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A context-sensitive grammar (CSG) is a formal grammar in which the left-hand sides and right-hand sides of any production rules may be surrounded by a context of terminal and nonterminal symbols. Context-sensitive grammars are more general than context-free grammars but still regular enough to be parsed by a linear bounded automaton.

The concept of context-sensitive grammar was introduced by Noam Chomsky in the 1950s as a way to describe the syntax of natural language where it is indeed often the case that a word may or may not be appropriate in a certain place depending upon the context. A formal language that can be described by a context-sensitive grammar is called a context-sensitive language.

Contents 1 Formal definition 2 Alternative definition 2.1 Example 3 Normal forms 4 Computational properties and Uses 5 See also 6 References

A formal grammar G = (N, Σ, P, S) 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).

In addition, a rule of the form

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

where ε 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 →ε 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.

Another definition of context-sensitive grammars defines them as formal grammars where all productions are of the form

α -> β where |α| ≤ |β| where |α| is the length of α.

Such a grammar is also called a monotonic or noncontracting grammar because none of the rules decreases the size of the string that is being rewritten.

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.

A simple monotonic grammar (which is not context-sensitive by the first definition) is

S → abc | aSBc

cB → Bc

bB → bb

where | is used to separate different options for the same non-terminal. This grammar generates the language , which is not context-free. Context-sensitive grammars can match an unlimited number of symbols to their partners, unlike context-free grammars, which can only match one symbol to its partner, so there is also a context-sensitive grammar for the language , but it's much more complex than the grammar above.

Every context-sensitive grammar which does not generate the empty string can be transformed into an equivalent one in Kuroda normal form. "Equivalent" here means that the two grammars generate the same language.

The decision problem that asks whether a certain string s belongs to the language of a certain context-sensitive grammar G, is PSPACE-complete. Indeed, there are even some context-sensitive grammars whose fixed grammar recognition problem is PSPACE-complete.

It has been shown that nearly all natural languages may in general be characterized by context-sensitive grammars, but the whole class of CSG's seems to be much bigger than natural languages. Worse yet, since the aforementioned decision problem for CSG's is PSPACE-complete, that makes them totally unworkable for practical use, as the general algorithm would take exponential time. Ongoing research on computational linguistics has focused on formulating other classes of languages that are "mildly context-sensitive" whose decision problems are feasible, such as tree-adjoining grammars, coupled context-free languages, and linear context-free rewriting systems. The languages generated by these formalisms properly lie between the context-free and context-sensitive languages.

• Chomsky hierarchy

• Introduction to Languages and the Theory of Computation by John C. Martin McGraw Hill 1996 (2nd edition)

Automata theory: formal languages and formal grammars
Chomsky hierarchy	Grammars	Languages	Minimal automaton
Type-0	Unrestricted	Recursively enumerable	Turing machine
n/a	(no common name)	Recursive	Decider
Type-1	Context-sensitive	Context-sensitive	Linear-bounded
n/a	Indexed	Indexed	Nested stack
Type-2	Context-free	Context-free	Nondeterministic Pushdown
n/a	Deterministic Context-free	Deterministic Context-free	Deterministic Pushdown
Type-3	Regular	Regular	Finite
Each category of languages or grammars is a proper subset of the category directly above it.