Exact Threshold Circuits

We initiate a systematic study of constant depth Boolean circuits built using exact threshold gates. We consider both unweighted and weighted exact threshold gates and introduce corresponding circuit classes. We next show that this gives a hierarchy of classes that seamlessly interleave with the well-studied corresponding hierarchies defined using ordinary threshold gates. A major open problem in Boolean circuit complexity is to provide an explicit super-polynomial lower bound for depth two threshold circuits. We identify the class of depth two exact threshold circuits as a natural subclass of these where also no explicit lower bounds are known. Many of our results can be seen as evidence that this class is a strict subclass of depth two threshold circuits - thus we argue that efforts in proving lower bounds should be directed towards this class.