Consider the following grammar where $S$ is the start symbol, and $a$ and $b$ are terminal symbols. $$S \rightarrow aSbS \mid bS \mid \epsilon$$ Which of the following statements is/are true?

Let us analyze each statement based on the grammar $S \to aSbS \mid bS \mid \epsilon$:

1. Ambiguity and Derivations for $abb$ (Statements A and B): Consider the string $w = abb$:

   • Derivation 1:

     $$S \Rightarrow aSbS \Rightarrow a(bS)bS \Rightarrow ab(\epsilon)b(\epsilon) = abb$$

   • Derivation 2:

     $$S \Rightarrow aSbS \Rightarrow a(\epsilon)bS \Rightarrow ab(bS) \Rightarrow abb(\epsilon) = abb$$

   Because there exist two distinct parse trees (and therefore two distinct leftmost/rightmost derivations) for the string $abb$, the string $abb$ has two distinct derivations and the grammar is ambiguous. Hence, (A) and (B) are true.

2. Derivation for $abab$ (Statement C): Let us find the rightmost derivation for $w = abab$:

   $$S \Rightarrow aSbS \Rightarrow aSb(aSbS) \Rightarrow aSb(aSb\epsilon) \Rightarrow aSb(a\epsilon b) \Rightarrow a(\epsilon)bab = abab$$

   There is no other valid derivation sequence to generate $abab$. Since it produces a unique parse tree, $abab$ has only one rightmost derivation. Thus, (C) is true.

3. Decidability (Statement D): The language generated by this context-free grammar is a context-free language (CFL). All context-free languages are decidable. Therefore, statement (D) is false.

Hence, the correct statements are A, B, and C.