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army1987 comments on An Intuitive Explanation of Solomonoff Induction - Less Wrong
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Only if there's discrete timing. If the time elapsed between two 1s can take uncountably many different values...
Can they? Both QFT and GR are based on real numbers, and the Standard Model has twenty-odd free parameters which are real numbers and hence in principle can require infinitely many bits to be fully specified. (Or do you mean something else by “the currently proposed laws”?)
That's a very good point.
It is where S.I. strays from how we applied Occam's razor.
If you lack knowledge of any discrete behaviour, it seems reasonable to assume that function is continuous, as a more likely hypothesis than it is made of steps of some specific size. The S.I. would produce models that contain specific step size, with highest probability for the shortest representable step size.
edit: how the hell is that 'wrong'?
As long as our best model of the laws of physics includes the concept of Planck time (Planck length), doesn't that mean that time (space) is discrete and that any interval of time (length of space) can be viewed as an integer number of Planck times (Planck lengths)?
We don't have anything remotely like a well-established theory of quantum gravity yet, so we don't know. Anyway, lack of observable frequency dispersion in photons from GRBs suggests space-time is not discrete at the Planck scale.[http://www.nature.com/nature/journal/v462/n7271/edsumm/e091119-06.html]
Doesn't quite work like this so far, maybe there will be some good discrete model but so far the Plank length is not a straightforward discrete unit, not like cell in game of life. More interesting still is why reals have been so useful (and not just reals, but also complex numbers, vectors, tensors, etc. which you can build out of reals but which are algebraic objects in their own right).
't Hooft has been quite successful in defining QM in terms of discrete cellular automata, taking "successful" to mean that he has reproduced an impressive amount of quantum theory from such a humble foundation.
This is answered quite trivially by simple analogy: second-order logics are more expressive than first-order logics, allowing us to express propositions more succinctly. And so reals and larger numeric abstractions allow some shortcuts that we wouldn't be able to get away with when modelling with less powerful abstractions.
This is also true of modern computers. This time is merely ignored with various hardware tricks to make sure things happen around the time we want them to happen. What it eventually comes down to is the order in which the inputs arrive at each neuron, regardless of how long it took to get there, and the order in which they come out, regardless of how long they take once sent, which in turn decides the order in which other neurons will get their inputs, and so forth.
You could very easily simulate any brain on an infinitely fast UTM by simply having the algorithm take that into account. (Hell, an infinitely fast UTM could simulate everything about the brain, including external stimuli and quantum effects, so it could be made infinitely accurate to the point of being the brain itself)