Has there been significant study of deterministic Hidden Markov Models? 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.
 A: There is a recent paper called "determenistic POMDPs revisited" by Blai Bonet:
http://arxiv.org/ftp/arxiv/papers/1205/1205.2659.pdf
A: There are some interesting hidden measurement models developed by D. Aerts. Its related to some spin ising models, where has been issues, with these extensions.https://en.wikipedia.org/wiki/Hidden-measurements_interpretation
Aerts, Diederik; Sassoli de Bianchi, Massimiliano, The unreasonable success of quantum probability. II: Quantum measurements as universal measurements, J. Math. Psychol. 67, 76-90 (2015). ZBL1354.81006.There was some difficulty in generalizing the mechanism to systems with more than three outcomes. He uses potential, elastic analogies, and a propensity like creative contextual aspect approach, distinct from most other hidden variable models. I recommend you read into it. 
It quite complex. He has recently published some large papers, where the issue of more then three outcomes problem was finally resolved I think, to their liking. 
This was a major complaint with the account, where its a model where the probabilities are not exactly redundant, the outcome which occurs is a deterministic or deterministic like function of the probabilities, amplitudes mod squared, or propensities if you will, and $\epsilon-d$ parameter,coloumbs, elastic charges,  and various other occult/resistance forces admixed with analogies to pulley system.
He used some kind of extended bloch sphere I think in the end to model., which for some time, was thought for some time, not possible.
You really need a line with three end points, or a some,multi-dimensional model to get it do what perhaps he wanted, a uniform distributed, 
that one and only outcome obtains for the most part, a super-set /higher probability, the outcome would still have occurred, notion (when working on the foundations of probabilities meaning as graded necessity)
mathematical compatibility with any probability value, and thus the values or distributions of the other parameters do not have to change, at least directly as function of the probability values. 
Despite the fact that the probability value has an active role in determining which outcome occurs
and what incidentally what I want with my model. It ended up being modelled on some kind of 4 dimensional equilateral triangle. 
To investigate  stastitical learning, the born rule, cognitive science, quantum logic, and quantum mechanics and economics, and of course probability theory.
And I dont think that its merely due to the nature of quantum mechanics. 
A: 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 $x_0$. 
