Model-Theoretic Characterizations of Boolean and Arithmetic Circuit Classes of Small Depth

In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes NC1, SAC1 and AC1 as well as their arithmetic counterparts #NC1, #SAC1 and #AC1. We build on Immerman’s characterization of constant-depth polynomial-size circuits by formulae of first-order logic, i.e., AC0 = FO, and augment the logical language with an operator for defining relations in an inductive way. Considering slight variations of the new operator, we obtain uniform characterizations of the three just mentioned Boolean classes. The arithmetic classes can then be characterized by functions counting winning strategies in semantic games for formulae characterizing languages in the corresponding Boolean class. 1998 ACM Subject Classification F.1.1 Models of Computation, F.1.3 Complexity Measures and Classes, F.4.1 Mathematical Logic