Ambiguous Grammar | Parse Tree | Important Points

Theory of Automata & Computation
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Ambiguous Grammar & Parse Tree-

 

We have discussed-

  • Ambiguous Grammar generates at least one string that has more than one parse tree.
  • Parse Tree is the geometrical representation of a derivation.

 

In this article, we will discuss important points about Ambiguous Grammar and Parse Tree.

 

Important Points-

 

Point-01:

 

  • There always exists a unique parse tree corresponding to each leftmost derivation and rightmost derivation.

 

If n parse trees exist for any string w, then there will be-

  • n corresponding leftmost derivations.
  • n corresponding rightmost derivations.

 

Point-02:

 

For ambiguous grammars,

  • More than one leftmost derivation and more than one rightmost derivation exist for at least one string.
  • Leftmost derivation and rightmost derivation represents different parse trees.

 

Point-03:

 

For unambiguous grammars,

  • A unique leftmost derivation and a unique rightmost derivation exist for all the strings.
  • Leftmost derivation and rightmost derivation represents the same parse tree.

 

Point-04:

 

  • There may exist derivations for a string which are neither leftmost nor rightmost.

 

Example

 

Consider the following grammar-

S → ABC

A → a

B → b

C → c

 

Consider a string w = abc.

Total 6 derivations exist for string w.

The following 4 derivations are neither leftmost nor rightmost-

 

Derivation-01:

 

S → ABC

→ aBC    (Using A → a)

→ aBc     (Using C → c)

→ abc     (Using B → b)

 

Derivation-02:

 

S → ABC

→ AbC    (Using B → b)

→ abC     (Using A → a)

→ abc     (Using C → c)

 

Derivation-03:

 

S → ABC

→ AbC    (Using B → b)

→ Abc     (Using C → c)

→ abc     (Using A → a)

 

The other 2 derivations are leftmost derivation and rightmost derivation.

 

Point-05:

 

  • Leftmost derivation and rightmost derivation of a string may be exactly same.
  • In fact, there may exist a grammar in which leftmost derivation and rightmost derivation is exactly same for all the strings.

 

Example

 

Consider the following grammar-

S → aS / ∈

 

The language generated by this grammar is-

L = { an , n>=0 } or a*

 

All the strings generated from this grammar have their leftmost derivation and rightmost derivation exactly same.

Let us consider a string w = aaa.

 

Leftmost Derivation-

 

S → aS

→ aaS       (Using S → aS)

→ aaaS     (Using S → aS)

→ aaa∈

→ aaa

 

Rightmost Derivation-

 

S → aS

→ aaS       (Using S → aS)

→ aaaS     (Using S → aS)

→ aaa∈

→ aaa

 

Clearly,

Leftmost derivation = Rightmost derivation

Similar is the case for all other strings.

 

Point-06:

 

  • For a given parse tree, we may have its leftmost derivation exactly same as rightmost derivation.

 

Point-07:

 

  • If for all the strings of a grammar, leftmost derivation is exactly same as rightmost derivation, then that grammar may be ambiguous or unambiguous.

 

Example

 

Consider the following grammar-

S → aS / ∈

  • This is an example of an unambiguous grammar.
  • Here, each string have its leftmost derivation and rightmost derivation exactly same.

 

Now, consider the following grammar-

S → aS / a / ∈

  • This is an example of ambiguous grammar.
  • Here also, each string have its leftmost derivation and rightmost derivation exactly same.

 

Consider a string w = a.

 

 

Since two different parse trees exist, so grammar is ambiguous.

 

Leftmost derivation and rightmost derivation for parse tree-01 are-

S → a

 

Leftmost derivation and rightmost derivation for parse tree-02 are-

S → aS

S → a∈

S → a

 

Next Article- Language of Grammar

 

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Summary
Ambiguous Grammar | Parse Tree | Important Points
Article Name
Ambiguous Grammar | Parse Tree | Important Points
Description
Important Points about Ambiguous Grammar and Parse Tree- Ambiguous grammar is a grammar whose all strings have exactly one parse tree. Parse tree is a geometrical representation of a derivation.
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Gate Vidyalay
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