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Chapter 5: 7E (page 239)

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

Short Answer

Expert verified

It is proved that A is decidable.

Step by step solution

01

Defining Turing Recognizability and Decidability

Turing Recognizability

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

Decidability

A language is called 鈥楧ecidable鈥� if it鈥檚 able to construct a Turing machine which accepts and halts on every string.

02

Proving A as decidable

Let鈥檚 assume that A猢�mA炉:, then A炉猢�mAby similar mapping reduction.

Now, it鈥檚 given that ifAis Turing-recognizable.

From this, it is also clear thatAisCo-Turing-recognizable.

SinceAis Turing-recognizable and Co-Turing-recognizable.

Therefore, A is decidable.

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Most popular questions from this chapter

Show that the Post Correspondence Problem is decidable over the unary alphabet鈭�=1.

Let 螕=0,1,蟽be the tape alphabet for all TMs in this problem. Define the busy beaver function BB:N鈫�N as follows. For each value of K, consider all K-state TMs that halt when started with a blank tape. LetBBk be the maximum number of1s that remain on the tape among all of these machines. Show thatBB is not a computable function.

Question: Consider the problem of determining whether a two-tape Turing machine ever writes a nonblank symbol on its second tape when it is run on input w. Formulate this problem as a language and show that it is undecidable.

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: Consider the problem of determining whether a single-tape Turing machine ever writes a blank symbol over a nonblank symbol during the course of its computation on any input string. Formulate this problem as a language and show that it is undecidable.
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