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TR05-032 | 16th March 2005 00:00

#### Reviewing Bounds on the Circuit Size of the Hardest Functions

TR05-032
Authors: Gudmund Skovbjerg Frandsen, Peter Bro Miltersen
Publication: 17th March 2005 08:55
Keywords:

Abstract:

In this paper we review the known bounds for $L(n)$, the circuit size
complexity of the hardest Boolean function on $n$ input bits. The
best known bounds appear to be $$\frac{2^n}{n}(1+\frac{\log n}{n}-O(\frac{1}{n})) \leq L(n) \leq\frac{2^n}{n}(1+3\frac{\log n}{n}+O(\frac{1}{n}))$$ However, the bounds do not seem to be
explicitly stated in the literature. We give a simple direct
elementary proof of the lower bound valid for the full binary basis,
and we give an explicit proof of the upper bound valid for the basis
$\{\neg,\wedge,\vee\}$.

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