I am making a scheme that respresents some closure properties (union, intersection, complement and concatenation) for regular languages, context-free languages, decidable languages and RE languages. Below you can see my scheme.


I am trying to find all possible internal closure properties of these different languages that will always be valid. E.g., I know that $$\text{Regular language } \cup \text{ CFL } = \text{ CFL } $$

$$\text{Regular language } \cap \text{ CFL } = \text{ CFL } $$ $$\text{Regular language } ^* \text{ CFL } = \text{ CFL } $$

What other internal properties are there between these languages?

  • $\begingroup$ Asking for all possible closure properties -- or asking for all "internal properties" (whatever that is) -- seems too broad to me. Any community votes on that? I suggest you consult closure-properties and en.wikipedia.org/wiki/Regular_language#Closure_properties and en.wikipedia.org/wiki/Context-free_language#Closure and math.stackexchange.com/q/1443162/14578, then see if there is anything not already covered by those and whether you have any specific question. $\endgroup$
    – D.W.
    Dec 31, 2020 at 7:59
  • $\begingroup$ We discourage "please check whether my answer is correct" questions, as only "yes/no" answers are possible, which won't help you or future visitors. See here and here. Usually what works better is to ask about a specific conceptual issue you're uncertain about. $\endgroup$
    – D.W.
    Dec 31, 2020 at 8:00
  • $\begingroup$ @D.W. How is this a "yes/no" question? I am clearly asking for the internal properties between these different languages. Online, you can only find the closure properties for each language, but you can't find the properties between them. Therefore, I think your arguments don't suffices enough. $\endgroup$
    – NimaJan
    Dec 31, 2020 at 10:36
  • $\begingroup$ It looks to me like the post has two questions: (a) is my table correct?, and (b) what other "internal properties" are there?. (a) is a check-my-answer, and (b) looks like it might be too broad to me. Perhaps the community will be OK with the question; that's up to them. $\endgroup$
    – D.W.
    Dec 31, 2020 at 20:59

1 Answer 1


I am afraid that there is no better general result than the obvious one.

Assume that $\mathcal K$ and $\mathcal L$ are two families of languages with $\mathcal K\subset \mathcal L$ such that $\mathcal L$ is closed under the operation $\circ$. Then for $K\in\mathcal K$ and $L\in \mathcal L$ we have $K\circ L\in \mathcal L$.

This is obvious, as both $K,L\in \mathcal L$. Usually we can choose simple examples that we can't do better. For instance, choose a "difficult" language in $\mathcal L$, so $L\in \mathcal L \setminus \mathcal K$. Then $L=\varnothing \cup L$, $L=\Sigma^*\cap L$, and $L=\{\lambda\}\cdot L$ show that we cannot do better than $\mathcal L$.


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