Consider below languages:
- $L_1=\{<M>|M$ is a regular expression which generates at least one string containing an odd number of 1's$\}$
- $L_2=\{<G>|G$ is context free grammar which generates at least one string of all 1's$\}$
Its given that both above languages are decidable, but no proof is given. I tried guessing. $L_1$ is decidable, its a set of regular expressions containing
- odd number of $1$'s, or
- even number of $1$'s and $1^+$ or
- $1^*$
So we just have to parse regular expression for these characteristics. Is this right way to prove $L_1$ is decidable?
However, can we have some algorithm to check whether given input CFG accepts at least one string of all 1's? I am not able to come up with and hence not able prove how $L_2$ is decidable.
