Functional characterizations of uniform log-depth and polylog-depth circuit families

The classes of functions computable by uniform log-depth (NC¹ ) and polylog-depth circuit families are characterized as closures of a set of base functions. (The former is equivalent to ALOGTIME, the latter to polylogarithmic space.) The closures involve the `safe´ composition of S. Bellantoni and S. Cook (1992) as well as a safe divide-and-conquer recursion: a simple change to the definition of the latter distinguishes between log and polylog depth. The proofs proceed, in one direction, by showing that safe composition and divide-and-conquer recursion preserve growth rate and circuit depth bounds, and in the other, by simulating alternating Turing machines with divide-and-conquer recursion