The main difficulty in finding minimal Boolean polynomials for given switching functions comes from the evaluation of the table of prime implicants.
We show the following results:
Switching functions with “don't care”-points and those without yield essentially the same class of tables of prime implicants.
A polynomial, which is minimal with respect to the costfunction, which counts the entries of conjunctions and disjunctions, must not be a polynomial with a minimal number of prime implicants.
Each binary matrix with at least one 1 in each row and column is the prime implicant table of some switching-function. Moreover this function can be constructed such that its prime implicants have arbitrarily prescribed costs.
Finally we make some remarks about the complexity of algorithms, which—given the graph of a switching function—find a minimal polynomial of this function.
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Paul, W.J. Boolesche Minimalpolynome und Überdeckungsprobleme. Acta Informatica 4, 321–336 (1975). https://doi.org/10.1007/BF00289615