Comparing the Complexity of Cut-Elimination Methods

We investigate the relative complexity of two different methods of cut-elimination in classical first-order logic, namely the methods of Gentzen and Tait. We show that the methods are incomparable, in the sense that both can give a nonelementary speed-up of the other one. More precisely we construct two different sequences of LK-proofs with cuts where cut-elimination for one method is elementary and nonelementary for the other one. Moreover we show that there is also a nonelementary difference in complexity for different deterministic versions of Gentzen's method.