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I am not well versed in this topic so please pardon any ambiguous notation. I am trying to describe the language of this grammar in set-theoretic notation.

The Grammar is given by:

$ S \rightarrow ASA|B$

$ A \rightarrow a|b$

$ B \rightarrow BC|\epsilon$

$ C \rightarrow a|b|c$

I think I can accurately describe this language in english:

If $w=xyz$ is a string and $xyz$ are substrings of $w$ then

$|x| = |z|$, and $x$ and $z$ are made up of symbols from $\{a,b\}^*$.

Furthermore, $y$ is made up of symbols from $\{a,b,c\}^*$

Mores simply, the strings are made up of two ends that are the same $|length| \ge 0$ and made up of symbols $a,b$. And the middle part of the string is made up of any symbols from $a,b,c$ and also has a $|length| \ge 0$

How do I more formally and accurately describe the language of the grammar $L(G)$? Is it possible to do this in a set-theoretic notation?

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  • $\begingroup$ Your question smells like XY problem. Why do you require "in set theoretic notation where the length of strings need to be remembered"? I would bet all you wanted was to describe the language generated by the given grammar. $\endgroup$ – Apass.Jack Oct 31 '18 at 4:09
  • $\begingroup$ @Apass.Jack Didn't I already describe the language of the Grammar in my question? All I am asking is what a more formal description of the language would look like (what form would it take), and if that formal description would take a set-theoretic notation, or if it even can. $\endgroup$ – SeesSound Oct 31 '18 at 4:14
  • $\begingroup$ The language of the give grammar can be described simply as "all words over the alphabet $\{a,b,c\}$", which is a formal description. I would say the set-theoretic notation of regular languages is the regular expression. That languages is $(a\mid b\mid c)^*$. Are those what you wanted? $\endgroup$ – Apass.Jack Oct 31 '18 at 4:29
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Your grammar generates $\Sigma^*$, where $\Sigma = \{a,b,c\}$.

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  • $\begingroup$ I'm in disbelief that it is this simple. What about the parts of the grammar that are saying that the outsides are all $a's$ and $b's$, why are you ignoring those rules? Why wouldnt they just make the Grammar as: $S \rightarrow aS|bS|cS|\epsilon$? Is this an equivalent grammar? $\endgroup$ – SeesSound Oct 31 '18 at 4:41
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    $\begingroup$ The grammar can definitely be simplified. They are just (figuratively) pulling your leg. $\endgroup$ – Yuval Filmus Oct 31 '18 at 4:48

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