A058759 -id:A058759 - 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

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.

Original entry on oeis.org

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.

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

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