GATE CS 2004 Q45 — Syntax-directed Translation

To find the value of the expression $2 \# 3 \& 5 \# 6 \& 4$, we first determine the precedence, associativity, and operations associated with the operators $\#$ and $\&$ from the given syntax-directed translation (SDT) rules.

1. Precedence and Associativity

• Precedence: Since $E \rightarrow E \# T$ and $E \rightarrow T$, the operator $\#$ is evaluated at a higher level in the parse tree (closer to the root), meaning it has lower precedence. The operator $\&$ is defined under $T \rightarrow T \& F$, meaning it has higher precedence.
• Associativity: Both productions $E \rightarrow E_1 \# T$ and $T \rightarrow T_1 \& F$ are left-recursive, which means both operators $\#$ and $\&$ are left-associative.

2. Associated Operations

• The rule for $\#$ is $E.value = E_1.value * T.value$, which means $\#$ performs multiplication ($*$).
• The rule for $\&$ is $T.value = T_1.value + F.value$, which means $\&$ performs addition ($+$).

3. Evaluation of the Expression

The given expression is: $$2 \# 3 \& 5 \# 6 \& 4$$

Since $\&$ (addition) has higher precedence than $\#$ (multiplication), we evaluate the $\&$ operations first:

• $3 \& 5 \rightarrow 3 + 5 = 8$
• $6 \& 4 \rightarrow 6 + 4 = 10$

Substituting these values back, the expression simplifies to: $$2 \# 8 \# 10$$

Since $\#$ is left-associative, we evaluate from left to right:

• $(2 \# 8) \# 10$
• $= (2 * 8) * 10$
• $= 16 * 10 = 160$

Thus, the final value at the root of the parse tree is 160.