Question

For any positive integer x, let x^R be the integer whose binary representation isthe reverse of the binary representation of x. (Assume no leading in the binary representation of x.) Define the function $R^{+}:N\to N$where$R^{+}\left(x\right)=x+x^R.$
a. Let $A_2=\left\{⟨x,y⟩\left|R^{+}\right(x)=y\right\}$.

Show$A_2\in L.$
b. Let $A_2=\left\{⟨x,y⟩\left|R^{+}\right(R^{+}\left(x\right))=y\right\}$. Show$A_3\in L$.

Short answer

(a), the machine has been performed from the computation between x and x^R. So, the language $A_2\in L$has been proved.

(b), the language $A_3\in L$has been proved by using computation.

Step-by-step solution

Reverse of integer x

x is one of the positive integers and the reverse of integer x is donated by the symbol x^Rin the binary representation.

Function R⁺defined in such a way that the integer x is a natural number.

$R^{+}\left(x\right)=x+x^R$

By replacing ones with zeros and zeros with ones, the binary representation is reversed.

To Explain the A2={⟨x,y⟩|R+(x)=y} and to show the A3∈L

Consider the Turing machineM, which computes the inverse of every positive integer x. When the integer x is multiplied by the inverse of the integer x, the result is also x.

This is due to the fact that when a binary number1is added to a number 0, the result is always 1. The binary representation of x is transformed here, however the binary representation x of must not include 0.

Only x values with a binary representation of 1 will be accepted by a Turing machine. Even a Turing machine will accept inverse of x values with a binary representation of 0.

Following that, the machine computes the difference between x and x^R.

Thus, the language $A_2\in L$has been proved.

To Explain the Turing Machine A2={⟨x,y⟩|R+(R+(x))=y}and show A3∈L

Consider the Turing machine M, which computes the inverse of every positive integer x. When the integer is multiplied by the inverse of the integer x, the result is also x.

This is due to the fact that when a binary number1 is added to a number 0, the result is always 1. The binary representation of x is transformed here, however the binary representation of x must not include 0.

Only x values with a binary representation of 1 will be accepted by a Turing machine. Even a Turing machine will accept inverse of x values with a binary representation of 0. Following that, the machine computes the difference between x and x^R.

Thus, the language$A_3\in L$ has been proved.