Context-free language

In formal language theory, a context-free language is a language generated by some context-free grammar. The set of all context-free languages is identical to the set of languages accepted by pushdown automata.

Examples

An archetypical context-free language is $L=\{a^{n}b^{n}:n\geq 1\}$, the language of all non-empty even-length strings, the entire first halves of which are $a$'s, and the entire second halves of which are $b$'s. $L$ is generated by the grammar $S\to aSb~|~ab$, and is accepted by the pushdown automaton $M=(\{q_{0},q_{1},q_{f}\},\{a,b\},\{a,z\},\delta ,q_{0},\{q_{f}\})$ where $\delta$ is defined as follows:

$\delta (q_{0},a,z)=(q_{0},a)$
$\delta (q_{0},a,a)=(q_{0},a)$
$\delta (q_{0},b,a)=(q_{1},x)$
$\delta (q_{1},b,a)=(q_{1},x)$
$\delta (q_{1},\lambda ,z)=(q_{f},z)$

$\delta (\mathrm {state} _{1},\mathrm {read} ,\mathrm {pop} )=(\mathrm {state} _{2},\mathrm {push} )$
where $z$ is initial stack symbol and $x$ means pop action.

Context-free languages have many applications in programming languages; for example, the language of all properly matched parentheses is generated by the grammar $S\to SS~|~(S)~|~\lambda$. Also, most arithmetic expressions are generated by context-free grammars.

Closure properties

Context-free languages are closed under the following operations. That is, if L and P are context-free languages, the following languages are context-free as well:

• the union $L\cup P$ of L and P
• the reversal of L
• the concatenation $L\cdot P$ of L and P
• the Kleene star $L^{*}$ of L
• the image φ(L) of L under a homomorphism φ
• the image $\varphi ^{-1}(L)$ of L under an inverse homomorphism $\varphi ^{-1}$
• the cyclic shift of L (the language $\{vu:uv\in L\}$)

Context-free languages are not closed under complement, intersection, or difference. However, if L is a context-free language and D is a regular language then both their intersection $L\cap D$ and their difference $L\setminus D$ are context-free languages.

Nonclosure under intersection and complement

The context-free languages are not closed under intersection. This can be seen by taking the languages $A=\{a^{m}b^{m}c^{n}\mid m,n\geq 0\}$ and $B=\{a^{m}b^{n}c^{n}\mid m,n\geq 0\}$, which are both context-free. Their intersection is $A\cap B=\{a^{n}b^{n}c^{n}\mid n\geq 0\}$, which can be shown to be non-context-free by the pumping lemma for context-free languages.

Context-free languages are also not closed under complementation. For example, the language $C=\{a^{i}b^{j}c^{k}\mid i\neq j~\lor ~j\neq k~\lor ~i\neq k\}$ is context-free, because it can be expressed as a union of three languages that are obviously context-free. However, its complement ${\overline {C}}$ is not context-free. To see why, note that if we take the regular language $D=\{a^{i}b^{j}c^{k}\mid i,j,k\geq 0\}$, then their intersection is ${\overline {C}}\cap D=\{a^{n}b^{n}c^{n}\mid n\geq 0\}$.

Decidability properties

The following problems are undecidable for arbitrary context-free grammars A and B:

• Equivalence: is $L(A)=L(B)$?
• is $L(A)\cap L(B)=\emptyset$ ? (However, the intersection of a context-free language and a regular language is context-free, so if $B$ were a regular language, this problem becomes decidable.)
• is $L(A)=\Sigma ^{*}$ ?
• is $L(A)\subseteq L(B)$ ?

The following problems are decidable for arbitrary context-free languages:

• is $L(A)=\emptyset$ ?
• is $L(A)$ finite?
• Membership: given any word $w$, does $w\in L(A)$ ? (membership problem is even polynomially decidable - see CYK algorithm and Early's Algorithm)

Properties of context-free languages

• The reverse of a context-free language is context-free, but the complement need not be.
• Every regular language is context-free because it can be described by a regular grammar.
• The intersection of a context-free language and a regular language is always context-free.
• There exist context-sensitive languages which are not context-free.
• To prove that a given language is not context-free, one may employ the pumping lemma for context-free languages.

Converting a grammar to Greibach Normal Form and Chomsky normal form using CNF/GNF software

CNF / GNF : is a software make the automatic Convertion of a context-free grammar to:

1- Chomsky normal form grammar (CNF).

2- Greibach normal form grammar (GNF).

and also it show you the new grammar in each of the steps of the transformation.

A- Chomsky normal normal grammar (CNF)

1- reduced grammar.

2- epsilon-free grammar.

3- grammar without cycles and unit rules .

4-CNF.

B-Greibach normal form grammar (GNF).

1- reduced grammar.

2- epsilon-free grammar.

3- without cycles and unit rules.

4- non-left recursive grammar.

5- GNF.

with this application you can have CNF and GNF for any context-free grammar just in few steps.

just run the application and create new project and enter your grammar and click done and you will have :

CNF and GNF grammar and the new grammar in each of the steps of the transformation.

See also

• CNF/GNF
• Chomsky normal form software

External links

• Chomsky normal form software
• CNF/GNF Homepage
• CNF/GNF download

References

• Seymour Ginsburg (1966). The Mathematical Theory of Context-Free Languages. New York, NY, USA: McGraw-Hill, Inc.
• Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. Chapter 2: Context-Free Languages, pp. 91–122.