a$)$ Design a Turing machine to accept the following language over the alphabet $\{0,1\}$ :

contains twice as many $1$ s as $0s$

Show how the Turing machine processes the string $110011.$

b$)$ For a tape containing an integer $k$ in unary form, construct a $TM$ to replace its input by

$f(k)=k+4$

c$)$ For a tape containing an integers i and $j$ in unary form separated by a blank $b,$ construct a

TM to replace its input by $f(i,j)=i+j$

d$)$ Construct a TM to recognise inputs over the alphabet $\{a,b\}$ containing the substring bab

e$)$ Design a Turing machine to accept the following language over the alphabet $\{0,1,2\}$ :

$L=\{0^n{1}^n{2}^n|n1\}$

f$)$ Design a Turing machine to recognise the following language over the alphabet $\{0,1\}$ :

$L=\{w{w}^{-1}|win\{0,1{\}}^{**}\}$ and $w^{-1}$ is $w$ reversed.