Lecture # 9

Chap 4: Ambiguous Grammar

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Chomsky Hierarchy: Language Classification

• A grammar G is said to be– Regular if it is right linear where each production is of the

formA w B or A w

or left linear where each production is of the formA B w or A w

– Context free if each production is of the formA

where A N and (NT)*– Context sensitive if each production is of the form

A where A N, ,, (NT)*, || > 0

– Unrestricted

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Chomsky Hierarchy

L(regular) L(context free) L(context sensitive) L(unrestricted)

Where L(T) = { L(G) | G is of type T }That is: the set of all languages

generated by grammars G of type T

L1 = { anbn | n 1 } is context free

L2 = { anbncn | n 1 } is context sensitive

Every finite language is regular! (construct a FSA for strings in L(G))

Examples:

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Parse Trees

• The root of the tree is labeled by the start symbol• Each leaf of the tree is labeled by a terminal (=token)

or • Each interior node is labeled by a nonterminal• If A X1 X2 … Xn is a production, then node A has

immediate children X1, X2, …, Xn where Xi is a (non)terminal or ( denotes the empty string)

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Parse Tree for the Example Grammar

Parse tree of the string 9-5+2 using grammar G

list

digit

9 - 5 + 2

list

list digit

digitThe sequence ofleafs is called the

yield of the parse tree

Example of Parse Tree

• Suppose we have the following grammar E → E + E E → E * E E → ( E ) E → - E E → id

Perform Left most derivation, right most derivation and construct a parse tree for the string

id+id*id

Two possible Parse Trees using Leftmost derivation

Parse Tree via Right most derivation

Ambiguity

• Grammar is ambiguous if more than one parse tree is possible for some string as shown in the previous example. If there are more than one left most derivations or more than one right most derivations.

• Ambiguity is not acceptable– Unfortunately, it’s undecidable to check whether a given

CFG is ambiguous– Some CFLs are inherently ambiguous (do not have an

unambiguous CFG)

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Ambiguity (cont’d)

string string + string | string - string | 0 | 1 | … | 9

G = <{string}, {+,-,0,1,2,3,4,5,6,7,8,9}, P, string>

with production P =

Consider the following context-free grammar:

This grammar is ambiguous, because more than one parse treerepresents the string 9-5+2

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Two Parse Trees for the same string

string

string

9 - 5 + 2

string

string string

string

string

9 - 5 + 2

string

string string

Practice

• Show that the following grammar is ambiguous: (Find out strings and two parse trees)

1) S AB | aaB 2) S a | abSb |aAb A a | Aa A bS | aAAb Bb

3) S aSb | SS | ε

How to eliminate Ambiguity?

• If ambiguity is of the form: S α S β S | α1 |……| αn Rewrite: S α S β S’ | S’ S’ α1 |……| αn• Make the following grammar unambiguous: E E+E | E*E | (E) | id

Eliminate Left Recursion

If S S α for some α then there exists left recursion

If there are productions: S S α | β Rewrite: S β S’ S’ α S’ |

Example

• Remove Left Recursion: S A A S d | b