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Chapter 0: Q23P (page 1)

Show that A is decidable iff A≤m0*1*.

Short Answer

Expert verified

A is decidable as ..

Step by step solution

01

 Introduction to Decidability

A problem is decidable if there exist a Turing Machine exist which will halt in finite amount of time.

02

Proving A as decidable

So according to the statement in question,

If Then A must be decidable language. This is because 0*1* is a decidable language.

Now we A is decidable, then we must able to construct a corresponding Turing Machine R , which will decide A .

To prove we will construct a Turing Machine T that works as:

T= on input (R , w)

  • · Run R on string w .
  • · If R accepts, output: 01

Else

If R rejects, output: 10

From above constructed TM T , we can conclude:
w∈A⇔outputfromT∈0*1*

So we have got mapping reduction of A on 0*1*.

This implies that: A is decidable iff

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Most popular questions from this chapter

We generally believe that PATH is not NP-complete. Explain the reason behind this belief. Show that proving PATH is not NP-complete would prove P ≠ NP

Let J={w|eitherw=0xfor some,x∈ATM orw=1yfor some y∈ATM¯}. Show that neither JnorJis Turing-recognizable.

Myhill–Nerode theorem. Refer to Problem 1.51 . Let L be a language and let X be a set of strings. Say that X is pairwise distinguishable by L if every two distinct strings in X are distinguishable by L. Define the index of L to be the maximum number of elements in any set that is pair wise distinguishable by L . The index of L may be finite or infinite.

a. Show that if L is recognized by a DFA with k states, L has index at most k.

b. Show that if the index of L is a finite number K , it is recognized by a DFA with k states.

c. Conclude that L is regular iff it has finite index. Moreover, its index is the size of the smallest DFA recognizing it.

Find the error in the following proof that 2 = 1. Consider the equation a = b. Multiply both sides by a to obtain a2 = ab. Subtract b2from both sides to get a2 - b2 = ab - b2. Now factor each side, (a+b) (a-b) = b (a-b),and divide each side by (a-b)to get a + b = bFinally, letequal 1, which shows that 2 = 1

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.
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