talking about finite state computers and finite time let me repeat my previous post:



/q still enjoying math or tempted to go back to economy?

Just to be clear, you're saying that no computer can meet the asymptotic density requirement if it attempts to construct a real number, right? While that's true, it's quite a bit different from what I was talking about. Borel's theorem is essentially the law of large numbers applied to a suitable coin flip sequence. To prove that result, you need to be working on an infinite product space (at least if you want to view the maps to the nth decimal place as random variables) and the full joint independence matters. What I'm talking about is substantially weaker. The nth random variable is allowed to be completely dependent on all lower indexed random variables, so long as the relationship is not computable. My hunch is that that is probably not possible either, but it's much harder to prove.

I'm extremely happy in math. Not going back.

This post was edited by darkfire on Nov 2 2013 11:26pm

yes, i know that what you stated is quite different

but what is really randomness? how can one well define it?
the statement you quoted, does it refer to sequence of numbers or sequence of digits within or of a number?
if the latter, there is some sort of conundrum: does any number base do or does it need to apply to all bases
just take pi, while the sequence of its digits is probably random (at least in base 10) there are algorithms which can calculate the n-th hexadecimal digit without having calculated any of the previous ones

btw, my question was tongue in cheek (was tempted to put a behind it)

I do not know exactly what randomness is. I don't even know of anyone who has a good definition. The formal definition of a random variable is at once too restrictive and too open. The requirement that you have a full countable sequence of mathematically independent non-trivial variables on a single measure space is wildly too restrictive for reality (I am reasonably confident that no such collection exists in reality), but surely if it did, that would be 'random'. If we want a more reasonable definition, then the best I can say is that I do know of some properties that anything 'random' should have. Put another way, I know necessary conditions and I know sufficient conditions, but I do not know of any necessary and sufficient conditions.

The thing I love most about probability is that things which are not truly random can interact in such a way that they are extremely well approximated by idealized random variables. You see this a lot in deterministic systems that essentially obey the Central Limit Theorem. If the approximation is so good that we essentially cannot distinguish the output from the output of randomness, why not pretend it was random to begin with?

From a probabilists perspective, generating a random real number is the same as randomly generating countably many digits is the same as randomly generating a sequence of numbers. It actually doesn't matter which view you choose. If you can produce any one sequence of mathematically independent random variables for which you know the marginal distributions, then you can produce any sequence of mathematically independent random variables you want. I'm not sure what you mean by saying that pi is random base 10. It is completely deterministic. The value of pi is computable to any degree of accuracy in any base given sufficient time and computing power. It may be true that the asymptotic density of each digit is correct base 10, but that is substantially weaker than true randomness.



I should have guessed

This post was edited by darkfire on Nov 3 2013 08:23am

You think it would be easy to come up with a good one, but they always have problems. For example:

A process is random if it is not possible to predict the results ahead of time.

Well lets say we can't predict it yesterday, so it is random. But since yesterday we have learned more about the process and can now predict it... Does that mean the process it isn't random today but it was random yesterday? hehe...

This post was edited by Azrad on Nov 3 2013 08:50am

there is a lot of work being done on the 'randomness' of the (sequences) of digits within pi
so it just depends of what you consider 'random'
taking you statement again:
can any fine time finite state computer 'predict' the next digits in pi just taking as input the first (how many you want) billions of sequences?
emphasis on 'predict' not 'calculate'

btw, what i love about probability is how confusing certain concept are for the 'normal' person
just look at all those attempts of creationist to disprove evolutionary concepts with probability approaching or being zero
one has to hope that borel's ghost haunts them every night