Lower Bounds and Separations for Constant Depth Multilinear Circuits
We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a super-polynomial separation between the size of product-depth d and product-depth d + 1 multilinear circuits (where d is constant). That is, there exists a polynomial f such that
There exists a multilinear circuit of product-depth d + 1 and of polynomial size computing f.
Every multilinear circuit of product-depth d computing f has super-polynomial size.
Keywords.Arithmetic circuits constant depth circuits separations
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© Birkhäuser Verlag, Basel 2009