Exact Identi cation of Read-once Formulas Using Fixed Points ofAmpli cation

In this paper we describe a new technique for exactly identifying certain classes of read-once Boolean formulas. The method is based on sampling the input-output behavior of the target formula on a probability distribution that is determined by the xed point of the formula's ampli cation function (de ned as the probability that a 1 is output by the formula when each input bit is 1 independently with probability p). By performing various statistical tests on easily sampled variants of the xedpoint distribution, we are able to e ciently infer all structural information about any logarithmic-depth formula (with high probability). We apply our results to prove the existence of short universal identi cation sequences for large classes of formulas. We also describe extensions of our algorithms to handle high rates of noise, and to learn formulas of unbounded depth in Valiant's model with respect to speci c distributions. Most of this research was carried out while all three authors were at MIT Laboratory for Computer Science with support provided by ARO Grant DAAL03-86-K-0171, DARPA Contract N00014-89-J-1988, NSF Grant CCR-88914428, and a grant from the Siemens Corporation. R. Schapire received additional support from AFOSR Grant 89-0506 while at Harvard University. S. Goldman is currently supported in part by a G.E. Foundation Junior Faculty Grant and NSF Grant CCR-9110108.

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