On the multiplicative complexity of Boolean functions over the basis [formula omitted]

The multiplicative complexity c ∧(f) of a Boolean function f is the minimum number of AND gates in a circuit representing f which employs only AND, XOR and NOT gates. A constructive upper bound, c ∧(f)=2 (n/2)+1−n/2−2 , for any Boolean function f on n variables ( n even) is given. A counting argumen...

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Bibliographic Details
Published in:Theoretical computer science Vol. 235; no. 1; pp. 43 - 57
Main Authors: Boyar, Joan, Peralta, René, Pochuev, Denis
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 17.03.2000
Elsevier
Subjects:
Algebraic logic
Exact sciences and technology
Logic and foundations
Mathematical logic, foundations, set theory
Mathematics
Sciences and techniques of general use
Lower bound
Upper bound
XOR gate
Algebraic complexity
Multiplicative complexity
Boolean function
Boolean algebra
Symmetric function
Hamming weight
Complexity
ISSN:0304-3975, 1879-2294
Online Access:Get full text
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Summary:The multiplicative complexity c ∧(f) of a Boolean function f is the minimum number of AND gates in a circuit representing f which employs only AND, XOR and NOT gates. A constructive upper bound, c ∧(f)=2 (n/2)+1−n/2−2 , for any Boolean function f on n variables ( n even) is given. A counting argument gives a lower bound of c ∧(f)=2 (n/2)− O(n) . Thus we have shown a separation, by an exponential factor, between worst-case Boolean complexity (which is known to be Θ(2 nn −1 )) and worst-case multiplicative complexity. A construction of circuits for symmetric Boolean functions on n variables, requiring less than n+3 n AND gates, is described.
ISSN:0304-3975
1879-2294
DOI:10.1016/S0304-3975(99)00182-6