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Chapter 0: Q18P (page 1)

Show that EQTM is recognizable by a Turing machine with an oracle for ATM.

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Step by step solution

01

Turing Recognizable.

A language L is said to be Turing Recognizable if and only if there exist any Turing Machine (TM) which recognize it i.e. Turing Machine halt and accept strings belong to language and will reject or not halt on the input strings that doesn鈥檛 belong to language.

02

The proof is given below.

To prove above statement, we will assume an oracle Turing Machine T for ATM.

Let us also consider two Turing Machines P and Q such that they will be decider for TM T and T will be decider for ATM. This in turn makes ATM as Turing Recognizable for EQTM.

We will express above sentence as follow:

(P,w)鈭�ATM鈬�T(M,w)鈭�EQTM

We will construct Turing Machine P and Q from above expression:

T = on inputP,w:string w will be made run on P

  • Construct TM P and Q such that:

鈬扨= on input: Accept in all input

鈬� Q= on input: w run on P

If accepts at w, then accept for any input

  • Output: 鈥筆,Q鈥�

Thus we can conclude that our assumed TM works as decider for ATM and so ATM will be decider for EQTM.

This proves that EQTM is recognizable by a Turing machine with an oracle for ATM .

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Most popular questions from this chapter

In both parts, provide an analysis of the time complexity of your algorithm.

a. Show thatEQDFA鈭�P.

b. Say that a languageAisstar-closedifA=A*. Give a polynomial time algorithm

to test whether a DFArecognizes a star-closed language. (Note that EQNFAis not

known to be in P.)

In Corollary 4.18, we showed that the set of all languages is uncountable. Use this result to prove that languages exist that are not recognizable by an oracle Turing machine with an oracle for ATM.

Write formal descriptions of the following sets.

  1. The set containing the numbers1,10, and100
  2. The set containing all integers that are greater than5
  3. The set containing all natural numbers that are less than5
  4. The set containing the string aba
  5. The set containing the empty string
  6. The set containing nothing at all

If Cis a set withcelements, how many elements are in the power set of C? Explain your answer.

Let 蠁eqbe defined as in Problem 6.10. Give a model of the sentence
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