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 first-order 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 non-elementarily related. We show that we can overcome these difficulties by extending LJ with an analytic cut rule.

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