Permanent Does Not Have Succinct Polynomial Size Arithmetic Circuits of Constant Depth

  • Maurice Jansen
  • Rahul Santhanam
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)


We show that over fields of characteristic zero there does not exist a polynomial p(n) and a constant-free succinct arithmetic circuit family {Φ n }, where Φ n has size at most p(n) and depth O(1), such that Φ n computes the n×n permanent. A circuit family {Φ n } is succinct if there exists a nonuniform Boolean circuit family {C n } with O(logn) many inputs and size n o(1) such that that C n can correctly answer direct connection language queries about Φ n - succinctness is a relaxation of uniformity.

To obtain this result we develop a novel technique that further strengthens the connection between black-box derandomization of polynomial identity testing and lower bounds for arithmetic circuits. From this we obtain the lower bound by explicitly constructing a hitting set against arithmetic circuits in the polynomial hierarchy.


Boolean Function Complexity Class Integer Sequence Arithmetic Circuit Polynomial Size 
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 2011

Authors and Affiliations

  • Maurice Jansen
    • 1
  • Rahul Santhanam
    • 1
  1. 1.School of InformaticsThe University of EdinburghScotland, UK

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