DOOM still used fixed point arithmetic at the time for speed, so it could be a large set but not as big as floats. You could also drop a few bits of precision off the LSB and make your search space that much smaller.
That would approximate infinity quite fast once you move around a few steps :-)
I guess limiting rotation to 64 steps would be enough, but even then the state space grows exponentially. So it's not really useful in the end, but that fits the project, I guess.
But you have to multiply that by the possible locations on the map and the enemy states. I imagine it would be possible to get a few hits at the starts of levels but I suspect that the vast majority of the frames for any play-through have never been seen before.