Is the Chomsky-hierarchy outdated? The Chomsky(–Schützenberger) hierarchy is used in textbooks of theoretical computer science, but it obviously only covers a very small fraction of formal languages (REG, CFL, CSL, RE) compared to the full Complexity Zoo Diagram. Does the hierarchy play any role in current research anymore? I found only little references to Chomsky here at cstheory.stackexchange, and in Complexity Zoo the names Chomsky and Schützenberger are not mentioned at all.

Is current research more focused on other means of description but formal grammars? I was looking for practical methods to describe formal languages with different expressiveness, and stumbled upon growing context sensitive language (GCSL) and visibly pushdown languages (VPL), which both lie between the classic Chomsky languages. Shouldn't the Chomsky hierarchy be updated to include them? Or is there no use of selecting a specific hierarchy from the full set of complexity classes? I tried to select only those languages that can be fit in gaps of the Chomsky hierarchy, as far as I understand:

REG (=Chomsky 3) ⊊ VPL ⊊ DCFL ⊊ CFL (=Chomsky 2) ⊊ GCSL ⊊ CSL (=Chomsky 1) ⊊ R ⊊ RE

I still don't get where "mildly context-sensitive languages" and "indexed languages" fit in (somewhere between CFL and CSL) although there seems to be of practical relevance for natural language processing (but maybe anything of practical relevance is less interesting in theoretical research ;-). In addition you could mention GCSL ⊊ P ⊂ NP ⊂ PSPACE and CSL ⊊ PSPACE ⊊ R to show the relation to the famous classes P and NP.

I found on GCSL and VPL:


Robert McNaughton: An Insertion into the Chomsky Hierarchy?. In: Jewels are Forever, Contributions on Theoretical Computer Science in Honor of Arto Salomaa. S. 204-212, 1999
http://en.wikipedia.org/wiki/Nested_word#References (VPL)


I'd also be happy if you know any more recent textbook on formal grammars that also deal with VPL, DCLF, GCSL and indexed grammars, preferable with pointers to practical applications.
In short: yes.

More particularly: Chomsky was one of the first to formalize a hierarchy relating languages, grammars, and automata. This insight is still very relevant and is taught in all intro courses on automata theory. However, the specific hierarchy Chomsky came up with and the names for the elements of the hierarchy aren't really significant anymore. We've since invented numerous formalisms which fall between levels of Chomsky's hierarchy, above it, or below it. And the names Chomsky used aren't particularly interesting, i.e. they aren't based on an interesting measure of complexity or anything, they're just numbers. Should mildly context sensitive languages be Type-1.5 or Type-1.7 or Type-1.3? Who cares. "Mildly context sensitive" is a much more informative name.

The Complexity Zoo is a bit different because it's full of all sorts of conditional equivalences and the like. A more modern hierarchy for automata theory wouldn't be linear (e.g., compare CFG vs PEG) but it would still have a well-known topology. To get a perspective on modern automata theory you should look at work on parser combinator libraries and some of the stuff on unification and type theory (though those both branch out far afield).