Evaluating Assertion A and Reason R

We need to determine the correctness of the Assertion (A) and Reason (R) regarding Linear Bounded Automata (LBA) and the language $L = \{a^n b^n c^n : n \ge 0\}$.

Assertion A Analysis

• The language $L = \{a^n b^n c^n : n \ge 0\}$ represents strings where the number of 'a's, 'b's, and 'c's are equal and follow the order a, then b, then c.
• This language is a standard example of a Context-Sensitive Language (CSL).
• Linear Bounded Automata (LBAs) are a type of Turing machine whose tape length is bounded by a linear function of the input length.
• It is a known result in automata theory that LBAs accept precisely the class of context-sensitive languages.
• Therefore, Assertion A is correct as $L$ is a CSL and is accepted by an LBA.

Reason R Analysis

• Reason R states that Linear Bounded Automata recognize exactly the class of context-sensitive languages.
• This is a fundamental definition and property of LBAs in the Chomsky hierarchy.
• Therefore, Reason R is correct.

Relationship between A and R

• Assertion A states a specific instance: a CSL ($L$) is accepted by an LBA.
• Reason R provides the general principle: LBAs accept all CSLs.
• Since $L$ is a CSL (established in A's analysis) and LBAs accept all CSLs (stated in R), Reason R directly explains why Assertion A is true.

Conclusion

Both Assertion A and Reason R are correct statements. Furthermore, Reason R provides the theoretical basis for why Assertion A is true.

Thus, both A and R are correct and R is the correct explanation of A.