A056287 -id:A056287 - cp's OEIS frontend

A057241 Circuit cost of the hardest Boolean function of n variables; metric: 2-input AND-gates cost 1, NOT is free, fanout is free, inputs are free, no feedback allowed.

0, 0, 3, 6, 10

Offset: 0

Richard C. Schroeppel, Jan 10 2001

a(5) <= 23. Boole expansion (Knuth page 52).
XOR costs 3.
a(5) <= 20. Modify procedure used to calculate a(5) in A056287 to add XOR.
a(5) <= 18. Let f = g(xi = h), then cost(f) <= cost(h) + cost(g). This is a generalization of minimum memory operations (xi = xj op xk -- Knuth page 101). Add this for min(cost(g),cost(h)) <= 3.
a(5) <= 17. Add cost(f(xi,xj = xi op1 xj, xi op2 xj)) <= cost(f) + 2 (Knuth page 105).
a(5,17) <= 187 where a(5,c) is the number of functions classes that cost c.
a(5,17) <= 1 (David Ian Stevenson).
a(5) >= 13 Stockmeyer limit for symmetric function S124 (Knuth exercise 7.1.2 - 80 and answer to 20 which clarifies Richard C. Schroeppel's hardest functions).

D. E. Knuth, The Art of Computer Programming, Volume 4A, Addison Wesley, 2011.

Richard C. Schroeppel, Comments and examples
David Ian Stevenson, Shorter chains for 185 function classes
David Ian Stevenson, Table of a(5,c)

Cf. A056287, A058759.

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)

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A057241 Circuit cost of the hardest Boolean function of n variables; metric: 2-input AND-gates cost 1, NOT is free, fanout is free, inputs are free, no feedback allowed.

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A178939 Maximal AND-OR-XOR formula complexity (operator count) for n-input Boolean functions.

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