# Threshold circuits of bounded depth

**A. Hajnal, W. Maass, P. Pudlak, M. Szegedy, and G. Turan**

### Abstract:

We examine a powerful model of parallel computation: polynomial size threshold
circuits of bounded depth (the gates compute threshold functions with
polynomial weights). Lower bounds are given to separate polynomial size
threshold circuits of depth 2 from polynomial size threshold circuits of
depth 3 and from probabilistic polynomial size circuits of depth 2. With
regard to the unreliability of bounded depth circuits, it is shown that the
class of functions computed reliably with bounded depth circuits of
unreliable A, v , 1 gates is narrow. On the other hand, functions computable
by bounded depth, polynomial-size threshold circuits can also be computed by
such circuits of unreliable threshold gates. Furthermore we examine to what
extent imprecise threshold gates (which behave unpredictably near the
threshold value) can compute nontrivial functions in bounded depth and a
bound is given for the permissible amount of imprecision. We also discuss
threshold quantifiers and prove an undefinability result for graph
connectivity.

**Reference:** A. Hajnal, W. Maass, P. Pudlak, M. Szegedy, and G. Turan.
Threshold circuits of bounded depth.
*J. Comput. System Sci.*, 46:129-154, 1993.