On TC^0, AC^0, and Arithmetic Circuits

摘要

Continuing a line of investigation that has studied the function classes #P, #SAC^1, #L, and #NC^1, we study the class of functions #AC^0. One way to define #AC^0 is as the class of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates. In contrast to the preceding function classes, for which we know no nontrivial lower bounds, lower bounds for #AC^0 follow easily from established circuit lower bounds. One of our main results is a characterization of TC^0 in terms of #AC^0: A language A is in TC^0 if and only if there is a #AC^0 function f and a number k such that x in A iff f(x) = 2^{|x|^k}. Using well known naming conventions this yields: TC^0 = PAC^0 = C_=AC^0. Another restatement of this characterization is that TC^0 can be simulated by constant-depth arithmetic circuits, with a single threshold gate. We hope that perhaps this characterization of TC^0 in terms of AC^0 circuits might provide a new avenue of attack for proving lower bounds. Our characterization differs markedly from earlier characterizations of TC^0 in terms of arithmetic circuits over finite fields Using our model of arithmetic circuits, computation over finite fields yields ACC^0. We also prove a number of closure properties and normal forms for #AC^0.