Tag: Derivation in Automata

Ambiguous Grammar | Parse Tree | Important Points

Theory of Automata & Computation

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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Parse Tree | Derivations | Automata

Theory of Automata & Computation

Parse Tree-

 

  • The process of deriving a string is called as derivation.
  • The geometrical representation of a derivation is called as a parse tree or derivation tree.

 

 

1. Leftmost Derivation-

 

  • The process of deriving a string by expanding the leftmost non-terminal at each step is called as leftmost derivation.
  • The geometrical representation of leftmost derivation is called as a leftmost derivation tree.

 

Example-

 

Consider the following grammar-

S → aB / bA

S → aS / bAA / a

B → bS / aBB / b

(Unambiguous Grammar)

 

Let us consider a string w = aaabbabbba

Now, let us derive the string w using leftmost derivation.

 

Leftmost Derivation-

 

S   → aB

→  aaBB                   (Using B → aBB)

→ aaaBBB                (Using B → aBB)

→ aaabBB                (Using B → b)

→ aaabbB                (Using B → b)

→ aaabbaBB            (Using B → aBB)

→ aaabbabB            (Using B → b)

→ aaabbabbS          (Using B → bS)

→ aaabbabbbA        (Using S → bA)

→ aaabbabbba         (Using A → a)

 

 

2. Rightmost Derivation-

 

  • The process of deriving a string by expanding the rightmost non-terminal at each step is called as rightmost derivation.
  • The geometrical representation of rightmost derivation is called as a rightmost derivation tree.

 

Example-

 

Consider the following grammar-

S → aB / bA

S → aS / bAA / a

B → bS / aBB / b

(Unambiguous Grammar)

 

Let us consider a string w = aaabbabbba

Now, let us derive the string w using rightmost derivation.

 

Rightmost Derivation-

 

S   → aB

→  aaBB                    (Using B → aBB)

→ aaBaBB                 (Using B → aBB)

→ aaBaBbS               (Using B → bS)

→ aaBaBbbA             (Using S → bA)

→ aaBaBbba              (Using A → a)

→ aaBabbba              (Using B → b)

→ aaaBBabbba          (Using B → aBB)

→ aaaBbabbba          (Using B → b)

→ aaabbabbba           (Using B → b)

 

 

NOTES

  • For unambiguous grammars, Leftmost derivation and Rightmost derivation represents the same parse tree.
  • For ambiguous grammars, Leftmost derivation and Rightmost derivation represents different parse trees.

 

Here,

  • The given grammar was unambiguous.
  • That is why, leftmost derivation and rightmost derivation represents the same parse tree.

 

Leftmost Derivation Tree = Rightmost Derivation Tree

 

Also Read- Ambiguous Grammar

 

Properties Of Parse Tree-

 

  • Root node of a parse tree is the start symbol of the grammar.
  • Each leaf node of a parse tree represents a terminal symbol.
  • Each interior node of a parse tree represents a non-terminal symbol.
  • Parse tree is independent of the order in which the productions are used during derivations.

 

Yield Of Parse Tree-

 

  • Concatenating the leaves of a parse tree from the left produces a string of terminals.
  • This string of terminals is called as yield of a parse tree.

 

PRACTICE PROBLEMS BASED ON DERIVATIONS AND PARSE TREE-

 

Problem-01:

 

Consider the grammar-

S → bB / aA

A → b / bS / aAA

B → a / aS / bBB

 

For the string w = bbaababa, find-

  1. Leftmost derivation
  2. Rightmost derivation
  3. Parse Tree

 

Solution-

 

1. Leftmost Derivation-

 

S   → bB

→ bbBB              (Using B → bBB)

→ bbaB              (Using B → a)

→ bbaaS            (Using B → aS)

→ bbaabB          (Using S → bB)

→ bbaabaS        (Using B → aS)

→ bbaababB      (Using S → bB)

→ bbaababa       (Using B → a)

 

2. Rightmost Derivation-

 

S   → bB

→ bbBB                (Using B → bBB)

→ bbBaS              (Using B → aS)

→ bbBabB            (Using S → bB)

→ bbBabaS          (Using B → aS)

→ bbBababB        (Using S → bB)

→ bbBababa        (Using B → a)

→ bbaababa         (Using B → a)

 

3. Parse Tree-

 

 

  • Whether we consider the leftmost derivation or rightmost derivation, we get the above parse tree.
  • The reason is given grammar is unambiguous.

 

Problem-02:

 

Consider the grammar-

S → A1B

A → 0A / ∈

B → 0B / 1B / ∈

 

For the string w = 00101, find-

  1. Leftmost derivation
  2. Rightmost derivation
  3. Parse Tree

 

Solution-

 

1. Leftmost Derivation-

 

S   → A1B

→ 0A1B              (Using A → 0A)

→ 00A1B            (Using A → 0A)

→ 001B              (Using A → ∈)

→ 0010B            (Using B → 0B)

→ 00101B          (Using B → 1B)

→ 00101             (Using B → ∈)

 

2. Rightmost Derivation-

 

S   → A1B

→ A10B                (Using B → 0B)

→ A101B              (Using B → 1B)

→ A101                (Using B → ∈)

→ 0A101              (Using A → 0A)

→ 00A101            (Using A → 0A)

→ 00101               (Using A → ∈)

 

3. Parse Tree-

 

 

  • Whether we consider the leftmost derivation or rightmost derivation, we get the above parse tree.
  • The reason is given grammar is unambiguous.

 

To gain better understanding about Derivations and Parse Tree,

Watch this Video Lecture

 

Next Article- Elimination of Left Recursion

 

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