Arithmetic Versions of Constant Depth Circuit Complexity Classes

The boolean circuit complexity classes AC ⊆ AC[m] ⊆ TC ⊆ NC have been studied intensely. Other than NC, they are defined by constant-depth circuits of polynomial size and unbounded fan-in over some set of allowed gates. One reason for interest in these classes is that they contain the boundary marking the limits of current lower bound technology: such technology exists for AC and some of the classes AC[m], while the other classes AC[m] as well as TC lack such technology. Continuing a line of research originating from Valiant’s work on the counting class ♯P , the arithmetic circuit complexity classes ♯AC and ♯NC have recently been studied. In this paper, we define and investigate the classes ♯AC[m] and ♯TC. Just as the boolean classes AC[m] and TC give a refined view of NC, our new arithmetic classes, which fall into the inclusion chain ♯AC ⊆ ♯AC[m] ⊆ ♯TC ⊆ ♯NC, refine ♯NC. These new classes (along with ♯AC) are also defined by constant-depth circuits, but the allowed gates compute arithmetic functions. We also introduce the classes DiffAC[m] (differences of two AC[m] functions), which generalize the class DiffAC studied in previous work. We study the structure of three hierarchies: the ♯AC[m] hierarchy, the DiffAC[m] hierarchy, and a hierarchy of language classes. We prove class separations and containments where possible, and demonstrate relationships among the various hierarchies. For instance, we prove that the hierarchy of classes ♯AC[m] has exactly the same structure as the hierarchy of classes AC[m]: AC[m] ⊆ AC[m] iff ♯AC[m] ⊆ ♯AC[m] We also investigate closure properties of our new classes, which generalize those appearing in previous work on ♯AC and DiffAC. ∗Department of Computer Science, Cornell University, Ithaca, NY 14853. E-mail: [email protected].