Yes!, strangely enough I have thought about this.. Here is the idea:
Model the Maxwell's Demon as a computer program. Then, How does Demon came to know particle's speed and position before opening the door for selection?
Suppose that the demon can't measure the speed at which particles hit the door, why? because that would change particle's speeds, so demon have to figure out before open it, without looking, without measuring. To be fair we will let the demon know the rules of the game in advance, i.e. feed the demon with laws of motion, interactions of particles, and initial conditions, enough of the physics/dynamic model.
Now model the gas of particles also as a computer program that is running same code given to the demon for every particle, so the gas is computing a result from its initial conditions, the Demon doesn't know that result until it halt (if ever): namely "a particle with the right speed is at the door", the decision yes/no question we are asking to the system is "Have a particle the right position and enough speed?", if so, the door could be opened and the fast particle can go into the high temperature side of the room setting new initial conditions (will those consecutive problems has an answer? or will run forever?)
There will be a time when there is no particle with enough speed to cross the boundary, so, there will be a time when the code will run forever (do not halt) for almost any given threshold.
Demon wants to know the result that is computed by the gas, but the result is in a sense involved potentially within the source code of the particle's laws plus the initial conditions.. of course we need to run that program to know it. If Demon run the same program waiting for the right speed at the output, the program could halt or it could run forever (but we suppose also that demon has no more computational power than the gas, so it won't be able to decide the door opening on time ).
Daemon could try to figure out the program output (or if it will halt) by watching the source and inputs without running it but it's like trying to solve the Halting Problem, why? because Demon doesn't know what laws and initial condition will be feed so Demon should be prepared to solve for any set of law and initial conditions, and we know it's not possible in general, it will need an oracle, if it could it will be enough to build a demon to generate energy from nothing. (even knowing the laws and initial condition, both things already enough hard to know)
This thought experiment can link how a reduction in entropy, by mean of computers, could in some way bounded by Halting Problem, as a problem to anticipate in general the outcomes.
(Sometime all limits seems to be the same limit..)
More about Particle Laws
Particle's laws are not the main issue of this thought experiment, those laws could be quantum or classical, but we must have into account the fact of complexity of laws and initial conditions, the complexity of the arrange of particles is not bounded, and it could have a lot of added complexity (in an extreme example of initial conditions you could even insert a whole computer firing particles according to an internal source code and give that code to the daemon).