GATE CS 2016 Set 1 — Question 54
Go beyond PYQs with Success TrackerAI-powered personalised practice and doubt support. Unlimited practice on eligible plans; AI usage limits apply.MCQ+2 / -0.67MediumRecursive & RE LanguagesTuring Machines & ComputabilityTheory of ComputationUndecidability Reductions
Theory of Computation → Turing Machines & Computability → Recursive & RE Languages
Last updated
Question
Let be a recursive language and be a recursively enumerable but not recursive language. Let and be two languages such that reduces to , and reduces to (reduction means the standard many-one reduction). Which one of the following statements is TRUE?
Correct answer
(C) W is not recursively enumerable and Z is recursive.
Solution
Given:
Since is recursive, its complement is also recursive (recursive languages are closed under complementation).
We are given .
Property: If and is recursive, then is recursive.
Since is recursive, must be recursive.Analysis of :
We are given that is RE but not recursive. This implies that is undecidable.
Property: A language is recursive if and only if both and are RE.
Since is RE but not recursive, cannot be RE (otherwise would be recursive).
We are given .
Property: If and is RE, then is RE.
Assume is RE. Then must be RE. But we established that is not RE. This is a contradiction.
Therefore, cannot be RE (and consequently, not recursive).Conclusion:
is not recursively enumerable.
is recursive.This matches option (C).
1. is a recursive language.
2. is a recursively enumerable (RE) but not recursive language.
3. ( reduces to ).
4. ( reduces to ).
Analysis of :Since is recursive, its complement is also recursive (recursive languages are closed under complementation).
We are given .
Property: If and is recursive, then is recursive.
Since is recursive, must be recursive.Analysis of :
We are given that is RE but not recursive. This implies that is undecidable.
Property: A language is recursive if and only if both and are RE.
Since is RE but not recursive, cannot be RE (otherwise would be recursive).
We are given .
Property: If and is RE, then is RE.
Assume is RE. Then must be RE. But we established that is not RE. This is a contradiction.
Therefore, cannot be RE (and consequently, not recursive).Conclusion:
is not recursively enumerable.
is recursive.This matches option (C).
Continue learning with Success Tracker
A step still unclear? Work through it with support
Use Success Tracker to ask about the reasoning, then try another GATE CS question to check your understanding.
AI-powered practice· Unlimited practice on eligible plans
- PYQs with solutions
- Attempt available previous-year questions, then compare your reasoning with the worked solution. Coverage varies by stream.
- Practice that adapts
- Choose a topic, work on weaker areas and bookmark questions to revisit. Your attempts feed your progress tracking.
- AI doubt support
- Ask follow-up questions about a step or concept while practising, instead of stopping at the final answer.
Unlimited practice is available on eligible plans. Free practice and AI usage have limits; check the current plan allowances before choosing.
This page stays readable without an account. AI responses can be wrong; check them against the solution and source material.
More questions on Turing Machines & Computability
2026 Set 2 Q13Which one of the following statements is equivalent to the following assertion? Turing machine …2026 Set 1 Q25Consider the following grammar where is the start symbol, and and are terminal symbols.…2026 Set 1 Q26Let be a nondeterministic finite automaton (NFA) with 6 states over a finite alphabet. Which of…2026 Set 2 Q29Which of the following grammars is/are ambiguous?2026 Set 2 Q47Consider the following two finite automata and . [figure] Which of the following…