Let's say that we have a regular language $L_1$ that is described by a right regular grammar. Now, assume we have a left regular grammar, can we describe the same language $L_1$ or would the left regular grammar produce the reverse of all strings present in $L_1$?

My understanding is that the left regular language would produce a new language $L_2$ containing the reverse of all strings of $L_1$. But since we know that reverse of a language is regular, both right regular and left regular grammars are capable of describing regular languages

Is my understanding correct?


1 Answer 1


Yes, your understanding is correct:

  • Reversing the rules of your right regular grammar for $L_1$ will produce a left regular grammar for $L_2$, the language consisting of the reverses of the strings in $L_1$.
  • Every right regular grammar can be reversed in this way to form a left regular grammar, and vice versa.
  • This does not give you any way to construct a right regular grammar for $L_1$.
  • Consequently, to show that every language that can be described with a right regular grammar can also be described by a left regular one, you need a different argument.
  • To show that, it is indeed sufficient to show (by different arguments) that
    • the right regular grammars can indeed describe all regular languages, and
    • the reverse of a regular language is always regular.

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