$e(exp)$::$=n|e|e_{1}o+e_2$

$\frac{e_1{e}_1^{\text{'}}}{e_{1}o+e_2{e}_1^{\text{'}}o+e_2},\frac{e_2{e}_2^{\text{'}}}{no+e_{2}no+e_2^{\text{'}}},\frac{mis\text{t}\text{h}\text{e}\text{s}\text{u}\text{m}of{n}_1\text{a}\text{n}\text{d}{n}_2}{n_{1}o+n_{2}m}$

$\frac{e{e}^{\text{'}}}{ee},\frac{mis\text{t}\text{h}\text{e}\text{n}\text{e}\text{g}\text{a}\text{t}\text{i}\text{o}\text{n}ofn}{nm}$

Figure I: Syntax and operational semantics of is a meta$-$variable

for integers$)$

Question $4$

Let $L_4$ be the language whose syntax and semantics are defined in Figure I.

Write a reduction sequence of $(4o+9)o+(5)$ to a value in $L_4.$

Question $5$

Language $L_5$ is the language with adding conditional branches "ifo $e_1$ then $e_2$ else $e_3"$

to $L_4,$ i$.$e$.,$ the syntax of $L_5$ is given as follows:

$e(exp)$::$=n|e|e_{1}o+e_2|if0e_1$ then $e_2$ else $e_3$

Conditional branch "ifo $e_1$ then $e_2$ else $e_3"$ evaluates to $e_2$ if $e_1$ evaluates to $0$ and

evaluates to $e_3$ otherwise. For example,

ifo $(ifo0$ then $1$ else $2)$ then $3$ else $4$ ifo $1$ then $3$ else $44.$

Define the small$-$step operational semantics $($rules$)$ of conditional branches.

Question $6$

Define the big$-$step operational semantics of $L_5.$