If the smallest length is defined (Planck length), then could the universe be a cellular automata, and objects don’t really move, but merely shift positions in a state machine?

Or have they proven it otherwise (i.e. measured a distance between two objects that is not an integer number of Planck lengths apart)?

Stephen Wolfram thinks so. See his book A New Kind of Science.

My own opinion is: a cellular automaton is a mathematical construct, and we don’t know yet whether the universe is a mathematical construct or whether it is merely described by mathematical constructs. If the latter, then the universe is not a cellular automaton. If the former, it could be, but we don’t know yet.

Quantum mechanics in particular is problematic to describe with cellular automata. Wolfram thinks he can do so, but his beliefs are not widely held. Regarding the Planck length: the units of the cellular automata are going to have to be a lot smaller than that, even, since the automaton will have to simulate quarks, strings, etc. which are all potentially much smaller than the Planck length.

Regarding “have they proven it otherwise”: this has not been proven either way yet. Also, just because the universe may fundamentally be quantized does not necessarily imply that it is a cellular automaton.

I recommend reading the book. Actually, no I don’t, it’s incredibly wordy and poorly written. I recommend Googling a good summary of the book.

I think it is possible to hypothesise a cellular automaton where apparent objects exist that are smaller than the individual cells - they might be described by the contents of more than one cell, but behave (in terms of everything we can measure about them) as if they are very much smaller.

Controvert - have you read Permutation City by Greg Egan?

Alternatively the cellular automaton might have a much smaller granularity, but just be set up in such a way that the ruls do not permit stable configurations to move less than a certain distance per cycle.

I think the idea of the universe as a mathematical simulation is quite a compelling one (I started a thread about it some time ago), based only on my poor understanding of how matter and energy doesn’t seem to exist in any form other than probabilities, when you look closely.

Ummm… I thought it was a fantastic book and although I realise it is a work of fiction, I think it explored the philosophical concepts of ‘real’ AI and consciousness rather neatly. However, the ‘dust’ hypothesis is interesting, but bogus I think.