context free grammer

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Context Free Grammars

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Context-Free Grammars: Concepts and

Notation

A context-free grammar G = (Vt, Vn, S, P)

A finite terminal vocabulary Vt

The token set produced by scanner

A finite set of nonterminal vacabulary Vn Intermediate symbols

A start symbol S Vn that starts all derivations Also called goal symbol

P, a finite set of productions (rewriting rules) of the

form A X1X2 Xm AVn, Xi VnVt, 1i m

A is a valid production

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Notation (Contd)

Other notations

Vacabulary V of G,

V= VnVt

L(G), the set of string s derivable from S Context-free language of grammar G

Notational conventions

a,b,c, denote symbols in Vt

A,B,C, denote symbols in Vn

U,V,W, denote symbols in V

,,, denote strings in V*

u,v,w, denote strings in Vt*

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Example:

E E + E | E * E | (E)| -E| id

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The String(id+id) is a sentence of the above grammer

E -E -(E) -(E+E) -(id+E) -(id+id)


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Notation (Contd)

Derivation One step derivation

If A , then A

One or more steps derivation

Zero or more steps derivation

If S , then is said to be sentential formof the CFG

SF(G) is the set of sentential forms of grammar G L(G) = {x Vt

*|S x}

L(G)=SF(G)Vt* (A Sentence)

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Notation (Contd)

Left-most derivation, a top-down parsers lm , lm , lm

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E -E -(E) -(E+E) -(id+E) -(id+id)

lm lmlmlmlm


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Notation (Contd)

Right-most derivation (canonical derivation) rm , rm , rm Buttom-up parsers

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Notation (Contd)

A parse tree rooted by the start symbol

Its leaves are grammar symbols or

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Parse tree for(id+id) E

E

E

E E

(

+

)

id id

-


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Two Parse trees for id+id*id

E

E + E

id E * E

id id

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(a)

E

E * E

E + E id

id id

(b)


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Ambiguity

Grammer produces more than one tree

More than one leftmost/rightmost

derivations for the same sentence

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Notation (Contd)

A phase of a sentential form is a sequence of

symbols descended from a single nonterminal

in the parse tree

Simple or prime phrase

Thehandle of a sentential form is the left-

most simple phrase

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Notation (Contd)

Regular grammars

is of CFGs

Limited to productions of the form

A aB

C

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Errors in Context-Free Grammars

CFGs are a definitional mechanism. Theymay have errors, just as programs may.

Flawed CFG

1. Useless nonterminals Unreachable

Derive no terminal string

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SA|BAa

BBb

Cc

Nonterminal C cannot be reached form S

Nonterminal B derives no terminal string

S is the start symbol.


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Another example Ambiguous:

Grammars that allow different parse trees for the sameterminal string

It is impossible to decide whether a given CFG isambiguous

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It is impossible to decide whether a given

CFG is ambiguous

For certain grammar classes, we can prove thatconstituent grammars are unambiguous

A general comparison algorithm applicable

to all CFGs is known to be impossible

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Parsers and Recognizers

Two general approaches to parsing

Top-down parser

Expanding the parse tree (via predictions) in adepth-first manner

Preorder traversal of the parse tree

Predictive in nature

lm

LL

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Parsers and Recognizers (Contd)

Buttom-down parser

Beginning at its bottom (the leaves of the tree,

which are terminal symbols) and determining theproductions used to generate the leaves

Postorder traversal of the parse tree

rm

LR

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Parsers and Recognizers (Contd)

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To parsebegin SimpleStmt; SimpleStmt; end $


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context free grammer

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