Permanent does not have succinct polynomial size arithmetic circuits of constant depth

Authors:
Maurice Jansen;Rahul Santhanam
Affiliations:
School of Informatics, The University of Edinburgh;School of Informatics, The University of Edinburgh
Venue:
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Year:
2011

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Permanent does not have succinct polynomial size arithmetic circuits of constant depth

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Abstract

We show that over fields of characteristic zero there does not exist a polynomial p(n) and a constant-free succinct arithmetic circuit family {φn}, where φn has size at most p(n) and depth O(1), such that φn computes the n × n permanent. A circuit family {φn} is succinct if there exists a nonuniform Boolean circuit family {Cn} with O(log n) many inputs and size no(1) such that that Cn can correctly answer direct connection language queries about φn - 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.