Q$3(10$ points$)$

Consider a $1n$ strip of cells containing white, grey, and black stones with the following rules:

$($i$)$ Every cell contains exactly one stone.

$($ii$)$ There is at least one black stone on the strip.

$($iii$)$ A grey stone cannot be adjacent to a black stone.

$($iv$)$ All cells to the left of a white stone are occupied by white stones.

For each $i=1,2,$dots,$n,$ let $w_i,g_i,b_i$ be boolean variables. We interpret $w_i=1$ to mean there is a white stone on cell i and

$w_i=0$ to mean there is not a white stone on cell $i.$ Similarly for $g_i($grey$)$ and $b_i($black$).$

Write down a boolean formula $F$ such that for any assignment of values $(0$ or $1)$ to the variables, $F$ evaluates to true if and only if

the rules above are respected.

You may use the following notations, similar to the sigma notation used for sums:

$vv{v}_i=1^n{x}_i=x_{1}vv{x}_{2}vvdotsvv{x}_n$

${?}^{{?}^{?}}{?}^{{?}_{?}}(i=1{)}^n{x}_i=x_1^{{?}^{?}}{x}_2^{{?}^{?}}dot{s}^{{?}^{?}}{x}_n$