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Boolesche Minimalpolynome und Überdeckungsprobleme


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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  1. 1.

    Aho, A. V., Hopcroft, J. E., Ullman, J. P.: The design and analysis of computer algorithms. Reading (Mass.): Addison Wesley 1974

  2. 2.

    Cobham, A., North, J. H.: Extension of the integer programming approach to the minimization of Boolean functions. Thomas J. Watson Research Ctr., IBM Corporation, Yorktown Heights (N.Y.), IBM Research Report RC-915, 1963

  3. 3.

    Cook, S.A.: The complexity of theorem-proving procedures. Proc. Third Annual ACM Symposium on Theory of Computing, Shakerheights, Ohio, 1971, p. 151–158

  4. 4.

    Garey, M. R., Johnson, D. S., Stockmeyer, L.: Some simplified NP-complete problems. Proc. Sixth Annual ACM Symposium on Theory of Computing, Seatle (Wash.), 1974, p. 47–63

  5. 5.

    Gimpel, J. F.: A method of producing a Boolean function having an arbitrarily prescribed prime implicant table. IEEE-EC 14, 485–488 (1965)

  6. 6.

    Hotz, G.: Schaltkreistheorie. Berlin-New York: de Gruyter 1974

  7. 7.

    Karp, R. M.: Reducibility among combinatorial problems. In: Miller, R. E., Thatcher, J.W. (eds.), Complexity of computer computations. Plenum Press 1972

  8. 8.

    Quine, W. V.: The problem of simplifying truth functions. American Mathematical Monthly 61, 521–531 (1952)

  9. 9.

    Roth, J. P.: Algebraic topological methods for the synthesis of switching systems. I. Transactions of the American mathematical Society 88, 301–326 (1958)

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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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