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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

Say that a variable A in CFLrole="math" localid="1659808454707" G is usable if it appears in some derivation of some stringw∈G . Given a CFGG and a variable A, consider the problem of testing whetherA is usable. Formulate this problem as a language and show that it is decidable.

Question:LetINFINITEDFA={(A)|AisaDFAandLAisaninfinitelanguage}.Show thatINFINITEDFA is decidable.

Let ∑={0,1}.Show that the problem of determining whether a CFG generates some string in 1*is decidable. In other words, show that

{G|G  is  a  CFG over 0,1 and 1*∩LG≠∅}is a decidable language.

Let PALDFA={M|M is a DFA that accepts some palindrome}. Show thatPALDFAis decidable.

Prove that the class of decidable languages is not closed under homomorphism
See all solutions

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