Context-sensitive Grammar - Formal Definition

Formal Definition

A formal grammar G = (N, Σ, P, S) (this is the same as G = (V, T, P, S), where N/V is the Non-terminal Variable, and Σ/T is the Terminal) is context-sensitive if all rules in P are of the form

αAβ → αγβ

where A ∈ N (i.e., A is a single nonterminal), α,β ∈ (N U Σ)* (i.e., α and β are strings of nonterminals and terminals) and γ ∈ (N U Σ)+ (i.e., γ is a nonempty string of nonterminals and terminals).

Some definitions also add that for any production rule of the form u → v of a context-sensitive grammar, it shall be true that |u|≤|v|. Here |u| and |v| denote the length of the strings respectively.

In addition, a rule of the form

S → λ provided S does not appear on the right side of any rule

where λ represents the empty string is permitted. The addition of the empty string allows the statement that the context sensitive languages are a proper superset of the context free languages, rather than having to make the weaker statement that all context free grammars with no →λ productions are also context sensitive grammars.

The name context-sensitive is explained by the α and β that form the context of A and determine whether A can be replaced with γ or not. This is different from a context-free grammar where the context of a nonterminal is not taken into consideration. (Indeed, every production of a context free grammar is of the form V → w where V is a single nonterminal symbol, and w is a string of terminals and/or nonterminals (w can be empty)).

If the possibility of adding the empty string to a language is added to the strings recognized by the noncontracting grammars (which can never include the empty string) then the languages in these two definitions are identical.