I'm struggling with a problem that I believe I've managed to reduce to a question of Kolmogorov complexity for infinite strings, but since I'm not an expert in this field, I'm not sure about the correctness of the reasoning.
Let us consider an alphabet $A$, and the set of all infinite strings $A^\omega$. Intuitively, given a function $U$ and a set of inputs $P$ such that $U : P \to A^\omega$, there exists a string $s \in A^\omega$ such that for all $p \in P$ such that $U(p) = s$, the size of $p$ is infinite.
My reasoning is based on the Incomputability of Kolmogorov complexity section of the Wikipedia page:
Theorem: There exist strings of arbitrary large Kolmogorov complexity. Formally: for each n ∈ ℕ, there is a string s with K(s) ≥ n
In other words, if I want to be able to generate all infinite sequences of a given alphabet, then I need to consider at least one infinite inputs (and not only an infinite set of inputs). Is this correct?
I'm not quite sure what you mean by an infinite program, but there is certainly an infinite sequence of symbols that is computed by no finite program.
This follows from a simple counting argument: Assuming $|A|\ge 2$, there are uncountably many infinite sequences of symbols, but there are only countably many finite programs.