Hey everyone,
   While I was looking at http://trac.sagemath.org/sage_trac/ticket/13556,
I noticed what I believe to be a greater underlying problem with sage in
that infinite sequences are not handled (gracefully). In particular, sage
tries to construct infinite sequences/lists/tuples when given an infinite
iterable set `L` (ex. `QQ` or `(1..)` or `Partitions()`)
{{{
sage: list(L)
sage: tuple(L)
sage: Sequence(L)

I doubt we could do anything about `list` or `tuple` since they are python
built-ins, but what about `Sequence`? My first thought is to raise a
`ValueError` if we give it an infinite list, and I think we need a method
for the ellipsis iterator which checks if it is finite.

   I also noted that `FreeModule` does not support infinite dimensions
by (naively) calling `FreeModule(QQ, oo)`. However I believe we can still
work in infinite dimensions by working in `CombinatorialFreeModule(QQ,
ZZ)`. In either case, this may only be suitable if we consider it as a
direct sum (and implementing using the sparse free modules)...

Thoughts?

Best,
Travis

CombinatorialFreeModule(QQ, basis) works fine if 'basis' is infinite:

sage: CombinatorialFreeModule(QQ, ZZ)
Free module generated by Integer Ring over Rational Field
sage: CombinatorialFreeModule(QQ, Partitions())
Free module generated by Partitions over Rational Field

Just don't ask for a list of the basis elements ;)

--
John

Hi,

In combinatorics on words, we often deal with infinite sequences, and there
is code in Sage to deal with these things. After that, I don't know if the
question you want to ask to those objects are the same as the problems
considered in combinatorics on words (like, computing the factor
complexity, amongst other things). But, at least, you can take a look at it
to see how it works.

So, an infinite word can be defined as a function from the non negative
integers to an certain alphabet :

sage: Word(lambda n: n^3 % 7)
word: 0116166011616601161660116166011616601161...

Or as an infinite iterator :

sage: Word(primes(0, infinity))
word:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101, 103,107,109,113,127,131,137,139,149,151,157,163,167,173,...

Well known infinite sequence on {0,1} include the Fibonacci sequence :

sage: words.FibonacciWord()
word: 0100101001001010010100100101001001010010...

The Fibonacci word is also the infinite fixed point of the following
morphism (another way of creating an infinite sequence in Sage) :

sage: m = WordMorphism('0->01,1->0')
sage: m.fixed_point('0')
word: 0100101001001010010100100101001001010010...

For any infinite word, there are some methods available, like get the i^th
term or slice it :

sage: P = Word(primes(0, infinity))
sage: P[1000]
7927
sage: P[100000:100006]
word: 1299721,1299743,1299763,1299791,1299811,1299817

In this last case, it returns a finite word :

sage: type(_)
<class 'sage.combinat.words.word.FiniteWord_iter_with_caching'>

Compute the finite differences, and so on...

sage: P.finite_differences()
word:
1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,6,6 ,4,6,6,...

Cheers,

Sébastien