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By 'deterministic Hidden Markov Models', I mean HMMs in which all state transition probabilities and output probabilities = 1 or 0.

Have models subject to this restriction received any significant study, and are there any useful results relating to them?

This is an area I know very little about, so it is quite possible that the models I am describing correspond to something which commonly goes by a different name.

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There is a recent paper called "determenistic POMDPs revisited" by Blai Bonet:

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If I read your post correctly, you are interested in deterministic dynamical systems on a state space $X$, that is, sequences $(x_t)$ such that $x_{t+1}=s(x_t)$ for every $t$, for some deterministic function $s:X\to X$, that are partially observed through a function $o:X\to Y$. Thus, you are considering the process $(y_t)$ on $Y$ defined by $y_t=o(x_t)$ for every $t$.

Obviously the only way some randomness could creep in is through the initial distribution of the state process $(x_t)$, that is, the distribution of $x_0$.

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