Some Pitfalls of LK-to-LJ Translations and How to Avoid Them

. In this paper, we investigate translations from a classical cut-free sequent calculus LK into an intuitionistic cut-free sequent calculus LJ. Translations known from the literature rest on permutations of inferences in classical proofs. We show that, for some classes of firstorder formulae, all minimal LJ-proofs are non-elementary but there exist short LK-proofs of the same formula. Similar results are obtained even if some fragments of intuitionistic first-order logic are considered. The size of the corresponding minimal search spaces for LK and LJ are also nonelementarily related. We show that we can overcome these difficulties by extending LJ with an analytic cut rule. 1 Introduction Characterizing classes of formulae for which classical derivability implies intuitionistic derivability was one topic in the Leningrad group around Maslov in the sixties. Such classes are called (complete) Glivenko classes which were extensively characterized by Orevkov [7]. More recently, people ar..