A057241 -id:A057241 - cp's OEIS frontend

A056287 Maximal AND-OR formula complexity (operator count) for n-input Boolean functions.

1, 3, 9, 15, 28

Offset: 1

N. J. A. Sloane, Jan 05 2001

a(n) = minimal number of edges in 2-terminal series-parallel switching network (where edges are labeled with the variables X_i and X_i') which achieves the worst f.

Consider all 2^2^n Boolean functions f of n variables X_1, ..., X_n; the X_i's and their negated values X_1', ..., X_n' are available and we must realize f using AND's and OR's of these 2n variables with the smallest total number of AND's and OR's; call the minimal total number of AND's and OR's used G(f); then a(n) = max G(f).

			For n=2 a worst f is X XOR Y, which can be realized by X AND Y' OR X' AND Y = XY' + X'Y.
For n=3 a worst f is X XOR Y XOR Z, which can be realized by (X*Z'+X'*Z+Y')*(X*Z+X'*Z'+Y).
For n=4 a worst f is W XOR X XOR Y XOR Z, which can be realized by ((X XOR Z)'+(W XOR Y)')*((X XOR Z)+(W XOR Y)) = (X*Z'+X'*Z+W'*Y+W*Y')*(X*Z+X'*Z'+W*Y+W'*Y').
For n=5 there are three worst f's up to permutation and negation of input variables. They have 32-bit truth tables 0x16686997, 0x16696997 and 0x1669e996 (in hexadecimal).

Russ Cox, Notes on computing a(4)
Russ Cox, Minimum Boolean Formulas

Cf. A058759, A057241, A178939.

a(3) and a(4) computed by Russ Cox, Jan 03 2001

a(5) computed by Russ Cox and Alexander D. Healy, Jul 12 2010

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).

Original entry on oeis.org

1, 4, 8, 13

Offset: 1

N. J. A. Sloane, Jan 01 2001

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.

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].

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

Cf. A056287, A057241.

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.

Additional comments from Vladeta Jovovic, Jan 01 2001

a(5) <= 28 (Povarov)

cp's OEIS Frontend

A056287 Maximal AND-OR formula complexity (operator count) for n-input Boolean functions.

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