On TC, AC, and Arithmetic Circuits

Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC [CMTV96], we study the class of functions #AC. One way to define #AC 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 ∗Part of this research was done while visiting the University of Ulm under an Alexander von Humboldt Fellowship. †Supported in part by NSF grants CCR-9509603 and CCR-9734918. Portions of the work were performed while this author was a visiting scholar at the Institute of Mathematical Sciences, Chennai, India ‡Supported in part by a Rutgers University Graduate Excellence Fellowship and by NSF grants CCR-9509603 and CCR-9734918. function classes, for which we know no nontrivial lower bounds, lower bounds for #AC follow easily from established circuit lower bounds. One of our main results is a characterization of TC in terms of #AC: A language A is in TC if and only if there is a #AC function f and a number k such that x ∈ A⇐⇒ f(x) = 2|x|k . Using the naming conventions of [FFK94, CMTV96], this yields: TC = PAC = C=AC. Another restatement of this characterization is that TC can be simulated by constant-depth arithmetic circuits, with a single threshold gate. We hope that perhaps this characterization of TC in terms of AC circuits might provide a new avenue of attack for proving lower bounds. Our characterization differs markedly from earlier characterizations of TC in terms of arithmetic circuits over finite fields [RT92, BFS92]. Using our model of arithmetic circuits, computation over finite fields yields ACC. We also prove a number of closure properties and normal forms for #AC.