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

1, 3, 9, 15, 28

Offset: 1

Author

N. J. A. Sloane, Jan 05 2001

Keywords

• nonn
• nice
• hard
• more

Comments

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).

Examples

			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).
		

Links

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

Crossrefs

Cf. A058759, A057241, A178939.

Extensions

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