Complexity of Symmetric Functions in Perceptron-Like Models (1992) [5 citations — 2 self]
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Abstract:
We examine the size complexity of the symmetric boolean functions in two circuit models containing threshold gates: the d-perceptron model [BRS, ABFR] (a single threshold function of constant-depth AND/OR circuits) and the parity-threshold model studied by Bruck [Br] (a single threshold function of exclusive-ORs). These models are intermediate between the well-understood model of constant-depth AND/OR circuits and the still mysterious model of general constantdepth threshold circuits. In the d-perceptron model, we give an if and only if condition for a symmetric boolean function to be computable by a quasi-polynomial size d-perceptron: we show that a symmetric boolean function can be computed by a quasi-polynomial size d-perceptron iff it has only poly-log many sign changes, i.e. the number of times the function changes output value as the number of inputs on varies from zero through n ( we call this parameter the degree of the symmetric function) is bounded above by log c n for some c. This extends the work of Fagin et al. [FKPS] which gave a very nice characterization of symmetric functions computable by AC 0 circuits. An interesting consequence of our result is that a recent construction of Beigel [Be] is optimal. In the parity-threshold model, we find a similar parameter as a measure of size complexity, the odd-even degree, or number of output value changes as the number of inputs on varies through the odd numbers from 0 through n and then through the even numbers. We observe that poly-log odd-even degree implies quasipolynomial size, conjecture the converse, and prove the converse in the presence of a certain technical condition on the function's Fourier coefficients. In particular, we prove that the modulo-q function for any constant q? 2 has more than quasi-polynomial size.
