Is this language context-free or not?

Question

I have a problem to solve, the problem is:

Is the language of strings

$$L=\{0^x1^y:x\nmid y\}$$

context free?

I suspect it isn't, I spent some time trying to make a grammar that could generate that language, but I couldn't and I don't know how to prove it.

Answer 1

Here's a set of production rules that produce the strings, which seems to fail the Wikipedia definition of context-free grammar.

$S \rightarrow 1 S$

$1S \rightarrow S0$

$S0 \rightarrow \emptyset$

As far as the restriction you mentioned on the produced strings, I don't know how that factors in. Hopefully this can help you.