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?