1. Consider the following attributed tree grammar for type checking a program AST. For simplicity, it hard codes, declarations for "X" and "Y", the identifier "X" and "Y", and the constants 2 and 3. Note also, that it encodes lists of declarations and lists of statements by simply having each declarations/statements last child be the next declaration.

start: start ::= declarations statements
^ statements.env = declarations.env
^ start.type_ok ::= statements.type_ok
decl_stop: declarations ::= Îμ
^ declarations.env = nil
decl_x_int: declarations1 ::= declarations2
^ declarations1.env =
decl_y_int: declarations1 ::= declarations2
^ declarations1.env =
decl_x_bool: declarations1 ::= declarations2
^ declarations1.env =
decl_y_bool: declarations1 ::= declarations2
^ declarations1.env =
stmt_stop: statements ::= Îμ
^ statements.type_ok = true
stmt_if: statements1 ::= expr statements2 statements3 statements4
^ statements1.type_ok = (expr.type == BOOL)
and statements2.type_ok
and statements3.type_ok
and statements4.type_ok
^ expr.env = statements1.env
^ statements2.env = statements1.env
^ statements3.env = statements1.env
^ statements4.env = statements1.env
stmt_prepareint: statements1 ::= expr statements2
^ statements1.type_ok = (expr.type == INT) and statements2.type_ok
^ expr.env = statements1.env
^ statements2.env = statements1.env
<: expr1 ::= expr2 expr3
^ expr1.type = if expr2.type == INT and expr3.type == INT then BOOL else ERR
^ expr2.env = expr1.env
^ expr3.env = expr1.env
+: expr1 ::= expr2 expr3
^ expr1.type = if expr2.type == INT and expr3.type == INT then INT else ERR
^ expr2.env = expr1.env
^ expr3.env = expr1.env
*: epxr1 ::= expr2 expr3
^ expr1.type = if expr2.type == INT and expr3.type == INT then INT else ERR
^ expr2.env = expr1.env
^ expr3.env = expr1.env

2: expr ::= Îμ
^ expr.type = INT

3: expr ::= Îμ
^ expr.type = INT
id_x: expr ::= Îμ
^ expr.type = lookup(X, expr.env)
id_y: expr ::= Îμ
^ expr.type = lookup(Y, expr.env)
With the auxilary function lookup:
lookup(X, env) = if env = nil
then ERR
else let = env in
if Y = X then n else lookup(X, env2)
end

(a) Of the attributes, declarations.env, statements.env, statements.type_ok, expr.env, and expr.type, which are inherited? Which are synthesized?

(b) Is the grammar L-attributed?

(c) Consider the following tree:

start
|
+-decl_x_int
| |
| +-decl_y_bool
| |
| +-decl_stop
|
+-stmt_if
|
+-<
| |
| +-id_x
| |
| +-3
|
+-stmt_prepare
| |
| +-id_y
| |
| +-stmt_stop
|
+-stmt_stop
|
+-stmt_stop

Show how the tree might be type-checked. Annotate the nodes in the tree with numbers indicating the order in which you find outd them? Below, using the same numbering, show the attribute values computed at each step of the evaluation.

(d) Is there a type error in the program represented by this AST? If so, what is it?

2. Consider the TL13' type rules in the posted lecture notes.

(a) Derive a proof tree showing that the following program is well-typed according to those rules:

program begin while true do prepareInt 25 ; end ; end

That is, the judgement:

[] |- program begin while true do prepareInt 25 ; end ; end

(b) Derive a proof tree justifying the judgement that following statement sequence is well-typed under the environment [][x -> int][b -> bool]:

while b do prepareInt x end ;

That is, the judgement:

[][x -> int][b -> bool] |- while b do prepareInt x end ;

(c) Attempt to derive proof tree for the judgment [][x -> bool] |- x * 5 + 2 : int. What premise cannot be satisfied because it doesn't match any rule? Show your incomplete tree circling the unjustified leaf premise (at the top of the tree).