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Chapter 0: Q23P (page 1)

Show that A is decidable iff A鈮�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)

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

Else

If R rejects, output: 10

From above constructed TM T , we can conclude:
w鈭�A鈬�outputfromT鈭�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

Myhill鈥揘erode 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.

Consider the undirected graph G=(V,E)whereV, the set of nodes, is{1,2,3,4}
andE, the set of edges, is{{1,2},{2,3},{1,3},{2,4},{1,4}}. Draw the graphG. What are the degrees of each node? Indicate a path from node 3 to node 4 on your drawing ofG.

Give a counter example to show that the following construction fails to prove that the class of context-free languages is closed under star. Let A be a CFL that is generated by the CFG G=(V,鈭�,R,S). Add the new ruleS鈫�SS and call the resulting grammar. This grammar is supposed to generate A*.

Let MODEXP={ha,b,c,pi|a,b,c,andpare positive binary integers such that ab鈮�cmodp}.

Show that MODEXP鈭�P. (Note that the most obvious algorithm doesn鈥檛 run in polynomial time. Hint: Try it first where b is a power of 2 .)

Consider the language B=L(G), where Gis the grammar given in

Exercise 2.13. The pumping lemma for context-free languages, Theorem 2.34,

states the existence of a pumping length p for B . What is the minimum value

of p that works in the pumping lemma? Justify your answer.
See all solutions

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