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Chapter 0: Q9P (page 1)

Use the recursion theorem to give an alternative proof of Rice鈥檚 theorem in Problem 5.28.

Short Answer

Expert verified

Answer:

Rice鈥檚 Theorem is proved.

Step by step solution

01

Rice Theorem

Let鈥檚 first recall the Rice鈥檚 Theorem which was in Problem 5.28.

Let be a language consisting of Turing machine descriptions, then P must fulfil the following conditions:

P is nontrivial鈥攊t contains some, but not all, TM descriptions.

P is a property of the TM鈥檚 language鈥攚henever we have

<M1>鈭�Piff<M2>鈭�P.Here, M1andM2areanyTMs.

02

Proving rice Theorem

Now, let us assume as Turing Machine that is decider of P and P satisfy both the conditions of Rice鈥檚 Theorem which is mentioned above. The second condition says that there are two Turing Machine M1 and M2 , where M1 鈭�P and M2 鈧�P.

Now we will use M1 and M2 to construct a Turing Machine R as follows:

R=on input(w)

  • Obtain own description using recursion theorem.
  • Run X on <R>
  • If X accepts <R> , run M2 on w

Else

If rejects , run M1 on w


<R> 鈭�P, then X will accepts =<R> and L(R) = L(M2)


But since, <B> 鈧�P so this makes as contradict statement. This is due to the fact that P will agree on Turing Machines that have same language.


Also, <R> 鈧�P can also be conclude from above. This shows our assumption was wrong.


So every property which satisfies Rice鈥檚 Theorem are undecidable.

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Most popular questions from this chapter

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.

Examine the following formal descriptions of sets so that you understand which members they contain. Write a short informal English description of each set.

  1. {1,3,5,7,...}
  2. {...,-4,-2,0,2,4,...}
  3. {n|n=2mfor someminN}
  4. {n|n=2mfor someminN, andn=3kfor somekinN}
  5. {w|wis a string of0sand1sandwequals the reverse ofw}
  6. {n|nis an integer andn=n+1}

Show that if A鈮�TBand B鈮�TC, thenA鈮�TC .

A queue automaton is like a push-down automaton except that the stack is replaced by a queue. A queue is a tape allowing symbols to be written only on the left-hand end and read only at the right-hand end. Each write operation (we鈥檒l call it a push) adds a symbol to the left-hand end of the queue and each read operation (we鈥檒l call it a pull) reads and removes a symbol at the right-hand end. As with a PDA, the input is placed on a separate read-only input tape, and the head on the input tape can move only from left to right. The input tape contains a cell with a blank symbol following the input, so that the end of the input can be detected. A queue automaton accepts its input by entering a special accept state at any time. Show that a language can be recognized by a deterministic queue automaton iff the language is Turing-recognizable.

Show that A is decidable iff A鈮�m0*1*.
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