Question # $1$

For each case below, draw a TM that computes the indicated function. In the first five parts, the function is from $N$ to $N.$ In each of these parts, assume that the TM uses unary notation$-$i$.$e$.,$ the natural number $n$ is represented by the string $1^n.$

a$.f(x)=x+2$

b$.f(x)=2x$

c$.f(x)=x^2$

d$.f(x)=$ the smallest integer greater than or equal to $lo{g}_2(x+1)($i$.$e$.,f(0)=0,f(1)=1,f(2)=f(3)=2,f(4)=$dots$=f(7)=3,$ and so on$).$

e$.E$:$\{a,b{\}}^{**}\{a,b{\}}^{**}\{0,1\}$ defined by $E(x,y)=1$ if $x=y,E(x,y)=0$ otherwise.

f$.p_l$:$\{a,b{\}}^{**}\{a,b{\}}^{**}\{0,1\}$ defined by $p_l(x,y)=1$ if $p_l(x,y)=0$