context-sensitive language

A context-sensitive language is a language over some alphabet generated by generated by some known as a context-sensitive grammar.

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

1. either has the form

   $$uXv\to uwv,$$

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

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,

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

where $T=\mathrm{\Sigma }-N$ is the set of terminals, and $\sigma \stackrel{*}{\to }v$ means that the word $v$ over $\mathrm{\Sigma }$ 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.

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

Remarks.

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 $V$: $|U|\le |V|$. 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. The minimal automaton required for processing a context-sensitive languages is a bounded non-deterministic Turing machine (bounded linear automaton).

3. The family of context-sensitive languages has the following closure properties: union, intersection, concatenation, Kleene star, reversal, and complementation (proved in 1988). It is not closed under homomorphism.

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