# Proving sets of regular expressions and context free grammars are decidable [duplicate]

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• Intersection between regular language and context-free language [closed] 1 answer
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Consider below languages:

1. $$L_1=\{|M$$ is a regular expression which generates at least one string containing an odd number of 1’s$$\}$$
2. $$L_2=\{|G$$ 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.