# context-sensitive language

Formally, a context-sensitive language is a formal grammar $G=(\Sigma,N,P,\sigma)$, such that given any production in $P$, it

1. 1.

either has the form

 $uXv\to uwv,$

where $X\in N$, $u,v,w\in\Sigma^{*}$, and $w\neq\lambda$, the empty word   ,

2. 2.

or is $\sigma\to\lambda$, provided that the start symbol $\sigma$ does not occur on the right side of any productions in $P$.

In other words, if $G$ does not contain the production $\sigma\to\lambda$, then any production will have the form in condition 1. On the other hand, if $G$ does contain $\sigma\to\lambda$, then for any production $uXv\to uwv$ of $G$, $\sigma$ does not occur in $uwv$.

The reason for including the second condition is to ensure the possibility that $\lambda$ may be generated by the grammar.

One may interpret the first condition as follows: the non-terminal symbol $X$ can only be transformed into the word $w$ if it is surrounded by $u$ and $v$, or that it is in the “context” of $uXv$. If in each production $uXv\to uwv$ of $G$, $u=v=\lambda$, (so that $X$ no longer has a “context”), then $G$ is a context-free grammar.

Given a context-sensitive grammar $G$, the context-sensitive language generated by $G$ is $L(G)$. In other words,

 $L(G):=\{v\in T^{*}\mid\sigma\lx@stackrel{{\scriptstyle*}}{{\to}}v\},$

where $T=\Sigma-N$ is the set of terminals, and $\sigma\lx@stackrel{{\scriptstyle*}}{{\to}}v$ means that the word $v$ over $\Sigma$ is produced after a finite number of applications of the productions in $P$ to $\sigma$.

With condition 2, we see that a context-sensitive language contains $\lambda$ iff it is generated by a context-sensitive grammar containing $\sigma\to\lambda$. With condition 2, every context-free language is context-sensitive. Without condition 2, every $\lambda$-free context-free language is $\lambda$-free context-sensitive.

$\{a^{n}b^{n}c^{n}\mid n\geq 0\}$ and $\{a^{2^{n}}\mid n\geq 0\}$ are examples of context-sensitive languages that are not context-free, the second of which is $\lambda$-free.

Remarks.

1. 1.

A formal grammar $G$ is said to be length-increasing if for every production $U\to V$ of $G$, the length of $U$ is at most the length of $V$: $|U|\leq|V|$. Every context-sensitive grammar not containing $\sigma\to\lambda$ (condition 2) is length-increasing. Conversely, any language generated by a length-increasing grammar is context-sensitive.

2. 2.

The minimal automaton required for processing a context-sensitive languages is a bounded non-deterministic Turing machine (bounded linear automaton).

3. 3.
4. 4.

In the Chomsky hierarchy, context-sensitive grammars are at Level 1. In fact, every context-sensitive language is recursive. The converse is not true, however.

5. 5.

Given a context-sensitive language $L$ and a word $w$, it is decidable whether $w\in L$.

## References

• 1 A. Salomaa, Formal Languages, Academic Press, New York (1973).
• 2 J.E. Hopcroft, J.D. Ullman, , Addison-Wesley, (1969).
 Title context-sensitive language Canonical name ContextsensitiveLanguage Date of creation 2013-03-22 16:28:40 Last modified on 2013-03-22 16:28:40 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 23 Author CWoo (3771) Entry type Definition Classification msc 68Q42 Classification msc 68Q45 Synonym type-1 language Synonym type-1 grammar Related topic ContextFreeLanguage Related topic LinearBoundedAutomaton Defines context-sensitive grammar Defines context-sensitive Defines length-increasing