Consider the following grammar (that admits a series of declarations, followed by expressions) and the associated syntax directed translation (SDT) actions, given as pseudo-code: $$P \rightarrow D^* E^*$$ $$D \rightarrow \text{int } \text{ID } \{\text{record that } \text{ID.lexeme} \text{ is of type int}\}$$ $$D \rightarrow \text{bool } \text{ID } \{\text{record that } \text{ID.lexeme} \text{ is of type bool}\}$$ $$E \rightarrow E_1 + E_2 \;\{\text{check that } E_1.\text{type} = E_2.\text{type} = \text{int}; \text{ set } E.\text{type} := \text{int}\}$$ $$E \rightarrow !E_1 \;\{\text{check that } E_1.\text{type} = \text{bool}; \text{ set } E.\text{type} := \text{bool}\}$$ $$E \rightarrow \text{ID } \{\text{set } E.\text{type} := \text{int}\}$$ With respect to the above grammar, which one of the following choices is correct?

Let's analyze the semantic rules in the given Syntax Directed Translation (SDT):

1. Declarations ($D$):

   • Variable types (int or bool) are recorded in the symbol table when declared.

2. Expressions ($E$):

   • In the base production $E \rightarrow \text{ID}$, the semantic action is: set E.type := int.

   • This hardcodes the type of any identifier appearing in an expression to int, completely bypassing the symbol table lookup.

3. Evaluation for Integer Expressions:

   • For any integer variable $\text{ID}$, $E \rightarrow \text{ID}$ sets $E.\text{type} = \text{int}$.

   • For addition $E \rightarrow E_1 + E_2$, it verifies that both operands are int and sets the result type to int. This works correctly for integer variables and expressions.

4. Evaluation for Boolean Expressions:

   • For any boolean variable $\text{ID}$, $E \rightarrow \text{ID}$ sets $E.\text{type} = \text{int}$ instead of bool.

   • When evaluating $E \rightarrow !E_1$, the condition check that E_1.type = bool will fail because $E_1.\text{type}$ is int. Therefore, boolean expressions involving variables will fail type checking.

Hence, the actions can only be used to type-check syntactically correct integer variable declarations and integer expressions.

Therefore, (B) is the correct choice.