Determination of the irredundant forms of a Boolean function using Walsh-Hadamard analysis and dyadic groups

Abstract
Transform methods and dyadic groups have been used for the classification of Boolean functions as well as for prime implicant determination. In a recent paper a prime implicant extraction method, based on Walsh-Hadamard transform methods, was presented. It processes the true minterms of the function separately, one at a time. In this paper this transform method is applied to the covering problem. Taking the prime implicants as binary variables a slight modification of the prime-implicant extraction method allows one to identify all complete covers by inspecting the elements of an inverse transform. Redundant forms can be detected and rejected easily. Another method for the determination of all irredundant covers classifies the 2m elements of the dyadic group of element length m as incomplete, redundant or irredundant covers, m beingthe number of prime implicants. A version for hand-worked problems is given, as well as a computer-oriented version.

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