Chapter 0: Q14P (page 1)

Show that for any two languages $A a n d B$ , a language J exists, where $A \leq_{T} J a n d B \leq_{T} J .$

Short Answer

Expert verified

Answer :

This proof of statement is given below.

Step by step solution

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.

Proof of problem.

Here, $\leq_{T}$ 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,

$A \in A a n d b \in B$

And let a' and b' be symbols such that: a',b' is not in the word $w \in A υ B .$

Now we will define such that:

$J = {a a' | a \in A} \cup {b b' | b \in B}$

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

M=on input<w>

Obtain query oracle if $w a' \cup w b \in J$
If above is TRUE, Then accept
Else.
Reject.

Since, ( $w \in A \Rightarrow w a' \in J) a n d (w \in B \Rightarrow' \in J)$

So, if $M_{1} a n d M_{2}$ are TM of A and B respectively, then M will simulate $M_{1} a n d M_{2}$ 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 $A \leq_{T} J B \leq_{T} J$

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Show that for any two languages $A a n d B$ , a language J exists, where $A \leq_{T} J a n d B \leq_{T} J .$

Short Answer

Step by step solution

Turing Machine

Proof of problem.

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

Short Answer

Step by step solution

Turing Machine

Proof of problem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Computer Science Textbooks

Computer Network

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Study anywhere. Anytime. Across all devices.

Show that for any two languages $A a n d B$ , a language J exists, where $A \leq_{T} J a n d B \leq_{T} J .$