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- W1640244183 abstract "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 then delay O(1) is sufficient, but otherwise delay $Omega(log n)$ is necessary; there are no intermediate growth rates. We also consider (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." @default.
- W1640244183 created "2016-06-24" @default.
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- W1640244183 date "2013-08-13" @default.
- W1640244183 modified "2023-09-27" @default.
- W1640244183 title "Gap Theorems for the Delay of Circuits Simulating Finite Automata" @default.
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