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Chapter 5: Q29P (page 241)

Show that both conditions in Problem 5.28 are necessary for proving that P is undecidable.

Short Answer

Expert verified

Language P is undecidable.

Step by step solution

01

Undecidability

A problem is undecidable if no Turing Machine exist which will halt in finite amount of time.

02

Conditions on  P

There were two conditions:

  • Condition (1): P is nontrivial鈥攊t contains some, but not all, TM descriptions.
  • Condition (2): P is a property of the TM鈥檚 language鈥攚henever,LM1=LM2, we haveM1鈭�P iffM2鈭�P .

In Condition (1), we have constructed Turing Machine that ensures that M is a valid Turning Machine description and accepts the input.

In Condition (2), we had constructed a Turing Machine that will reject all the strings.

03

Proving P undecidable

Now, ifis the property of the machine, then P would be 鈥楧ecidable鈥�, but rather these properties is due to two conditions given, that makes P as undecidable.

Hence, both conditions are necessary for proving that P is undecidable.

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Most popular questions from this chapter

Question: Consider the problem of determining whether a PDA accepts some string of the form {ww|w鈭�{0,1}*}. Use the computation history method to show that this problem is undecidable.

Define a two-headed finite automaton (2DFA) to be a deterministic finite automaton that has two read-only, bidirectional heads that start at the left-hand end of the input tape and can be independently controlled to move in either direction. The tape of a 2DFA is finite and is just large enough to contain the input plus two additional blank tape cells, one on the left-hand end and one on the right-hand end, that serve as delimiters. A 2DFA accepts its input by entering a special accept state. For example, a 2DFA can recognize the languageanbncn|n鈮�0 .

  • a. Let A2DFA={<M,x>|Mis a 2DFA and M acceptsx} . Show that A2DFA is decidable.
  • b. Let E2DFA={<M>|Mis a 2DFA and LM=鈭�}. Show that E2DFA is not decidable.

Say that a variable A in CFG G is necessary if it appears in every derivation of some string w鈭�G. Let NECESSARYCFG={G,AAisanecessaryvariableinG}.

  1. Show that NECESSARYCFG is Turing-recognizable.
  2. Show that NECESSARYCFG is undecidable.

Question: Let T={<M>|MisaTMthatacceptswRwheneveritacceptsw}. Show that T is undecidable.

Show that if A is Turing-recognizable and A猢�mA炉, then A is decidable.
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