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Introduction to Automata

Ambiguity in Context-Free Grammar

We use a general approach to identify ambiguity in Context-free grammar as there is still no algorithm that checks it.

Based on the number of derivation trees, ambiguity Grammar is classified into two terms. One is ambiguous grammar, and the other is Unambiguous grammar.

Types of Ambiguity in Context Free Grammar

1. Ambiguous Grammar

A grammar is said to be ambiguous grammar if any string generated by it produces more than one.

  • Parse tree
  • Or syntax tree
  • Or leftmost derivation
  • Or rightmost derivation

Examples of Ambiguous Grammar

Example 01

Check whether the following grammar is ambiguous or not for string w = ab

S → A / B

A → aAb / ab

B → abB / ∈

Solution

Now, we draw more than one parse tree to get the string w = ab.

Examples of ambiguous Grammar 01

The original string (w =ab)  can derived through two different parse trees. So,  the given grammar is ambiguous.

Example-02

Check whether the following grammar is ambiguous or not for string w = aabbccdd

S → AB / C

A → aAb / ab

B → cBd / cd

C → aCd / aDd

D → bDc / bc

Solution

Now we draw more than one parse tree to get string w = aabbccdd.

Examples of ambiguous Grammar 02

The original string (w =aabbccdd)  can derived through two different parse trees. So,  the given grammar is ambiguous.

2. Unambiguous Grammar

A grammar is said to be unambiguous grammar if every string generated by it produces exactly one.

  • Parse tree
  • Or syntax tree
  • Or leftmost derivation
  • Or rightmost derivation

Note: So, If we try to derive more than one tree of unambiguous grammar, then all trees will be similar

Examples of Unambiguous Grammar

Example 01

For string “aab” the following grammar is unambiguous

S → AB  

A → Aa | a  

B → b  

Solution

Let’s draw the leftmost and rightmost derivations of the above grammar to get the string “aab.”

Examples of unambiguous Grammar 01

Because all parse trees, syntax trees, and left or right derivations will be similar for the above grammar of string “aab.” So, the above grammar is unambiguous.

Example 02

For string “id+id*id,” the following grammar is unambiguous

  • E → E + T  
  • E → T  
  • T → T * F  
  • T → F  
  • F → id  

Solution

Because all parse trees, syntax trees, and left or right derivations will be similar for the above string grammar “id+id*id.” As given below

Examples of unambiguous Grammar 2

Removal of Ambiguity from CFG

Conversion from ambiguous grammar into an unambiguous grammar is not always possible.

Some methods are used to remove ambiguity from CFG. Methods are given below.

  • By fixing the grammar
  • By adding the precedence rules
  • By adding grouping rules
  • By using semantics