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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 INFINITEPDA={(M)|MisaPDAandL(M)isaninfinitelanguage}.

Show thatINFINITEPDAis decidable.

LetBALDFA={(M)|Mis a DFA that accepts some string containing an equal number of 0s and 1s}. Show that BALDFA is decidable.

Prove that EQDFA is decidable by testing the two DFAs on all strings up to a certain size. Calculate a size that works.

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A useless state in a pushdown automaton is never entered on any input string. Consider the problem of determining whether a pushdown automaton has any useless states. Formulate this problem as a language and show that it is decidable
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