I have a problem with the recognition of languages. Given a certain language, for example ancb2n, 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?
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 { ancb2n | 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 ({ ancnbn | n > 0 }) is not context-free (but is context-sensitive). (That's not the same as { ancn+mbm | n > 0 }, which is context-free.)
