Context Free Language1

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Context Free Language

CSE, 2ndYear, 4thSemester

Copyright to: Abhishek Mukhopadhyay3/25/2014 7:47:05 PM


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Introduction

If we follow the Chomsky classification oflanguage, according the Type-II grammar,

But there is no restriction in the right side. Wesaw in regular grammar that certain

restriction on grammatical forms; eliminatingrules of the form and makethe arrangement easier.


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Context Free Language

A language L is said to be context-free if and only if

there is a context free grammar G such that L = L (G).Examples of Context-Free Languages

The grammar G = ({S} , {a, b}, S, P), with productions

S > aSa, S -> bSb, S >A,

is context-free. A typical derivation in this grammar is

S =t> aSa => aaSaa => aabSbaa => aabbaa.

The language is context-free, but this is not regular


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Parse Tree

Definition: Let G = (V, T, P, S) be a CFG. A tree is a

derivation (or parse) tree if:

Every vertex has a label from V U T U {}

The label of the root is S

If a vertex with label A has children with labels X1, X2,, Xn, from

left to right, then

A> X1, X2,, Xn

must be a production in P

If a vertex has label , then that vertex is a leaf and the only childof its parent

More Generally, a derivation tree can be defined with any

non-terminal as the root.


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Example:

S> AB S A

A> aAA

A> aA A B a A

A> a

B> bB a A A b aAA

B> ba

yield = aAab yield = aaAA

Notes:

Root can be any non-terminal

Leaf nodes can be terminals or non-terminals

A derivation tree with root S shows the productions used to obtain a sentential form


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Observation:Every derivation corresponds to one derivation tree.

S => AB S

=> aAAB

=> aaAB A B

=> aaaB

=> aaab a A A b

a a

Observation:Every derivation tree corresponds to one or more derivations.

S => AB S => AB S => AB

=> aAAB => Ab => Ab

=> aaAB => aAAb => aAAb

=> aaaB =>aAab => aaAb

=> aaab => aaab => aaab

Definition:A derivation is leftmost (rightmost)if at each step in the derivation a

production is applied to the leftmost (rightmost) non-terminal in the sentential form.

The first derivation above is leftmost, second is rightmost, the third is neither.


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Observation: Every derivation tree corresponds to exactly one leftmost (and

rightmost) derivation.

S => AB S

=> aAAB

=> aaAB A B

=> aaaB

=> aaab a A A b

a a

Observation: Let G be a CFG. Then there may exist a string x in L(G) that has

more than 1 leftmost (or rightmost) derivation. Such a string will also havemore than 1 derivation tree.


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Example:Consider the string aaab and the preceding grammar.

S> AB S => AB S

A> aAA => aAAB

A> aA => aaAB A BA> a => aaaB

B> bB => aaab a A A b

B> b

a a

S => AB S

=> aAB

=> aaAB A B

=> aaaB

=> aaab a A b

a A

a

The string has two left-most derivations, and therefore has two distinct parse trees.


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Context-Free Grammars and

Programming Languages

NEXT CLASS is followed by

1. Construction of reduced grammar

2. Elimination of NULL production3. Elimination of UNIT production

4. Pumping Lemma


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Construction of reduced grammarOne invariably wants to remove productions from a grammar that can never take part

in any derivation. For example, in the grammar whose entire production set is

the production S > A clearly plays no role, as A cannot be transformed into a

terminal string. While A can occur in a string derived from S, this can never lead to asentence. Removing this production leaves the language unaffected and is a

simplification by any definition.


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Reduced Grammar
http://localhost/var/www/apps/conversion/tmp/scratch_9/Reduced_grammar.docxhttp://localhost/var/www/apps/conversion/tmp/scratch_9/Reduced_grammar.docx


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Remove -productions

Algorithm. Remove -productions from grammars for

langauges without .

1. Find nonterminals that derive .

2. For each productionA w construct all productionsA w where w is obtained from

w by removing one or more occurrences of the

nonterminals from Step 1.3. Combine the original productions with those of step

2 and eliminate any -productions.


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Example

Remove -productions from the grammar

S ABc

A aA |

B bB | .

Solution.

Step 1: The nonterminalsA and B derive . Step 2: From the production S ABc we construct S Bc | Ac | c.

From the productionA aA we constructA a.

From the production B bB we construct B b.

Step 3: S ABc | Bc | Ac | c

A aA | aB bB | b.


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Problem(Remove -productions)

Remove -productions from

S ABc | Ab | c

A ABa |

B Bbc | .

Solution.

S ABc | Ab | c| Bc | Ac | b

A ABa | Ba | Aa | a

B Bbc | bc.


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Dealing with loops

A unit productionis a production of the form

whereA1andA2are both variables Example

A1 A2

S 0S1 | 1S0S1 | TT S | R |

R 0SR

grammar: unit productions:

S T

R


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Removal of unit productions

If there is a cycle of unit productions

delete it and replace everything withA1

Example

A1 A2 ... Ak A1

S 0S1 | 1S0S1 | T

T S | R |

R 0SR

S T

R

S 0S1 | 1S0S1

S R |

R 0SR

Tis replaced by Sin the {S, T}cycle


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Removal of unit productions

For other unit productions, replace every chain

by productionsA1

,... , Ak

Example

A1 A2 ... Ak

S R 0SRis replaced by S 0SR, R 0SR

S 0S1 | 1S0S1

| R |

R 0SR

S 0S1 | 1S0S1

| 0SR|

R 0SR


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Chomsky Normal Form

A Context free grammar is in CNF if every

production is in form, Aa or ABC and S

, L(G).

Construction of CNF

First remove and unit production

Then construct according to the above

mentioned production rules


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Context Free Language1

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