computer science

$1.)$ Give an implementation level description of a Turing machine. You can check for an empty tape cell as a $\_$ character. Convert from decimal to Unary: You are given a $2-$digit integer on a tape and you need to replace it with that many $1\text{'}$s$.$ They do not have to be at the beginning of the tape. Examples: $"12"-->"111111111111","07",-->"1111111","00"-->"\_-".$

A Turing machine implementation for the conversion from decimal to unary would begin with a blank tape. The machine would begin in the leftmost cell and scan to the right untili it reaches a cell containing data. It would then read the value in that cell and write the appropriate number of $15$ in the adjacent cells. Finally, it would return to the leftmost cell and repeat the process until the tape ran out.

For instance, if the input was $"12,"$ the machine would write $"1"$ in the first cell before moving to the second. It would then write $"1111"$ in the following four cells before returning to the initial cell. Then, it would put $"1"$ in the first cell and go to the second. It would then write $"1111"$ in the next four cells before returning to the first cell and repeating the process until the end of the tape is reached.

Similarly, if the input were $"07",$ the machine would write $"1"$ in the first cell before on to the second. It would then write $"11111111"$ in the following seven cells before returning to the first cell. Then, it would put $"1"$ in the first cell and go to the second. It would then write $"1111111"$ in the next seven cells before returning to the first cell and repeating the process until the end of the tape was reached.

Lastly, given the input $"00,"$ the computer would perform no action because there are no non$-$blank cells.

$2.)$ Give the state transition diagram $($formal description$)$ for the Turing machine in Problem $1.$

Question $2[36$ marks$]$

A mass of consumers is uniformly distributed along the interval $0,1.$ Two firms, A and $B,$ are located at points $0$ and $1$ respectively. We denote by $p_i$ the price of firm iinA,$B.$ A consumer located at point xin$[0,1]$ obtains utility $U_A(x)=u-p_A-t{x}^2$ if he consumes from firm $A,$ and $U_B(x)=u-p_B-t(1-x{)}^2$ if he consumes from firm $B.$ In the following, we assume that the gross utility $u$ is sufficiently high, so that the market will be covered and all consumers will get positive utility in equilibrium. Both firms have a cost function equal to $T_i(q_i)=(1+x)q_i,$ where you should substitute $x$ for the last number of your student ID number.

$($a$)$ Find the demand function for both firms.

$($b$)$ Assume firms set their prices simultaneously. Solve for the Nash equilibrium prices, and compute the equilibrium profits. $[6$ marks code to Check if two strings are Anagram or not