乔姆斯基语言：如何识别它们？

Question

I have a problem with the recognition of languages. Given a certain language, for example a ⁿ cb ²ⁿ , n > 0 , how do I determine quickly what type belongs according Chomsky?

My idea was to determine the grammar that generates it and then up to the language but it is a long process. I think there's another way to recognize it by eye, without writing grammars or automata. can someone help me?

Answer 1

Unfortunately, associating an arbitrary language with a level of the Chomsky hierarchy is, in the general case, undecidable. (See Rice's Theorem .)

Of course, it is easy to categorize a given grammar, since the Chomsky hierarchy is defined by a simple syntactic analysis of the grammar itself. However, languages do not have unique grammars; the existence of (for example) a Type 2 (context-free) grammar for a language does not mean that there is not a Type 3 (regular) grammar for the same language.

So there are no shortcuts.

However, there is a lot to be said for experience. The language { a n cb 2n | n > 0 } { a n cb 2n | n > 0 } is context-free (and not regular), as are all languages of similar forms. That it is context free is demonstrated by the grammar

L → c
L → a L b b

and the fact that it is not regular can be proven using the pumping lemma for regular languages . (The linked Wikipedia article contains, as an example of the use of the lemma, a proof for a similar language which should be easy to adapt.)

On the other hand, a language which requires three equal counts ( { a n c n b n | n > 0 } ) is not context-free (but is context-sensitive). (That's not the same as { a n c n+m b m | n > 0 } , which is context-free.)