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A056287 Maximal AND-OR formula complexity (operator count) for n-input Boolean functions 4
1, 3, 9, 15, 28 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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).
LINKS
EXAMPLE
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).
CROSSREFS
Sequence in context: A131801 A280963 A122819 * A099409 A002127 A357764
KEYWORD
nonn,nice,hard,more
AUTHOR
N. J. A. Sloane, Jan 05 2001
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
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)