## Average Case Completeness

- Yuri Gurevich
- J. Comput. Syst. Sci.
- 1991

- Published 1995

In contrast to machine models like Turing machines or random access machines, circuits are a rigid computational model. The internal information ow of a computation is xed in advance, independent of the actual input. Therefore, in complexity theory only worst case complexity measures have been used to analyse this model. In JRS 94] we have deened an average case measure for the time complexity of circuits. Using this notion tight upper and lower bounds could be obtained for the average case complexity of several basic Boolean functions. In this paper we will examine the asymptotic average case complexity of the set of n-input Boolean functions. In contrast to the worst case a simple counting argument does not work. We prove that almost all Boolean function require at least n?log n?llog n expected time even for the uniform probability distribution. On the other hand, we show that there are signiicant subsets of functions that can be computed with a constant average delay. Finally, worst case and average case complexity of Boolean will be compared. We show that for each function that is not computable by circuits of depth less than d, the expected time complexity will be at least d ? log n ? log d with respect to an explicitely deened probability distribution. We obtain a nontrivial bound on the complexity of such a distribution.

@inproceedings{Jakoby1995MedizinischeUA,
title={Medizinische Universitt at L Ubeck 1 Malign Distributions for Average Case Circuit Complexity},
author={Andreas Jakoby and Christian Schindelhauer},
year={1995}
}