Gap Theorems for the Delay of Circuits Simulating Finite Automata

We study the delay (also known as depth) of circuits that simulate finite automata, showing that only certain growth rates (as a function of the number $n$ of steps simulated) are possible. A classic result due to Ofman (rediscovered and popularized by Ladner and Fischer) says that delay $O(\log n)$ is always sufficient. We show that if the automaton is "generalized definite", then delay O(1) is sufficient, but otherwise delay $\Omega(\log n)$ is necessary; there are no intermediate growth rates. We also consider "physical" (rather than "logical") delay, whereby we consider the lengths of wires when inputs and outputs are laid out along a line. In this case, delay O(n) is clearly always sufficient. We show that if the automaton is "definite", then delay O(1) is sufficient, but otherwise delay $\Omega(n)$ is necessary; again there are no intermediate growth rates. Inspired by an observation of Burks, Goldstein and von Neumann concerning the average delay due to carry propagation in ripple-carry adders, we derive conditions for the average physical delay to be reduced from O(n) to $O(\log n)$, or to O(1), when the inputs are independent and uniformly distributed random variables; again there are no intermediate growth rates. Finally we consider an extension of this last result to a situation in which the inputs are not independent and uniformly distributed, but rather are produced by a non-stationary Markov process, and in which the computation is not performed by a single automaton, but rather by a sequence of automata acting in alternating directions.