The following grammar generates strings of symbols over the alphabet $\{1,0,.\}$:

S $->$ T$.$T

S $->$ T

T $->$ TB

T $->$ B

B $->0$

B $->1$

$($The variables are S$,$ T$,$ and B; S is the starting variable; the terminals are $0,1,$ and $.).$

The grammar generates binary numbers $($like $0,00,1,101,00111100,$ etc.$)$ or binary fractions $($like $0\mathrm{.}01,1\mathrm{.}00,0111\mathrm{.}11001,$ etc.$).$

Write a syntax$-$directed definition $($SDD$)$ of type L to compute the property value.S$,$ whose value is the numerical value of the string derived from S$.$

For example, the numerical value of $0$ is $0,$ the numerical value of $00$ is $0,$ the numerical value of $1$ is $1,$ the numerical value of $101$ is $5,$ the numerical value of $00111100$ is $60,$ the numerical value of $0\mathrm{.}01$ is $0\mathrm{.}25($a quarter$),$ the numerical value of $1\mathrm{.}00$ is $1,$ the numerical value of $0111\mathrm{.}11001$ is $7\mathrm{.}78125(7+$ half $+$ quarter $+1/32).$

Define additional properties as necessary.

Explain in detail, in prose, the SDD you constructed.