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Chapter 0: Q14P (page 1)

Show that for any two languages AandB , a language J exists, where A鈮�TJandB鈮�TJ.

Short Answer

Expert verified

Answer :

This proof of statement is given below.

Step by step solution

01

Turing Machine

A Turing Machine is computational model concept that runs on the unrestricted grammar of Type-zero. It accepts recursive enumerable language. It comprises of an infinite tape length where read and write operation can be perform accordingly.

02

Proof of problem.

Here, 鈮�T is Turing Reducible.

So, we have to prove that A are B Turing Reducible to J.

Let there be strings a and b such that,

A鈭�Aandb鈭�B

And let a' and b' be symbols such that: a',b' is not in the word w鈭�A蠀B.

Now we will define such that:

J={aa'|a鈭�A}鈭�{bb'|b鈭�B}

Now we will construct an oracle Turing Machine M as follows:

M=on input<w>

  • Obtain query oracle if wa'鈭�wb鈭�J
  • If above is TRUE, Then accept
    Else.
  • Reject.


Since, (w鈭�A鈬�wa'鈭�J)and(w鈭�B鈬�'鈭�J)


So, if M1andM2are TM of A and B respectively, then M will simulate M1andM2 simultaneously.


And TM M decide J from above.

Thus we can say that, A and b are Turing recognizable to J .

Therefore, for any two languages A and B, a language J exists, where A鈮�TJB鈮�TJ

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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.

Show that the function K(x) is not a computable function.

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 G=(V,鈭�,R,S)that is generated by the CFG . Add the new rule S鈫�SSand call the resulting grammar. This grammar is supposed to generate A* .

Show how to compute the descriptive complexity of strings K(x) with an oracle for ATM.

LetMAX-CLIQUE={(G,k)|alargestcliqueinGisofsizeexactlyk}. Use the result of Problem 7.47 to show that MAX-CLIQUEis DP-complete.
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