Uniform Circuit Complexity

Abstract

In the previous chapters we have studied some models of parallel machines. In this chapter, we are going to define some parallel complexity classes. In modelling parallel computation, we wish to model the situation in which the number of processors is greater than the length of the input; however it is clear that any “real world” parallel computer should have a feasible number of processors. For this reason most of the theoretical research on parallelism has focused on the case in which the number of processors is bounded by a polynomial on the size of the input. As we have mentioned in the Chapters one and two, the number of theoretical models of parallel computation is very large; we shall define complexity classes in terms of uniform families of circuits, which seems to be a unifying framework for most of the parallel models, as well as a natural measure to count the size of the hardware. In the second section, we introduce the basic definitions and properties. In the following section we prove the robustness of parallel complexity classes with respect to other models of parallel computers studied in Chapters 1 and 2; in particular we prove the equivalence of uniform circuits with vector machines. Then we discuss other measures of uniformity and its incidence in the definition of parallel classes. In section 4, we establish the relationship of uniform circuits with alternating machines, and prove a characterization of the defined parallel classes in terms of alternating machines.