On the minimum number of negations leading to super-polynomial savings

Authors:
Stasys Jukna
Affiliations:
Universität Frankfurt, Institut für Informatik, Robert-Mayer-Str. 11-15, D-60054 Frankfurt, Germany and Institute of Mathematics and Informatics, Akademijos 4, LT-2600 Vilnius, Lithuania
Venue:
Information Processing Letters
Year:
2004

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Abstract

We show that an explicit sequence of monotone functions fn : {0, 1}n → {0, 1}m (m ≤ n) can be computed by Boolean circuits with polynomial (in n) number of And, Or and Not gates, but every such circuit must use at least log n - O(log log n) Not gates. This is almost optimal because results of Markov [J. ACM 5 (1958) 331] and Fisher [Lecture Notes in Comput. Sci., Vol. 33, Springer, 1974, p. 71] imply that, with only small increase of the total number of gates, any circuit in n variables can be simulated by a circuit with at most ⌈log(n + 1)⌉ Not gates.