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Improved Bounds on the Weak Pigeonhole Principle and Infinitely Many Primes from Weaker Axioms

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Book cover Mathematical Foundations of Computer Science 2001 (MFCS 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2136))

Abstract

We show that the known bounded-depth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a size-depth trade-off upper bound: there are proofs of size \( n^{O(d(\log (n))^{2/d} )} \) and depth O(d). This solves an open problem of Maciel, Pitassi and Woods (2000). Our technique requires formalizing the ideas underlying Nepomnjaščij’s Theorem which might be of independent interest. Moreover, our result implies a proof of the unboundedness of primes in IΔ 0 with a provably weaker ‘large number assumption’ than previously needed.

Supported by the CUR, Generalitat de Catalunya, through grant 1999FI 00532. Partially supported by ALCOM-FT, IST-99-14186.

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Author information

Authors and Affiliations

  1. Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya, Barcelona, Spain

    Albert Atserias

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  1. Albert Atserias
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Editors and Affiliations

  1. Mathematical Institute, AS CR, Žitná 25, 115 67, Praha 1, Czech Republic

    Jiří Sgall

  2. Faculty of Mathematics and Physics Institute for Theoretical Computer Science (ITI), Charles University, Malostranské náměstí 25, 118 00, Praha 1, Czech Republic

    Aleš Pultr  & Petr Kolman  & 

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Atserias, A. (2001). Improved Bounds on the Weak Pigeonhole Principle and Infinitely Many Primes from Weaker Axioms. In: Sgall, J., Pultr, A., Kolman, P. (eds) Mathematical Foundations of Computer Science 2001. MFCS 2001. Lecture Notes in Computer Science, vol 2136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44683-4_14

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  • DOI: https://doi.org/10.1007/3-540-44683-4_14

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  • Print ISBN: 978-3-540-42496-3

  • Online ISBN: 978-3-540-44683-5

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