Average case complexity of unbounded fanin circuits

Several authors have shown that the PARITY-function cannot be computed by unbounded fanin circuits of small depth and polynomial size. Even more, constant depth k circuits of size exp(n^⊖(1/k) ) give wrong results for PARITY for almost half of all inputs. We generalize these results in two directions. First, we obtain similar tight lower bounds for the average case complexity of circuits, measuring the computational delay instead of the static circuit depth. Secondly, with respect to average delay of unbounded fanin circuits we completely classify all parallel prefix functions, for which PARITY is just one prominent example. It is shown that only two cases can occur: a parallel prefix functions f either has the same average complexity as PARITY, that is the average delay has to be of order O(log n/ loglog s) for circuits of size s, or f can be computed with constant average delay and almost linear size there is no complexity level in between. This classification is achieved by analyzing the algebraic structure of semigroups that correspond to parallel prefix functions. It extends methods developed for bounded fanin circuits by the first author in his Ph.D. Thesis