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Select all languages that are known to be decidable.1 point{x}, where x is some specific string{}{x#y#xy | x and y are strings over {a, b}}HALTS = {(M, x) | M halts on input x}{x: x = aba and Goldbach conjecture is True}

Question

Select all languages that are known to be decidable.1 point{x}, where x is some specific string{}{x#y#xy | x and y are strings over {a, b}}HALTS = {(M, x) | M halts on input x}{x: x = aba and Goldbach conjecture is True}

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Solution 1

The languages that are known to be decidable from the given options are:

  1. {x}, where x is some specific string: This language is decidable because it only contains one specific string. A Turing machine can decide this language by accepting if the input is x and rejecting otherwise.

  2. {}: This language is decidable because it contains no strings. A Turing machine can decide this language by always rejecting.

  3. {x#y#xy | x and y are strings over {a, b}}: This language is decidable because a Turing machine can decide it by checking if the input string can be split into three parts such that the first part is the same as the third part and the second part is the same as the fourth part.

  4. {x: x = aba and Goldbach conjecture is True}: This language is decidable because it only contains the string 'aba' if the Goldbach conjecture is true. A Turing machine can decide this language by accepting if the input is 'aba' and rejecting otherwise.

The language HALTS = {(M, x) | M halts on input x} is not decidable. This is the Halting Problem, which is known to be undecidable. A Turing machine cannot decide whether an arbitrary machine M will halt on input x.

This problem has been solved

Solution 2

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