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

Maurice Jansen, Rahul Santhanam
2011 Lecture Notes in Computer Science  
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} using division by constants 1 , 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 n o(1) such that that Cn can correctly answer direct connection language queries about Φn -succinctness
more » ... s 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. 5 It is possible to give a uniform upper of E NP RP for L. 6 One of the referees of this paper suggested such an argument.
doi:10.1007/978-3-642-22006-7_61 fatcat:tjfzx3b6sngjzcyhqpmrdwcmqi