Algorithmic Structuring of Cut-free Proofs

The problem of algorithmic structuring of proofs in the sequent calculi LK and LKB LK where blocks of quanti ers can be introduced in one step is investigated where a distinction is made between linear proofs and proofs in tree form In this framework structuring coincides with the introduction of cuts into a proof The algorithmic solvability of this problem can be reduced to the question of k l compressibility Given a proof of of length k and l k Is there is a proof of of length l When restricted to proofs with universal or existential cuts this problem is shown to be undecidable for linear or tree like LK proofs corresponds to the undecidability of second order uni cation undecidable for lin ear LKB proofs corresponds to the undecidability of semi uni cation and decidable for tree like LKB proofs corresponds to a decidable subprob lem of semi uni cation