Grammars and languages: Chomsky’s hierarchy

N. Chomsky: Three models are there for the explanation of language.

Motivating illustration:

Sentence -> Noun Verb Noun, example: Bob loves Alice.

Sentence -> Sentence Conjunction Sentence, example: Bob loves Alice and Rome fights Carthage.

Grammar G(V, A, P, S). V: alphabet of the non-terminal symbols, variables and grammatical types;

A: alphabet of the terminal symbols, S ∈ V: start symbol, ‘sentence’;

P: unordered set of the productions of form L -> R, where L, R ∈ (V ∪ A)*

Rewriting step: for x, y, y’, z ∈ (V ∪ A)*, u -> v if u = xyz, v = xy’z and y -> y’ ∈ P

Derivation: “->*” is the transitive, reflexive closure of “->”, that is, u ->* v iff ∃ w0, w1 ... wj, with j ≥ 0, u = w0, w(i-1) -> wi,  wj = v

Language defined by G: L(G) = {w ∈ A* | S ->* w)

Different restrictions on the productions define various types of grammars and corresponding languages:

Type 0: Phrase structure grammar: No restrictions

Type 1: Context sensitive: |L| ≤ |R|, (exception: S -> ε is permitted when S never takes place on any right-hand side)

Type 2: context free: L ∈ V

Type 3: regular: L ∈ V, R = a or R = aX, where a ∈ A, X ∈ V

Exercise: Define a grammar for some very small subset of natural language, and exemplify it with illustrations. Try this for a subset of English and a semantically identical subset of the other language, and show that distinct natural languages encompass different grammars.

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