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Higher complexity search problems for bounded arithmetic and a formalized no-gap theorem

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Abstract

We give a new characterization of the strict \(\forall {\Sigma^b_j}\) sentences provable using \({\Sigma^b_k}\) induction, for 1 ≤ j ≤ k. As a small application we show that, in a certain sense, Buss’s witnessing theorem for strict \({\Sigma^b_k}\) formulas already holds over the relatively weak theory PV. We exhibit a combinatorial principle with the property that a lower bound for it in constant-depth Frege would imply that the narrow CNFs with short depth j Frege refutations form a strict hierarchy with j, and hence that the relativized bounded arithmetic hierarchy can be separated by a family of \(\forall {\Sigma^b_1}\) sentences.

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, 115 67, Praha 1, Czech Republic

    Neil Thapen

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  1. Neil Thapen
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Correspondence to Neil Thapen.

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Partially supported by institutional research plan AV0Z10190503 and grant IAA100190902 of GA AV ČR, grants LC505 (Eduard Čech Center) and 1M0545 (ITI) of MŠMT and by a grant from the John Templeton Foundation.

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Thapen, N. Higher complexity search problems for bounded arithmetic and a formalized no-gap theorem. Arch. Math. Logic 50, 665–680 (2011). https://doi.org/10.1007/s00153-011-0240-0

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