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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|∃yx,y∈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|∃yx,y∈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|∃yx,y∈Dis tested using the constructed Turing machine.

If x,y∈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|∃yx,y∈D.

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Most popular questions from this chapter

Let ALLDFA={A|A  is a DFA andLA=∑*}. Show thatALLDFAis decidable.

Let A and B be two disjoint languages. Say that language C separates A and B if A⊆Cand B⊆C. Show that any two-disjoint co-Turing-recognizable languages are separable by some decidable language.

Let CCFG={G,k|G is CFG and  LG contains exactly k strings where k⩾0 or k=∞}Show that is decidable.

Let C={{G,x}|GisaCFGxisasubstringofsomey∈L(G)}. Show thatC is decidable.

Let A={M|M is  a  DFA  that  doesn't  accept  any  string  contianing  an  odd number of  1s}. Show that A is decidable.
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