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Archive for Mathematical Logic

, Volume 42, Issue 5, pp 403–414 | Cite as

Degree complexity for a modified pigeonhole principle

  • Maria Luisa Bonet
  • Nicola Galesi
  • 31 Downloads

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].

Keywords

Simple Lemma Resolution Simulation Pigeonhole Principle Degree Complexity Polynomial Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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