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In his lectures on computer science, Feynman talks about finite state machines:

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he present a simple delay finite state machine

Let me now give a specific example of an FSM that actually does something, albeit something pretty trivial - a delay machine. You feed it a stimulus and, after a pause, it responds with the same stimulus. That's all it does. Figure 3.4 shows the "state diagram" of such a delay machine.

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I just don't see the utility of using two states here, why not just use a machine with one state?

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Your machine outputs the character it just received; Feynman's outputs the previous one.

More specifically, when Feynman's automaton receives the sequence of stimuli $s_1, s_2, \dots$, its output is $x, s_1, s_2, \dots$, where $x=0$ if the automaton starts in state $1$ and $x=1$, otherwise. Your automaton outputs $s_1, s_2, \dots$ given the same sequence of stimuli.

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