Can you explain me why we can't show decidability of problem using non-determinism ? I know that this problem (described below) is not decidability, however I can't understand why following reasoning is not working.
Problem Given a context free grammar $G$, does there exists a word $w∈Σ^∗$ such that $w^4∈L(G)$ ?
My reasoning: Lets show Turing machine which decide this problem. Let machine guesses some word $w$ (it is possible because word is finite and $Σ$ is finite) and check if (using CYK algorithm) $G$ generates $w^4$.
I totally don't understand why this approach doesn't work. After all, non-determinism is about guessing...