Solution

The correct answer is Both A and R are correct and R is the correct explanation of A.

Key Points

• Linear bounded automata (LBA) is a type of Turing machine that operates within space bounded by the input size.
• The language L = {aⁿbⁿcⁿ: n > 0} is a context-sensitive language because it requires equal numbers of a, b, and c in the correct sequence.
• Context-sensitive languages are precisely the class of languages recognized by LBAs.
• The assertion and reason align since LBAs recognize context-sensitive languages, and the given language is context-sensitive.

Additional Information

• Context-Sensitive Language:
  • A language is context-sensitive if it can be generated by a context-sensitive grammar, where production rules are of the form αAβ → αγβ, with γ ≠ ε.
  • Context-sensitive grammars can define languages that require constraints or dependencies between different parts of the input.
• Linear Bounded Automata:
  • An LBA is a restricted form of a Turing machine where the tape size is limited to the input size.
  • LBAs are used to recognize context-sensitive languages, which are more expressive than context-free languages but less powerful than general Turing-recognizable languages.
  • LBAs have practical applications in parsing and analyzing structured data with constraints.
• Relation Between Context-Sensitive Languages and LBAs:
  • All context-sensitive languages can be recognized by LBAs.
  • The language L = {aⁿbⁿcⁿ: n > 0} is an example of a context-sensitive language because it imposes a strict dependency between the counts of symbols.