There are research articles (e.g. Chakrapani & Palem) and devices (e.g. Lyric) that use a so-called probabilistic logic. I suppose the idea is that the outputs of such a device, given some inputs, will converge to some probability distribution. What is the difference between these devices and those using analog signals? That is, are these devices still considered digital, analog, mixed-signal?
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This paper seems to describe some novel kind of (probabilistic) boolean logic, and it is not about implementation. I only skimmed through the paper, but it seems to be just another one of those theories. There is, by the way, a simple reason why probabilistic logics don't give you what classical logics give you, namely, they are not truth functional (i.e. the value of A & B does not depend solely on the value of A and the value of B).
As for implementing such a thing on a chip: I think both are possible. If you do it digitally, then you're calculating probabilities, and you can just as well run some code on a CPU. I don't really know about analog implementations, but I guess any elementary analog component (transistor, opamp etc) can be seen as performing some kind of basic arithmetic operation on voltages and currents. Whether a circuit gives outputs that adhere to, or approximate, the Kolmogorov laws of probability, that's another question, but my guess is: it is somehow possible and maybe it has been done.