Cut normal forms and proof complexity Original Research Article

Statman and Orevkov independently proved that cut-elimination is of nonelementary complexity. Although their worst-case sequences are mathematically different the syntax of the corresponding cut formulas is of striking similarity. This leads to the main question of this paper: to what extent is it possible to restrict the syntax of formulas and — at the same time—keep their power as cut formulas in a proof? We give a detailed analysis of this problem for negation normal form (NNF), prenex normal form (PNF) and monotone formulas. We show that proof reduction to NNF is quadratic, while PNF behaves differently: although there exists a quadratic transformation into a proof in PNF too, additional cuts are needed; the elimination of these cuts requires nonelementary expense in general. By restricting the logical operators to {⋀,⋁,∀,∃}, we obtain the type of monotone formulas. We prove that the elimination of monotone cuts can be of nonelementary complexity (here generalized disjunctions in the antecedents of sequents play a central role). On the other hand, we define a large class of problems (including all Horn theories) where cut-elimination of monotone cuts is only exponential and show that this bound is tight. This implies that the elimination of monotone cuts in equational theories is at most exponential. Particularly, there are no short proofs of Statmanʹs sequence with monotone cuts.