Abstract.
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
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There exists a multilinear circuit of product-depth d + 1 and of polynomial size computing f.
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Every multilinear circuit of product-depth d computing f has super-polynomial size.
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The product-depth of a circuit is the largest number of product gates in a directed path in it. According to the standard definition of depth, we show a super-polynomial separation between the size of depth d and depth d+ 2 multilinear circuits.
Manuscript received 18 June 2008
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Raz, R., Yehudayoff, A. Lower Bounds and Separations for Constant Depth Multilinear Circuits. comput. complex. 18, 171–207 (2009). https://doi.org/10.1007/s00037-009-0270-8
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DOI: https://doi.org/10.1007/s00037-009-0270-8