4 ms·
What about uncomputable numbers like Chaitin's constant? Since we are subject to time, it makes it impossible to constrain its value on real line. But in timel
by rini17 5y ago
What about uncomputable numbers like Chaitin's constant? Since we are subject to time, it makes it impossible to constrain its value on real line.
But in timeless universe, it should be a known final quantity, similar to 1+1, no? Because all possible halting Turing machines are in halted state there and non-halting ones can't exist there(*). So if we could peek into such universe we could sample them and get Chaitin's approximation with any accuracy desired.
(*) or, all their states exist and form a graph with closed loop, which would also be observable
- thewakalix 5y agoEven if it were true that you can’t instantiate a non-terminating computation in a timeless universe, doesn’t that just make timeless physics less than Turing-equivalent? It doesn’t say anything about math.
- mannykannot 5y agoThat's an interesting question. Firstly, we can note that there are no Turing machines for computing uncomputable numbers in the temporal world, either. There are Turing machines which will not halt (and therefore not compute any number), and their graphs will either form cycles or extend indefinitely like the number line itself, but in neither case will these graphs contain a reachable halt state. (Having said that, I cannot wrap my mind around Babour's concept of an atemporal universe, either.)
- User23 5y agoThere are no Turing machines in the “temporal world” at all. We compute using devices much closer to some kind of bounded automata. An arbitrarily large, even growable, tape has more limited computational power than an infinite tape.
- mannykannot 5y agoCan you elabrate on why a growable tape has more limited computational power than an infinite tape (is the growth you have in mind constrained in some way?) Not that we could have the latter, or an unconstrained version of the former, in our temporal world, of course!
- User23 5y agohttps://cs.stackexchange.com/questions/127810/difference-between-linear-bound-automata-and-a-turing-machine https://cs.stackexchange.com/questions/127810/difference-bet...
- mannykannot 5y agoI see, so by 'growable' you don't mean arbitrarily growable or growable-as-needed without restriction. That would have overturned my intuitions in these matters.
- User23 5y agoI’m not confident in my ability to explain it more clearly. However I highly recommend reading Sipser[1] because his treatment of the subject, and many others, is as clear as can be. [1] https://www.goodreads.com/book/show/400716.Introduction_to_the_Theory_of_Computation https://www.goodreads.com/book/show/400716.Introduction_to_t...
- mannykannot 5y agoThanks - this is new to me.
- jerf 5y ago"But in timeless universe, it should be a known final quantity, similar to 1+1, no?" The word "known" is doing a lot of heavy lifting there, and bear examination for hidden assumptions. I would go with "defined". It is defined and concrete, but "known" adds a lot of questions in like who or what "knows" it. (I am skipping the question of "existence" on purpose because that's yet another question. The BusyBeaver(10,000,000)'th row of Pascal's Triangle is well-defined but whether it "exists" is yet more philosophy.)