Tableau-Based Theorem Proving

34 Reiner Hahnlereservation and it can safely be claimed that any competitive propositional proofprocedure embodies some variant of cut. On the rst-order level, however, additionalliterals introduced as cuts or lemmas create additional possibilities for branch closure.The negative eects from this increase of the search space can easily outweighthe possibility of nding shorter proofs.Example 4.7. Let S = f:p(x)_q(x)_r(x); p(a); :q(a); p(b); :q(b); :r(b)g. In thepartial connection tableau in Figure 13 the left branch was closed rst by extensionwith p(a) (where only p(b) leads to success). This forces extension with q(a) in themiddle branch and generation of some lemmas (framed literals). The lemmas allowto extend the right branch with another instance of the rst clause which wouldhave been impossible without them. Detection of the wrong rst extension is thuspossibly delayed a long time.In the example, a regularity check would help (r(a) becomes irregular on therightmost branch) and also restriction of lemma usage to reduction steps. Althoughthis helps somewhat, more complex examples create the same problems as before.true:p(x)q(x)r(x)p(a)p(x)p(x)fx=ag:q(a)id:p(y)fy=xg:q(x)q(y)idr(y).Figure 13. Search space increase caused by local lemmas.On the other hand, local lemmas are not strong enough on the rst-order level.In the clause tableau in Figure 5, for example, the lemma :q(c) can be folded upto the true node, but is useless to close the open branch on the right, because adierent instance is required. But in fact it is justied to derive even (8x):q(x)as a lemma. This is always possible when the proof of (that is: the tableau below)the lemma does not instantiate any variables that occur outside of it. It remains tobe seen whether such an optimization can be eciently implemented and does notblow up the search space beyond any usefulness.4.8. Tableaux and Logic ProgrammingSome calculi for extensions of logic programs have a natural interpretation as variantsof clause tableaux. This includes [47, 48, 40, 51, 3]. Their treatment isbeyond