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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

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.

Prove that the following two languages are undecidable.

  1. OVERLAPCFG={G,HGandHareCFGswhereLG鈭�LH6=鈭�}and are CFGs where. (Hint: Adapt the hint in Problem 5.21.)
  2. PREFIX-FREECFG={GGisaCFGwhereL(G)isprefix-free} .

Say that a CFG is minimal if none of its rules can be removed without changing the language generated. Let MINCFG = {GGisaminimalCFG}is a minimal CFG}.

  1. Show that MINCFG is T-recognizable.
  2. Show that MINCFG is undecidable.

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.

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.
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