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From: RichD on 24 Jun 2010 23:39
On Jun 24, Edward Green <spamspamsp...(a)netzero.com> wrote:
> > Strictly speaking, is there a difference betweem non-determinism
> > and randomness (in a dynamic system)?
>
> The two sound synonymous to me. Did you have some
> distinction in mind?

Colloquially, they are synonymous. But it
seems to me, there are places in the math
literature where some kind of distinction exists.
(I can't cite any off hand)

I might add, 'non-deterministic' and
'unpredictable' are also often interchanged,
but that's semantic and conceptual sloppiness.

--
Rich
From: RichD on 24 Jun 2010 23:48
On Jun 24, Robert Israel <isr...(a)math.MyUniversitysInitials.ca> wrote:
> > >Strictly speaking, is there a difference betweem non-
> > >determinism and randomness (in a dynamic system)?
>
> >Strange attractors are associated with non-deterministic
> >dynamical systems.
> >As fas as I'm aware, this is not the case for random systems.
> > So, the answer is yes!
>
> As far as I am aware, strange attractors are associated
> with deterministic systems...  A random discrete
> system might have
> X_{n+1} = f(X_n, R_n) where R_n form a sequence of
> independent random variables with a given distribution.  
> On the other hand, a non-deterministic system could be
> X_{n+1} = f(X_n, R_n) where R_n are arbitrary inputs (perhaps
> subject to some constraints, but not assumed to follow
> any particular probability distribution).

That's more or less what I had in mind.
A random system would be non-deterministic.
Except I'm considering the function itself
as a random variable.

But does it make sense to say the arbitrary
inputs to the non-deterministic system have
no prob. distribution, if they are unknown
and not repeatable? Isn't that simply ill-posed?

--
Rich

From: Bill Taylor on 25 Jun 2010 01:41
On Jun 25, 8:00 am, RichD <r_delaney2...(a)yahoo.com> wrote:
> Strictly speaking, is there a difference betweem non-determinism
> and randomness (in a dynamic system)?

This is not an answer to the question; but some wandering thoughts
on related matters.

Namely, what contexts ARE THERE in which non-deterministic
and random are both (more or less) defined, but NOT equivalent?

I can think of one right off. That is, in abstract machine theory.
For finite, pushdown or linear-bounded machines, both random
and non-deterministic can be clearly defined, and are different.
Different theorems are true of them.

Much more vaguely, in the interminable arguments about Free Will,
claims are often made that conscious human behaviour is (in part)
non-deterministic, or that it is (in part) random. And these
are typically considered to have different meanings.

But in math - what other contexts are there where the two are
precisley defined, and different?

-- Wondering Willy

** We do not say a stone is free
** merely because it is not in a cage.
From: Derek Holt on 25 Jun 2010 03:59
On 24 June, 21:00, RichD <r_delaney2...(a)yahoo.com> wrote:
> Strictly speaking, is there a difference betweem non-determinism
> and randomness (in a dynamic system)?
>

I would say that the difference is that a random system will generally
have an associated probability distribution that governs the
behaviour, whereas a non-deterministic system need not.

With a non-deterministic computation you are generally interested in
whether there exists a sequence of valid steps that successfully
executes the computation, rather than estimating the probability that
this will happen. For example, the "N" in "NP-complete" means non-
deterministic, and (roughly) a problem is in NP if there exists a
successful computational path of polynomially bounded length. Of
course, you may also be interested in the probability of finding a
solution using random methods, but that is a different issue.

Derek Holt.
From: illywhacker on 25 Jun 2010 05:11
On Jun 24, 10:00 pm, RichD <r_delaney2...(a)yahoo.com> wrote:
> Strictly speaking, is there a difference betweem non-determinism
> and randomness (in a dynamic system)?

Are you asking whether these words have different definitions in
certain mathematical contexts (dynamical systems), or are you asking
about dynamical systems in a physical sense? If you were to define
what you mean by 'dynamical system', your question would probably
already be answered.

As far as I know, word 'random', although used in a mathematical
context, is only strictly defined in the phrase 'random variable'.
Otherwise its use is loose, as other answers suggest.

If you are talking about physical systems, then these words have no
empirical meaning at all.

illywhacker;
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