$2^{nd}$ Question:

Two tapes. We wish to construct a $2-$tape Turing Machine

$M=(Q,,,,s,q_{\text{a}\text{c}\text{c}\text{e}\text{p}\text{t}},q_{\text{r}\text{e}\text{j}\text{e}\text{c}\text{t}})$

that decides the language $L=\{a^n{b}^n|n1\}.$ For simplicity, our machine's tapes are $2-$way infinite and it allows the stay$-$put motion $S.$

Initially, the input $w$ is written on tape $1,$ with head $1$ under its first symbol $($or under a blank if the input is empty$).$ At the end of its computation, the machine should be in state $q_{\text{a}\text{c}\text{c}\text{e}\text{p}\text{t}}$ if winL or in state $q_{\text{r}\text{e}\text{j}\text{e}\text{c}\text{t}}$ if $\{$:$w!$inL$\}.$ A constraint is that your machine should never print $b$ on tape $2.$

We choose $Q=\{s,q,q_{\text{a}\text{c}\text{c}\text{e}\text{p}\text{t}},q_{\text{r}\text{e}\text{j}\text{e}\text{c}\text{t}}\},=\{a,b\},=\{u\}$ and begin with the transitions:

$])$

$([*])$

$([S])$

$([b])$

$([a])$

$([R])$

$([b])$

$([b])$

$([L$

A transition is inessential if no input will ever cause $M$ to execute this transition, otherwise it is essential. Of the missing $15$ transitions of $M$ only $6($for $q)$ are essential. Write these $6$ transitions to complete $.$