Grammar Ambiguity-

 

Before you go through this article, make sure that you have gone through the previous article on Grammar Ambiguity.

 

We have discussed-

• There exists no general algorithm to remove the ambiguity from grammar.
• To check grammar ambiguity, we try finding a string that has more than one parse tree.
• If any such string exists, then the grammar is ambiguous otherwise not.

 

In this article, we will discuss how to convert ambiguous grammar into unambiguous grammar.

 

Converting Ambiguous Grammar Into Unambiguous Grammar-

 

• Causes such as left recursion, common prefixes etc makes the grammar ambiguous.
• The removal of these causes may convert the grammar into unambiguous grammar.
•  However, it is not always compulsory.

 

NOTE It is not always possible to convert an ambiguous grammar into an unambiguous grammar.

 

Methods To Remove Ambiguity-

 

The ambiguity from the grammar may be removed using the following methods-

 

 

• By fixing the grammar
• By adding grouping rules
• By using semantics and choosing the parse that makes the most sense
• By adding the precedence rules or other context sensitive parsing rules

 

Removing Ambiguity By Precedence & Associativity Rules-

 

An ambiguous grammar may be converted into an unambiguous grammar by implementing-

• Precedence Constraints
• Associativity Constraints

 

These constraints are implemented using the following rules-

 

Rule-01:

 

The precedence constraint is implemented using the following rules-

• The level at which the production is present defines the priority of the operator contained in it.
• The higher the level of the production, the lower the priority of operator.
• The lower the level of the production, the higher the priority of operator.

 

Rule-02:

 

The associativity constraint is implemented using the following rules-

• If the operator is left associative, induce left recursion in its production.
• If the operator is right associative, induce right recursion in its production.

 

PROBLEMS BASED ON CONVERSION INTO UNAMBIGUOUS GRAMMAR-

 

Problem-01:

 

Convert the following ambiguous grammar into unambiguous grammar-

R → R + R / R . R / R* / a / b

where * is kleen closure and . is concatenation.

 

Solution-

 

To convert the given grammar into its corresponding unambiguous grammar, we implement the precedence and associativity constraints.

 

We have-

• Given grammar consists of the following operators-

+ , . , *

• Given grammar consists of the following operands-

a , b

 

The priority order is-

(a , b) > * > . > +

where-

• . operator is left associative
• + operator is left associative

 

Using the precedence and associativity rules, we write the corresponding unambiguous grammar as-

 

E → E + T / T

T → T . F / F

F → F* / G

G → a / b

Unambiguous Grammar

 

OR

 

E → E + T / T

T → T . F / F

F → F* / a / b

Unambiguous Grammar

 

Problem-02:

 

Convert the following ambiguous grammar into unambiguous grammar-

bexp → bexp or bexp / bexp and bexp / not bexp / T / F

where bexp represents Boolean expression, T represents True and F represents False.

 

Solution-

 

To convert the given grammar into its corresponding unambiguous grammar, we implement the precedence and associativity constraints.

 

We have-

• Given grammar consists of the following operators-

or , and , not

• Given grammar consists of the following operands-

T , F

 

The priority order is-

(T , F) > not > and > or

where-

• and operator is left associative
• or operator is left associative

 

Using the precedence and associativity rules, we write the corresponding unambiguous grammar as-

 

bexp → bexp or M / M

M → M and N / N

N → not N / G

G → T / F

Unambiguous Grammar

 

OR

 

bexp → bexp or M / M

M → M and N / N

N → not N / T / F

Unambiguous Grammar

 

To gain better understanding about Conversion into Unambiguous Grammar-

Watch this Video Lecture

 

Next Article- Evaluating Expressions Based On Given Grammar

 

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Evaluating Expressions Based On Given Grammar-

 

There are two methods for evaluating the expressions based on given grammar-

 

 

1. Drawing a parse tree
2. Designing the rules for operators

 

Method-01: Drawing Parse Tree-

 

In this method, following steps are followed-

• We draw a Parse Tree for the given expression.
• We evaluate the parse tree to get the required value of the expression.

 

Drawback-

 

• Drawing a parse tree for large expressions is cumbersome and time taking.
• So, this method becomes difficult to follow when the expression is large.

 

Method-02: Designing Rules For Operators-

 

In this method, following steps are followed-

• We decide the priority and associativity of operators in the grammar.
• We parenthesize the given expression.
• We evaluate the expression to get the required value.

 

Advantage-

 

• This method is easy to follow whether the expression is small or large.
• So, this method is preferred.

 

Rules For Deciding Priority Of Operators-

 

For the given unambiguous grammar,

• The priority of operators is decided by checking the level at which the production is present.
• The higher the level of production, the lower the priority of operator contained in it.
• The lower the level of production, the higher the priority of operator contained in it.

 

Also Read- Converting Ambiguous into Unambiguous Grammar

 

Example-

 

Consider the grammar-

E → E x F / F + E / F

F → F – T / T

T → id

 

Here,

• id has the highest priority.

(since T → id is present at the bottom most level)

• x and + operators have the least priority.

(since E → E x F / F + E are present at the top most level)

• x and + operators have the same priority.

(since E → E x F / F + E are present at the same level)

• – operator has higher priority than x and + but lesser priority than id.

(since F → F – T is present at the middle level)

 

So, the priority order is-

id > – > ( x , + )

 

Rules For Deciding Associativity Of Operators-

 

For the given unambiguous grammar,

• The associativity of operators is decided by checking the Type Of Recursion in the production.
• If the production has left recursion, then the operator is left associative.
• If the production has right recursion, then the operator is right associative.
• If the production has both left and right recursion, then the operator is neither left associative nor right associative.

 

Example-

 

Consider the grammar-

E → E x F / F + E / F

F → F – F / T

T → id

 

Here,

• x operator is left associative.

(since E → E x F has left recursion in it)

• + operator is right associative.

(since E → F + E has right recursion in it)

• F – F is neither left associative nor right associative.

(since F → F – F has both left and right recursion in it)

 

PRACTICE PROBLEMS BASED ON EVALUATING EXPRESSIONS FOR GRAMMAR-

 

Problem-01:

 

Consider the given grammar-

E → E + T / T

T → F x T / F

F → id

 

Evaluate the following expression in accordance with the given grammar-

2 + 3 x 5 x 6 + 2

 

Solution-

 

Method-01:

 

• Let us draw a parse tree for the given expression.
• Evaluating the parse tree will return the required value of the expression.

 

The parse tree for the given expression is-

 

 

On evaluating this parse tree, we get the value = 94.

 

Method-02:

 

The priority order and associativity of operators on the basis of given grammar is-

id > x > +

where-

• x operator is right associative.
• + operator is left associative.

 

Now, we parenthesize the given expression based on the precedence and associativity of operators as-

( 2 + ( 3 x ( 5 x 6 ) ) ) + 2

 

Now, we evaluate the parenthesized expression as-

= ( 2 + ( 3 x ( 5 x 6 ) ) ) + 2

= ( 2 + ( 3 x 30 ) ) + 2

= ( 2 + 90 ) + 2

= 92 + 2

= 94

 

Problem-02:

 

Consider the given grammar-

E → E + T / E – T / T

T → T x F / T ÷ F / F

F → G ↑ F / G

G → id

 

Evaluate the following expression in accordance with the given grammar-

2 x 1 + 4 ↑ 2 ↑ 1 x 1 + 3

 

Solution-

 

Method-01:

 

• Let us draw a parse tree for the given expression.
• Evaluating the parse tree will return the required value of the expression.

 

The parse tree for the given expression is-

 

 

On evaluating this parse tree, we get the value = 21.

 

Method-02:

 

The priority order and associativity of operators on the basis of given grammar is-

id > ↑ > ( x , ÷ ) > ( + , – )

where-

• + , – , x , ÷ operators are left associative.
• ↑ operator is right associative

 

Now, we parenthesize the given expression based on the precedence and associativity of operators as-

( ( 2 x 1 ) + ( ( 4 ↑ ( 2 ↑ 1 ) ) x 1 ) ) + 3

 

Now, we evaluate the parenthesized expression as-

= ( ( 2 x 1 ) + ( ( 4 ↑ ( 2 ↑ 1 ) ) x 1 ) ) + 3

= ( ( 2 x 1 ) + ( ( 4 ↑ 2 ) x 1 ) ) + 3

= ( ( 2 x 1 ) + ( 16 x 1 ) ) + 3

= ( 2 + ( 16 x 1 ) ) + 3

= ( 2 + 16 ) + 3

= 18 + 3

= 21

 

Problem-03:

 

Consider the given grammar-

E → E + T / E – T / T

T → T x F / T ÷ F / F

F → F ↑ G / G

G → id

 

Evaluate the following expression in accordance with the given grammar-

2 ↑ 1 ↑ 4 + 3 x 5 x 6 ↑ 1 + 2 ↑ 3

 

Solution-

 

The priority order and associativity of operators on the basis of given grammar is-

id > ↑ > ( x , ÷ ) > ( + , – )

where + , – , x , ÷, ↑ operators are left associative.

 

Now, we parenthesize the given expression based on the precedence and associativity of operators as-

( ( ( 2 ↑ 1 ) ↑ 4 ) + ( ( 3 x 5 ) x ( 6 ↑ 1 ) ) ) + ( 2 ↑ 3 )

 

Now, we evaluate the parenthesized expression as-

= ( ( ( 2 ↑ 1 ) ↑ 4 ) + ( ( 3 x 5 ) x ( 6 ↑ 1 ) ) ) + ( 2 ↑ 3 )

= ( ( 2 ↑ 4 ) + ( ( 3 x 5 ) x ( 6 ↑ 1 ) ) ) + ( 2 ↑ 3 )

= ( 16 + ( ( 3 x 5 ) x ( 6 ↑ 1 ) ) ) + ( 2 ↑ 3 )

= ( 16 + ( ( 3 x 5 ) x 6 ) ) + ( 2 ↑ 3 )

= ( 16 + ( ( 3 x 5 ) x 6 ) ) + 8

= ( 16 + ( 15 x 6 ) ) + 8

= ( 16 + 90 ) + 8

= 106 + 8

= 114

 

Problem-04:

 

Consider the given grammar-

E → E ↑ T / T

T → T + F / F

F → G – F / G

G → id

 

Evaluate the following expression in accordance with the given grammar-

2 ↑ 1 ↑ 3 + 5 – 6 – 8 – 5 + 10 + 11 ↑ 2

 

Solution-

 

The priority order and associativity of operators on the basis of given grammar is-

id > – > + > ↑

where-

• + , ↑ operators are left associative.
• – is right associative.

 

Now, we parenthesize the given expression based on the precedence and associativity of operators as-

( ( 2 ↑ 1 ) ↑ ( ( ( 3 + ( 5 – ( 6 – ( 8 – 5 ) ) ) ) + 10 ) + 11 ) ) ↑ 2

 

Now, we evaluate the parenthesized expression as-

= ( ( 2 ↑ 1 ) ↑ ( ( ( 3 + ( 5 – ( 6 – ( 8 – 5 ) ) ) ) + 10 ) + 11 ) ) ↑ 2

= ( ( 2 ↑ 1 ) ↑ ( ( ( 3 + ( 5 – ( 6 – 3 ) ) ) + 10 ) + 11 ) ) ↑ 2

= ( ( 2 ↑ 1 ) ↑ ( ( ( 3 + ( 5 – 3 ) ) + 10 ) + 11 ) ) ↑ 2

= ( ( 2 ↑ 1 ) ↑ ( ( ( 3 + 2 ) + 10 ) + 11 ) ) ↑ 2

= ( ( 2 ↑ 1 ) ↑ ( ( 5 + 10 ) + 11 ) ) ↑ 2

= ( ( 2 ↑ 1 ) ↑ ( 15 + 11 ) ) ↑ 2

= ( ( 2 ↑ 1 ) ↑ 26 ) ↑ 2

= ( 2 ↑ 26 ) ↑ 2

= (2²⁶) ↑ 2

= (2²⁶)²

 

Problem-05:

 

Consider the given priority order of operators-

id > ( x , ÷ ) > ( + , – )

Given all the operators are left associative, construct the corresponding grammar.

 

Solution-

 

We apply the rules learnt above in reverse direction to obtain the required grammar.

The corresponding grammar is-

E → E + T / E – T / T

T → T x F / T ÷ F / F

F → id

 

Problem-06:

 

Consider the given priority order of operators-

id > ↑ > ( x , ÷ ) > ( + , – )

Given all the operators are left associative, construct the corresponding grammar.

 

Solution-

 

The corresponding grammar is-

E → E + T / E – T / T

T → T x F / T ÷ F / F

F → G ↑ F / G

G → id

 

Next Article- Important Points For Exams

 

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