Characteristic polynomial method for verification and test of combinational circuits

In a recent paper, Jain et al. [1992] probabilistically establish the equivalence of two given Boolean functions. They assign randomly selected integers to input variables and compute integer-valued transform functions. If the evaluations give the same value, the Boolean functions are shown to be identical with some probability of error, The error probability is reduced as the domain from which the integers are obtained is enlarged. Also, for a fixed domain, the probability of error can be reduced by taking multiple samples for inputs. In this paper, we assign randomly selected real numbers to input variables and show that when the characteristic polynomials of two Boolean functions give the same value, then the functions are identical with probability 1. It can be shown that when the inputs are sampled from the real domain [0,1], and are interpreted as probabilities of logic 1, then the corresponding value of the characteristic polynomial gives the probability of output logic 1 for the Boolean function.