Ronak ThakkarRoll no 32

M.Sc. Computer Science

Context Free Grammars

What are Context Free Grammars?In Formal Language Theory , a Context free

Grammar(CFG) is a formal grammar in which every production rule is of the form

V w Where V is a single nonterminal symbol and w

is a string of terminals and/or nonterminals (w can be empty)

The languages generated by context free grammars are knows as the context free languages

What does CFG do?A CFG provides a simple and mathematically

precise mechanism for describing the methods by which phrases in some natural language are built from smaller blocks, capturing the “block structure” of sentences in a natural way.

Important features of natural language syntax such as agreement and reference is are not the part of context free grammar , but the basic recursive structure of sentences , the way in which clauses nest inside other clauses, and the way in which list of adjectives and adverbs are swallowed by nouns and verbs is described exactly.

Formal Definition of CFGA context-free grammar G is a 4-tuple (V, , R, S), where:  V is a finite set; each element v V  is called a

non-terminal character or a variable. is a finite set of terminals, disjoint from , which

make up the actual content of the sentence.R is a finite relation from V  to (V U )* , where the asterisk

represents the Kleene star operation.

If (,) R, we write production

is called a sentential form

• S, the start symbol, used to represent the whole sentence (or program). It must be an element of  V.

Production rule notation

A production rule in R is formalized mathematically as a pair (,) , where is a non-terminal and is a string of variables and nonterminals; rather than using ordered pair notation, production rules are usually written using an arrow operator with  as its left hand side and as its right hand side: .

It is allowed for  to be the empty string, and in this case it is customary to denote it by ε. The form ε is called an ε-production.

Context-Free Languages•Given a context-free grammar G = (V,,R, S), the language generated or derived from G is the setL(G) = {w :S * w}

A language L is context-free if there is a context-free grammar G = (V,, R, S), such that L is generated from G. 

Example :Well-formed parenthesesThe canonical example of a context free

grammar is parenthesis matching, which is representative of the general case. There are two terminal symbols "(" and ")" and one nonterminal symbol S. The production rules are

S → SSS → (S)S → ()The first rule allows Ss to multiply; the second

rule allows Ss to become enclosed by matching parentheses; and the third rule terminates the recursion.

Parse TreeA parse tree of a derivation is a tree in which:

• Each internal node is labeled with a nonterminal

• If a rule A A1A2…An occurs in the derivation then A

is a parent node of nodes labeled A1, A2, …, An

S

a S

a S

S

e

b

Leftmost, Rightmost DerivationsA left-most derivation of a sentential form is one in

which rules transforming the left-most nonterminal are always applied

A right-most derivation of a sentential form is one in which rules transforming the right-most nonterminal are always applied

Ambiguous Grammar. A grammar G is ambiguous if there is a word w

L(G) having are least two different parse trees

Notice that a has at least two left-most derivations

S AS BS ABA aAB bBA eB e

Ambiguity & DisambiguationGiven an ambiguous grammar, would like

an equivalent unambiguous grammar.Allows you to know more about structure of a

given derivation.Simplifies inductive proofs on derivations.Can lead to more efficient parsing algorithms.In programming languages, want to impose a

canonical structure on derivations. E.g., for 1+23.

Strategy: Force an ordering on all derivations.

CFG SimplificationCan’t always eliminate ambiguity.

But, CFG simplification & restriction still useful theoretically & pragmatically.Simpler grammars are easier to understand.Simpler grammars can lead to faster parsing.Restricted forms useful for some parsing

algorithms.Restricted forms can give you more

knowledge about derivations.