$($Please read questions from the screenshot and give me solutions for question$3$ and $4)$

$3.$ Give a high level, detailed, description of a Turing machine which computes the projection function $U(n,i,x_1,x_2,$dots,$x_n)=x_i.$ Do not give a state diagram. The Turing

machine must begin with $1^n$#$1^i$#$x_1$#dots#$x_n$ and halt with just $x_i$ on its tape left shifted fully. If this was psuedocode, the Turing machine would compute the following algorithm:

def U$($n$,$ i$,$ A$[])$:

return A$[$i$]$

$4.$ This question tests your ability to construct and defend a rigorous formal argument. Recall one of the greatest applications of the Church$-$Turing Thesis. You

may be considerate of an algorithm $($to some extent$)$ and then, by the Church$-$Turing

Thesis, a device must exist $($to some extent$).$ Your assignment is to determine to

what extents does the Church$-$Turing Thesis apply. See the attached chart on canvas.

There are two axii. One of logical purism, neutrality, and rebellion. The other of ex$-$

istential purism, neutrality, and rebellion. I have provided an example for each of the

nine categories, organized into a three by three table. Write a rigorous and persuasive

argument as to which of the nine categories you can best apply the Church$-$Turing

Thesis. Note you are not arguing correctness of the provided example, but of its cat$-$

egory. Your argument should convince me your selection of the nine is correct, and

the other eight are incorrect. In presenting Turing's Thesis, Alan Turing operated on

some set of unstated premises of the intuitive nature of computation. Read his argu$-$

ment carefully and choose a category which best extends from his unstated, assumed

premises.