Archive for Mathematical Logic

, Volume 42, Issue 5, pp 403–414

Degree complexity for a modified pigeonhole principle

  • Maria Luisa Bonet
  • Nicola Galesi

Abstract.

 We consider a modification of the pigeonhole principle, M P H P, introduced by Goerdt in [7]. M P H P is defined over n pigeons and log n holes, and more than one pigeon can go into a hole (according to some rules). Using a technique of Razborov [9] and simplified by Impagliazzo, Pudlák and Sgall [8], we prove that any Polynomial Calculus refutation of a set of polynomials encoding the M P H P, requires degree Ω(log n). We also prove a simple Lemma giving a simulation of Resolution by Polynomial Calculus. Using this lemma, and a Resolution upper bound by Goerdt [7], we obtain that the degree lower bound is tight.

Our lower bound establishes the optimality of the tree-like Resolution simulation by the Polynomial Calculus given in [6].

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Maria Luisa Bonet
    • 1
  • Nicola Galesi
    • 1
  1. 1.Department de Llenguatges i Sistemes Informatics, Universitat Politecnica de Catalunya, Jordi Girona Salgado 1-3 Barcelona Spain. e-mail: bonet;galesi@lsi.upc.esES

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