Bounds for cut elimination in intuitionistic propositional logic

The central theorem of Gentzen's theory of proofs states that every deduction d (in classical or intuitionistic, propositional or quantifier logic) can be transformed into a deduction G(d) which does not make use of the cut rule. Avoiding the use of a particular proof rule will, obviously, have the effect that G(d) becomes longer than d, and Gentzen's algorithm for cut elimination establishes an upper bound for the length l(G(d)) of G(d). In this article, I shall construct a (different) cut free deduction J(d) for the case of intuitionistic propositional logic and derive considerably sharper upper bounds for l(J(d)). Also, I shall use the methods developed for this purpose in order to set up an effective decision method. Gentzen's upper bound for l(G(d)) depends on both the length l(d) and the cut degree g(d) of d, viz. the maximum of the degrees, increased by 1, of cut formulas used in d; it has the form