A is some decidable language. Select all statements that are true irrespective of what A is.1 pointHALTS ≤ HALTS'HALTS' ≤ HALTSA ≤ HALTS'HALTS' ≤ A
Question
A is some decidable language. Select all statements that are true irrespective of what A is.
HALTS ≤ HALTS'HALTS' ≤ HALTSA ≤ HALTS'HALTS' ≤ A
Solution
The correct statement irrespective of what A is: A ≤ HALTS'. This is because any decidable language A is Turing reducible to the Halting problem (HALTS'). The Halting problem is undecidable, meaning it is at least as hard as any problem in the class of decidable languages, which includes A.
The other statements are not necessarily true for all decidable languages A.
- HALTS ≤ HALTS': This is not necessarily true because it's saying the Halting problem is reducible to its complement, which is not a proven fact.
- HALTS' ≤ A: This is not necessarily true because it's saying the complement of the Halting problem is reducible to A. But A is a decidable language, and we can't guarantee that every decidable language can solve the complement of the Halting problem.
- HALTS' ≤ HALTS: This is not necessarily true because it's saying the complement of the Halting problem is reducible to the Halting problem, which is not a proven fact.
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