Lower Bounds for Approximations by Low Degree Polynomials Over Zm

نویسندگان

  • Noga Alon
  • Richard Beigel
چکیده

We use a Ramsey-theoretic argument to obtain the first lower bounds for approximations over Zm by nonlinear polynomials: A degree-2 polynomial over Zm (m odd) must differ from the parity function on at least a 1=2 1=2(logn) (1) fraction of all points in the Boolean n-cube. A degree-O(1) polynomial over Zm (m odd) must differ from the parity function on at least a 1=2 o(1) fraction of all points in the Boolean n-cube. These nonapproximability results imply the first known lower bounds on the top fanin of MAJ Æ MODm Æ ANDO(1) circuits (i.e., circuits with a single majority-gate at the output node, MODm-gates at the middle level, and constantfanin AND-gates at the input level) that compute parity: MAJ Æ MODm Æ AND2 circuits that compute parity must have top fanin 2(logn) (1) . Parity cannot be computed by MAJ Æ MODm Æ ANDO(1) circuits with top fanin O(1). Similar results hold for the MODq function as well.

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تاریخ انتشار 2001