The complexity of the representation of multiple-output Boolean functions

This paper considers cost of logic circuits that implement Boolean functions. The realization of Boolean functions is considered in the class of reversible logic circuits. Reversible circuits are constructed with elementary reversible circuits known as Toffoli gates or Toffoli basis. Traditional Boolean functions are not reversible except for two unary functions. However, Boolean functions can be modeled as so-called multiple-output functions for which the number of outputs is equal to the number of arguments and that are permutations on the set of arguments sets. In the paper, Boolean function is implemented as the multiple-output function that in turn is realized as a reversible circuit constructed in the Toffoli basis. The circuit implementing this function is not uniquely defined. Thus the complexity of the function can be defined as the complexity of the minimal circuit implementing this function. This paper presents results on the complexity of most complex functions and on Shannon function value for the Boolean functions in the class of reversible circuits implemented in a subset of Toffoli basis. The solution to the problem is reduced to solving the problem of finding the Shannon function value for the Boolean functions class in the class of extended Reed–Muller forms. A special sequence of functions is constructed for this class. We have proved that this sequence consists of the most complex functions and found the complexity of these functions.