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Boolesche Minimalpolynome und Überdeckungsprobleme

Summary

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:

  1. 1)

    Switching functions with “don't care”-points and those without yield essentially the same class of tables of prime implicants.

  2. 2)

    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.

  3. 3)

    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

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