Problem $2$: Context free Grammar & derivation $[20$ pts$]$ Context$-$free grammars $($CFGs$)$ can be useful in various use cases. One such use case is to check on the syntax of the programming language and hence even now CFGs are used in Natural Language Processing $($NLP$)$ to help models interpret languages. Further, modern$-$day IDEs are leveraging this use case to construct Syntax Highlighters.

Let's consider you are given a task to design the CFG for a very basic valid variable definition in $C++.$

A couple of examples $($not exhaustive$)$:

float $a=10\mathrm{.}0$;

double sum $=c+d$;

int $a=c++$;

$(10$ pts$)$ Create a context$-$free grammar for the given task. Assume the operators are $\{+,-,**\frac{,}{,}++,--\}$ and the data$-$types are int, float, double, long. For simplicity, we are excluding char or string data types. Further, you may also assume that all variable names are composed of only lowercase English alphabets $($a$-$z$)$ and digits $(0-9)$ but don't start with the digits and there are no special characters or capital letters in the variable name. Also, the operators $++$ and $--$ are only applicable with variable names.

$(10pts)$ Using the grammar generated in part $1$ provide the leftmost derivation of the expression: