Characteristic polynomial method for verification and test of combinational circuits

– This paper gives a new and efficient method of determining the equivalence of two given Boolean functions. We define a characteristic polynomial directly from the sum-of-product form of the logic function. The polynomial contains a real variable corresponding to each Boolean variable. Logical operations on the Boolean function correspond to arithmetic operations on the polynomial. We show that if the characteristic polynomials of two Boolean functions, when evaluated at the same randomly sampled values of their variables, produce identical result then the two corresponding Boolean functions are identical with probability 1. In a typical application, one characteristic function may be derived from the truth table specification while the other is obtained from a logic implementation. The proposed method is very efficient as it allows to prove correctness by just one evaluation of the two polynomials. We further show that when the real variables in the polynomial are restricted to the range [ 0 , 1 ], the value of the polynomial is the same as the probability of the Boolean function producing a true output. This result is applied to testing of combinational circuits. We derive the length of a random test sequence that will detect any fault in the circuit with any given arbitrarily high probability.