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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

Table of n, a(n) for n=1..5.

Russ Cox, Notes on computing a(4)

Russ Cox, Minimum Boolean Formulas

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

Cf. A058759, A057241, A178939.

Sequence in context: A131801 A280963 A122819 * A099409 A002127 A061810

Adjacent sequences:  A056284 A056285 A056286 * A056288 A056289 A056290

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 Alex Healy, Jul 12 2010

STATUS

approved

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Last modified May 29 21:28 EDT 2017. Contains 287257 sequences.