Which of the following languages is generated by the following grammar G over Sigma = { a , b }:

S -- aS | Sb | ab

Pick all that applies.

Question

Which of the following languages is generated by the following grammar G over Sigma = { a , b }:

S --> aS | Sb | ab

Pick all that applies.

BlackTom AI · Accepted Answer

Every derivation inserts a’s to the left of the nonterminal S (via S → aS) and b’s to the right of S (via S → Sb); the base production S → ab provides one a and one b in the middle. Thus all generated strings have all a’s before all b’s and contain at least one a and one b.

1. Relevant concepts
- Context-free grammar derivations: replacing a nonterminal using productions.
- Form of generated strings: track how repeated applications of aS and Sb extend the string.
- Language description equivalence: recognizing a^n b^m patterns.

2. Step-by-step reasoning
- Base derivation: S → ab produces the string "ab" = a^1 b^1.
- Left-extension: applying S → aS once yields aS, and repeating i times gives a^i S.
- Right-extension: applying S → Sb once yields S b, and repeating j times gives S b^j.
- Combining extensions: apply aS i times to get a^i S, then apply Sb j times to that same S to get a^i S b^j.
- Final step: replace the remaining S by ab to obtain a^i (ab) b^j = a^{i+1} b^{j+1}.

3. Intermediate-step formula
- For any i, j ≥ 0: S ⇒* a^i S b^j ⇒ a^i (ab) b^j = a^{i+1} b^{j+1}.
- Let p = i+1 and q = j+1; then p ≥ 1 and q ≥ 1 and every generated string is a^p b^q.

4. Why the correct answer is correct
- The grammar generates exactly the set { a^p b^q | p ≥ 1, q ≥ 1 }, i.e., strings that start with at least one a and end with at least one b with all a’s before all b’s. The production shapes (only a’s added on the left of S and only b’s on the right) prevent any a occurring after a b, so all a’s precede all b’s.

5. Why the other options are incorrect
- "Exactly one substring ab": this description would include strings like "baab" (which can have exactly one occurrence of the substring "ab") but that string is not of the form a^p b^q, so the option is too broad.
- "a^n b^m where at least one equals 1": excludes strings like a^2 b^2 (e.g., "aabb"), which the grammar does generate, so this option is too restrictive.
- "Contain at least one a and at least one b with no restriction on order": allows strings like "ba" or "aba" where a occurs after b, which cannot be derived from this grammar.

6. Conclusion
The correct characterization is: the set of all strings that start with at least one a and end with at least one b, with all a’s appearing before all b’s. This matches the provided answer.

Which of the following languages is generated by the following grammar G over Sigma = { a , b }: S --> aS | Sb | ab Pick all that applies.多项选择题

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