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Grammar Ambiguity | Check Ambiguous Grammar

Grammar Ambiguity-

Before you go through this article, make sure that you have gone through the previous article on Ambiguous Grammar.
  • There exists no algorithm to check whether any given grammar is ambiguous or not.
  • This general decision problem is undecidable-
“Whether a grammar is ambiguous or not?”
  • This is because it can be shown that this problem is equivalent to Post Correspondence Problem.

General Approach To Check Grammar Ambiguity-

To check whether a given grammar is ambiguous or not, we follow the following steps-

Step-01:

We try finding a string from the Language of Grammar such that for the string there exists more than one-
  • parse tree
  • or derivation tree
  • or syntax tree
  • or leftmost derivation
  • or rightmost derivation

Step-02:

If there exists at least one such string, then the grammar is ambiguous otherwise unambiguous.

PROBLEMS BASED ON CHECKING WHETHER GRAMMAR IS AMBIGUOUS-

Problem-01:

Check whether the given grammar is ambiguous or not-
S → SS
S → a
S → b

Solution-

Let us consider a string w generated by the given grammar-
w = abba
Now, let us draw parse trees for this string w.



Since two different parse trees exist for string w, therefore the given grammar is ambiguous.

Problem-02:

Check whether the given grammar is ambiguous or not-
S → A / B
A → aAb / ab
B → abB / ∈

Solution-

Let us consider a string w generated by the given grammar-
w = ab
Now, let us draw parse trees for this string w.

Since two different parse trees exist for string w, therefore the given grammar is ambiguous.

Problem-03:

Check whether the given grammar is ambiguous or not-
S → AB / C
A → aAb / ab
B → cBd / cd
C → aCd / aDd
D → bDc / bc

Solution-

Let us consider a string w generated by the given grammar-
w = aabbccdd
Now, let us draw parse trees for this string w.

Since two different parse trees exist for string w, therefore the given grammar is ambiguous.

Problem-04:

Check whether the given grammar is ambiguous or not-
S → AB / aaB
A → a / Aa
B → b

Solution-

Let us consider a string w generated by the given grammar-
w = aab
Now, let us draw parse trees for this string w.

Since two different parse trees exist for string w, therefore the given grammar is ambiguous.

Problem-05:

Check whether the given grammar is ambiguous or not-
S → a / abSb / aAb
A → bS / aAAb

Solution-

Let us consider a string w generated by the given grammar-
w = abababb
Now, let us draw parse trees for this string w.


Since two different parse trees exist for string w, therefore the given grammar is ambiguous.

Problem-06:

Check whether the given grammar is ambiguous or not-
E → E + T / T
T → T x F / F
F → id

Solution-

  • There exists no string belonging to the language of grammar which has more than one parse tree.
  • Since a unique parse tree exists for all the strings, therefore the given grammar is unambiguous.

Problem-07:

Check whether the given grammar is ambiguous or not-
S → aSbS / bSaS / ∈

Solution-

Let us consider a string w generated by the given grammar-
w = abab
Now, let us draw parse trees for this string w.


Since two different parse trees exist for string w, therefore the given grammar is ambiguous.

Problem-08:

Check whether the given grammar is ambiguous or not-
R → R + R / R . R / R* / a / b

Solution-

Let us consider a string w generated by the given grammar-
w = ab + a
Now, let us draw parse trees for this string w.

Since two different parse trees exist for string w, therefore the given grammar is ambiguous

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