The problem of Π2-cut-introduction

We describe an algorithmic method of proof compression based on the introduction of Π2-cuts into a cut-free LK-proof. This method extends previous research on the introduction of Π1-cuts and is based on a connection between proof theory and formal language theory. Given is a cut-free proof π of a sequent S and a so called schematic Π2-grammar G, a grammar formalizing the substitutions of quantifiers in the elimination of Π2-cuts and describing the instantiations for the generation of a Herbrand sequent of π. An algorithm is developed to automatically construct a Π2-cut A and a proof π ′ of S with one cut on A. Basically, the method inverts Gentzen’s method of cut-elimination. It is shown that the algorithm can achieve an exponential compression of proof length.