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Lower bounds for increasing complexity of derivations after cut elimination

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Abstract

We denote by C *k the formula

. In this paper for all k there is constructed a derivation of C *k with cut, the number of sequents in which depends linearly on k. On the other hand, it is impossible to give an upper bound which is a Kalmar elementary function of k for the number of sequents in any derivation of the formula C *k without cuts, or for the number of disjunctions in a refutation by the method of resolutions of systems of disjunctions corresponding to the negation of the formula C *k . In particular, one can construct a derivation with cut of the formula C *6 , in which there is contained no more than 253 sequents, but in seeking a derivation of C *6 by the method of resolutions it is necessary to construct more than 1019200 disjunctions. With the help of Skolemization and taking out of quantifiers with respect to the formula C *k there is constructed a formula ∃v0B +k (v0), which satisfies the following conditions: 1) one can construct a derivation with cuts of the formula ∃v0B +k (v0) in the constructive predicate calculus, the number of sequents in which depends linearly on k; 2) it is impossible to give an upper bound which is a Kalmar elementary function of k of the length of a term t such that the formula B +k (t) is derivable.

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Authors
  1. V. P. Orevkov
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 137–161, 1979.

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Orevkov, V.P. Lower bounds for increasing complexity of derivations after cut elimination. J Math Sci 20, 2337–2350 (1982). https://doi.org/10.1007/BF01629444

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  • DOI: https://doi.org/10.1007/BF01629444

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