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Chapter 4: 18P (page 212)

Let C be a language. Prove that C is Turing-recognizable if a decidable language D exists such that C={x|鈭�yx,y鈭�D}.

Short Answer

Expert verified

It can be proved that that C is Turing-recognizable if a decidable language D exists such that C=x|鈭�yx,y鈭�D.

Step by step solution

01

Explain Decidable language

A language is said to be decidable if the input of a language is accepted by a Turing machine.

02

Step 2: Prove that C is Turing-recognizable

Consider that the language C is recognized by the Turing machine M and consider that an input x is given to the Turing machine.

The language D on input x,yverifies whether the y decides an accepting computation of the input string x on the machine M.

Consider, that the D is a decider.

Construct the Turing machine that decides the decidability of the C. Construct the Turing machine that searches y and find it.

The given string C=x|鈭�yx,y鈭�Dis tested using the constructed Turing machine.

If x,y鈭�D,the Turing machine accepts, otherwise rejects.

Therefore, it can be proved that that C is Turing-recognizable if a decidable language D exists such that C=x|鈭�yx,y鈭�D.

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