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Use the string “ababab” to show that the following grammar is ambiguous: 𝑆 → 𝑆𝑆 | 𝑇 𝑇 → 𝑎𝑇𝑏 | 𝑎𝑏

Question

Use the string “ababab” to show that the following grammar is ambiguous:

S→SS ∣ T S \rightarrow SS \, | \, T T→aTb ∣ ab T \rightarrow aTb \, | \, ab

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Solution

The given grammar is ambiguous because the string "ababab" can be derived in two different ways. Here are the two different derivations:

  1. Derivation 1:

    • S -> SS (using the rule S -> SS)
    • S -> TS (replacing the first S in SS with T, using the rule S -> T)
    • S -> aTbS (replacing T in TS with aTb, using the rule T -> aTb)
    • S -> abS (replacing T in aTb with ab, using the rule T -> ab)
    • S -> abTS (replacing the first S in abS with T, using the rule S -> T)
    • S -> abaTb (replacing T in abTS with aTb, using the rule T -> aTb)
    • S -> abab (replacing T in abaTb with ab, using the rule T -> ab)

  2. Derivation 2:

    • S -> SS (using the rule S -> SS)
    • S -> TS (replacing the first S in SS with T, using the rule S -> T)
    • S -> aTbS (replacing T in TS with aTb, using the rule T -> aTb)
    • S -> abTS (replacing T in aTbS with ab, using the rule T -> ab)
    • S -> abTS (replacing the first S in abTS with T, using the rule S -> T)
    • S -> abaTb (replacing T in abTS with aTb, using the rule T -> aTb)
    • S -> abab (replacing T in abaTb with ab, using the rule T -> ab)

As you can see, the string "ababab" can be derived in two different ways from the given grammar, which shows that the grammar is ambiguous.

This problem has been solved

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