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Preservation theorems for bounded formulas

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Abstract

In this paper we naturally define when a theory has bounded quantifier elimination, or is bounded model complete. We give several equivalent conditions for a theory to have each of these properties. These results provide simple proofs for some known results in the model theory of the bounded arithmetic theories like CPV and PV1. We use the mentioned results to obtain some independence results in the context of intuitionistic bounded arithmetic. We show that, if the intuitionistic theory of polynomial induction on strict \(\Pi_2^{b}\) formulas proves decidability of \(\Sigma_1^b\) formulas, then P=NP. We also prove that, if the mentioned intuitionistic theory proves \({\rm LMIN}(\Sigma_{1}^{b})\), then P=NP.

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Authors and Affiliations

  1. Shahid Beheshti University (Department of Mathematics), Evin, Tehran, Iran

    Morteza Moniri

  2. Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5746, Tehran, Iran

    Morteza Moniri

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  1. Morteza Moniri
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Moniri, M. Preservation theorems for bounded formulas. Arch. Math. Logic 46, 9–14 (2007). https://doi.org/10.1007/s00153-006-0016-0

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  • DOI: https://doi.org/10.1007/s00153-006-0016-0

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