Malign Distributions for Average Case Circuit Complexity

In contrast to machine models like Turing machines or random access machines, circuits are a static computational model. The internal information ow of a computation is xed in advance, independent of the actual input. Therefore, the size and the depth are natural and simple measures for circuits and provide a worst case measure. We consider a model where an internal gate can be evaluated when the result is determined only by parts of its input. So we obtain a dynamic deenition of delay. In JRS94] 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. Here, we will examine the asymptotic average case complexity of the set of all n-ary Boolean functions. In contrast to worst case analyses a simple counting argument does not work. We prove that almost all Boolean function require at least n ?log n ?log log n expected time even for the uniform probability distribution. On the other hand, there are signiicant subsets of functions that can be computed with a constant average delay. Finally, we compare worst case and average case complexity of Boolean functions. 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. In addition, a nontrivial upper bound on the complexity of such a distribution will be obtained.