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:
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1)
Switching functions with “don't care”-points and those without yield essentially the same class of tables of prime implicants.
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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.
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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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Literatur
Aho, A. V., Hopcroft, J. E., Ullman, J. P.: The design and analysis of computer algorithms. Reading (Mass.): Addison Wesley 1974
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
Cook, S.A.: The complexity of theorem-proving procedures. Proc. Third Annual ACM Symposium on Theory of Computing, Shakerheights, Ohio, 1971, p. 151–158
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
Gimpel, J. F.: A method of producing a Boolean function having an arbitrarily prescribed prime implicant table. IEEE-EC 14, 485–488 (1965)
Hotz, G.: Schaltkreistheorie. Berlin-New York: de Gruyter 1974
Karp, R. M.: Reducibility among combinatorial problems. In: Miller, R. E., Thatcher, J.W. (eds.), Complexity of computer computations. Plenum Press 1972
Quine, W. V.: The problem of simplifying truth functions. American Mathematical Monthly 61, 521–531 (1952)
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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DOI: https://doi.org/10.1007/BF00289615