A strong problem reduction method based on function introduction

• Authors: M. Baaz;A. Leitsh
• Venue: ISSAC '90 Proceedings of the international symposium on Symbolic and algebraic computation
• Year: 1990

Although problem reduction is a very important tool in mathematical practice, relatively little attention has been paid to problem reduction in automated theorem proving. A systematical treatment of different problem reduction methods can be found in [BI71] and [Lo78]. (In [BI71] we find unary and binary reduction rules. In the unary case we reduce a problem E to a problem E', where E is provable (refutable) if E' is; for binary reduction we have to produce problems E1, E2 out of E such that E is provable (refutable) if both E1, E2 are provable (refutable).)In finding stronger reduction strategies, one has to give up completeness of the split of E into E1, E2; that means, we only require that the provability of E1 and E2 implies that of E.The authors therefore propose problem reduction based on a splitting rule of the form C → C', where C ∼ C1 &ngr; C2, C' ∼ C1 &ngr; C'2, C'2 ∼ C2 {× ← ƒ(y1,…yn)}, {x,y1,…yn} is the set of variables both in C1 and C2 and ƒ is a new function symbol up to this point not occurring in any clause. Note that the Herbrand universe is extended by ƒ and consequently C' is not derivable from C by usual resolution methods! As (∀y1) … (∀yn(∀x) (C1 &ngr; C2) → (∀y1) … (∀yn ((∀x) C1 &ngr; (∃x) (C2) is valid in first order predicate logic and C1 &ngr; C'2 is the Skolemization of the implied formula, splitting of this kind is correct. For reasons of efficiency (reduction of search space and restriction of compound terms) the authors restrict the application of the rule above to sequences of applications (called Q-reduction) completely separating the variables in C1 and C2.Finally the authors construct a sequence of clause sets Cn having resolution proofs exponential in n only, but application of the new reduction rule reduces the problem to two problems linear in n. Thus it turns out that the introduction of (elementary) quantificational rules into clause logic can strongly influence the structure of proofs and the performance of theorem provers.