Hi,

I’m looking for some guidence on how to answer the following questions:

Assume that a black box is outputing a sequence of 1′s and 0′s and that the output has a
starting point in time but appears to be unbounded.

Consider the infinite set of all possible finite models (i.e. models specifiable by
finite applications of a finite alphabet) which produce binary sequences. Is the set of
nontrivial, generative models which produce outputs that are past consistent with the
output of the black box after any fixed amount of time but which make mutually inconsistant
predictions about its future output finite or infinite?

By nontrivial I mean at very least excluding models of the form: past output + random
generative proceedure.

For example if the output of the box is 3.1415 in binary then possible models include:

pi
31415/10000
sqareroot(9.8691)
though I tend to think that the last two are trivial as well.

Are there any interesting ways to definite nontrivial?
What areas of mathemetics or logic deal with this kind of thing?
Thanks,

Mark