode L (except true) has L as one of its immediate successors.