Threshold Circuits of Small Majority-Depth

We investigate the complexity of computations with constant-depth threshold circuits. Such circuits are composed of gates that determine if the sum of their inputs is greater than a certain threshold. When restricted to polynomial size, these circuits compute exactly the functions in the class TC$ sp0$. These circuits are usually studied by measuring their efficiency in terms of their total depth. Using this point of view, the best division and iterated multiplication circuits have depth three and four, respectively. In this thesis, we propose a different approach. Since threshold gates are much more powerful than AND-OR gates, we allow the explicit use of AND-OR gates and consider the main measure of complexity to be the majority-depth of the circuit, i.e. the maximum number of threshold gates on any path in the circuit. Using this approach, we obtain division and iterated multiplication circuits of total depth four and five, but of majority-depth two and three. The technique used is called Chinese remaindering. We present this technique as a general tool for computing functions with integer values and use it to obtain depth-four threshold circuits of majority-depth two for other arithmetic problems such as the logarithm and power series approximation. We also consider the iterated multiplication problem for integers modulo q and for finite fields. The notion of majority-depth naturally leads to a hierarchy of subclasses of TC$ sp0$. We investigate this hierarchy and show that it is closely related to the usual depth hierarchy.