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A056287
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Maximal AND-OR formula complexity (operator count) for n-input Boolean functions
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4
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OFFSET
| 1,2
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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).
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LINKS
| Russ Cox, Notes on computing a(4)
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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).
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CROSSREFS
| Cf. A058759, A057241, A178939.
Sequence in context: A131822 A131801 A122819 * A099409 A002127 A061810
Adjacent sequences: A056284 A056285 A056286 * A056288 A056289 A056290
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KEYWORD
| nonn,nice,hard,more
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jan 05 2001
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EXTENSIONS
| a(3) and a(4) computed by Russ Cox (rsc(AT)swtch.com) Jan 03, 2001.
a(5) computed by Russ Cox (rsc(AT)swtch.com) and Alex Healy (ahealy(AT)post.harvard.edu), Jul 12, 2010.
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