On TC⁰, AC⁰, and arithmetic circuits

Abstract

Continuing a line of investigation that has studied the function classes P, we study the class of functions AC⁰. One way to define AC⁰ is as the class of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fanin addition and multiplication gates. In contrast to the preceding function classes, for which we know no nontrivial lower bounds, lower bounds for AC⁰ follow easily from established circuit lower bounds. One of our main results is a characterization of TC⁰ in terms of AC⁰: A language A is in TC⁰ if and only if there is a AC⁰ function f and a number k such that x∈A⇔f(x)=2^|x|k. Using the naming conventions, this yields: TC⁰=PAC⁰=C=AC⁰. Another restatement of this characterization is that TC⁰ can be simulated by constant-depth arithmetic circuits, with a single threshold gate. We hope that perhaps this characterization of TC⁰ in terms of AC⁰ circuits might provide a new avenue of attack for proving lower bounds. Our characterization differs markedly from earlier characterizations of TC⁰ in terms of arithmetic circuits over finite fields. Using our model of arithmetic circuits, computation over finite fields yields ACC⁰. We also prove a number of closure properties and normal forms for AC⁰.