A058759 Shannon switching function: a(n) is the least number k such that any switching (or Boolean) function of n variables can be realized as a two-terminal network of AND's and OR's in which the total number of occurrences of the variables X_1, X_1', ..., X_n, X_n' is no more than k (where the primes indicate complements).

1, 4, 8, 13

Offset: 1

Author

N. J. A. Sloane, Jan 01 2001

Keywords

• nonn
• nice
• more
• hard

Comments

The variables X_1, ..., X_n and their negated values X_1', ..., X_n' are available, we only use AND's and OR's and we wish to minimize the total number of appearances of X_1, X_1', ..., X_n, X_n'. What is the worst case?

To describe this another way: X_i and X_i' are the (front and back) contacts (or elements) of a two-terminal network. Let L(S) be the number of contacts in a network S and L(f) = min L(S), where minimum is taken over all networks S which realize the Boolean function f. Then a(n) = max L(f), where maximum is taken over all n-variable Boolean functions.

References

• M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965; see especially pp. 230-235 and 408 (for a(4)=13).
• O. B. Lupanov, On the synthesis of contact networks, Dokl. Akad. Nauk SSSR, vol. 119, no. 1, pp. 23-26, 1958.
• G. N. Povarov, Investigation of contact networks with minimal number of contacts, Ph. D. thesis, Moscow, 1954.
• S. Seshu and M. B. Reed, Linear Graphs and Electrical Networks, Addison-Wesley, 1961; see p. 247.
• C. E. Shannon, The synthesis of two-terminal switching networks, Bell Syst. Tech. J., 28 (1949), pp. 59-98. Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 588-627.
• Y. L. Vasilev, Minimal contact networks for 4-variable Boolean functions, Dokl. Akad. Nauk SSSR, vol. 127 (no. 2, 1959), pp. 242-245 [shows that a(4) = 13].

Links

• N. J. A. Sloane, Illustration of initial terms
• Index entries for sequences related to Boolean functions

Crossrefs

Cf. A056287, A057241.

Formula

For any epsilon > 0, a(n) > (1-epsilon)*2^n/n for sufficiently large n (Shannon). For any epsilon > 0, a(n) <= (1+epsilon)*2^n/n for sufficiently large n (Lupanov). Hence a(n) ~ 2^n/n as n tends to infinity.

Extensions

Additional comments from Vladeta Jovovic, Jan 01 2001

a(5) <= 28 (Povarov)