Arithmetic Constant-Depth Circuit Complexity Classes

Abstract

AbstractThe boolean circuit complexity classes AC 0 ⊆ AC 0[m] ⊆ TC 0 ⊆ NC 1 have been studied intensely. Other than NC 1, 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 0 and some of the classes AC 0[m], while the other classes AC 0[m] as well as TC 0 lack such technology.Continuing a line of research originating from Valiant’s work on the counting class \(\ensuremath{\sharp} P\), the arithmetic circuit complexity classes \(\ensuremath{\sharp} AC^0\) and \(\ensuremath{\sharp} NC^1\) have recently been studied. In this paper, we define and investigate the classes \(\ensuremath{\sharp} AC^0[m]\) and \(\ensuremath{\sharp} TC^0\), new arithmetic circuit complexity classes that are defined by constant-depth circuits and are analogues of the classes AC 0[m] and TC 0.KeywordsBoolean FunctionClosure PropertyCircuit ComplexityLanguage ClassArithmetic CircuitThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.