Approximate evaluations of characteristic polynomials of Boolean functions

Abstract

Motivated by combinational circuit verification and testing, we study the approximate evaluation of characteristic polynomials of Boolean functions. We consider an oracle model in which the values of the characteristic polynomials are approximated using the evaluations of the corresponding Boolean functions. The approximation error is defined in the worst case, average case and randomized settings. We derive lower bounds on the approximation errors in terms of the number of Boolean function evaluations. We design algorithms with an error that matches the lower bound. Let k(ε,n) denote the minimal number of Boolean function evaluations needed to reduce the initial error by a factor of ε where n is the number of Boolean variables. We show that k(ε,n) is exponential in n in the worst and average case settings, and that it is independent of n and of order ε−2 in the randomized setting.