Reviewing bounds on the circuit size of the hardest functions

  • Authors:

  • Gudmund Skovbjerg Frandsen;Peter Bro Miltersen

  • Affiliations:

  • Basic Research in Computer Science, Department of Computer Science, University of Aarhus, Aarhus N, Denmark;Basic Research in Computer Science, Department of Computer Science, University of Aarhus, Aarhus N, Denmark

  • Venue:
  • Information Processing Letters
  • Year:
  • 2005

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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 2n/n (1 + log n/n-O(1/n))≤L(n) ≤ 2n/n(1 + 3log n/n + O(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 {¬ ∧ ∨}.