This homework is due at **Oct 10, 11pm**. *It will not be accepted late*.

Make sure to follow the submission instructions. Link

Homework assignments must be done individually. Collaboration on homework assignments is
*not* allowed.

Part 1 is about the language of *set expressions* defined as follows:

A hash-separated list of zero or more countries enclosed in curly braces is a set expression.

- If S1 and S2 are both set expressions, then so are each of the following:
S1 ∪ S2

S1 ∩ S2

S1 + S2

( S1 )

In a set expression, parenthesis has the highest precedence; intersection and union have the same, second highest precedence; addition has the lowest precedence. Intersection, union and addition are all left associative.

Valid examples:

- {}
- {Australia}
- {England # Australia}
- ({England # Asia} ∩ {Australia}) ∪ {Antarctica}

Write an unambiguous CFG for the language of set expressions so that parse trees correctly reflect the precedences and associativities of the operators. The terminals used are already for tokens. Use lower-case names for nonterminals and use the following terminals:

COUNTRY // one country in a set ∪ // union ∩ // intersection + // plus ( // left paren ) // right paren { // left curly brace } // right curly brace # // hash

For Part 2 you will define a syntax-directed translation for the CFG given below, which defines a very simple programming language.

program → MAIN LPAREN RPAREN LCURLY list RCURLY list → list oneItem | epsilon oneItem → decl | stmt decl → BOOL ID SEMICOLON | INT ID SEMICOLON stmt → ID ASSIGN exp SEMICOLON | IF LPAREN exp RPAREN stmt | LCURLY list RCURLY exp → exp PLUS exp | exp LESS exp | exp EQUALS exp | ID | BOOLLITERAL | INTLITERAL

Write a syntax-directed translation for the CFG given above, so that the
translation of an input program is the *set* of variables used somewhere in the program.
Note: here the term *use* is in contrast to *declaration*;
any appearance of a variable that is not a variable declaration counts as a *use* of that variable. For the
example code in Question 2, the translation should be `{ x, a, b }`.

Your translation rules should use the following notation:

`{ }`is an empty set`{ ID.value }`is a set containing the variable whose name is the value associated with this`ID`token`S1`∩`S2`is the intersection of sets`S1`and`S2``S1`∪`S2`is the union of sets`S1`and`S2``S1 - S2`is the set of all items that are in`S1`but not in`S2`

Note that you should not try to use something like "`{ a, b }`"
to mean a set with two elements;
instead, use set union to combine
two sets that each contain one element.

Use the notation that was used in class and in the on-line readings;
i.e., use `nonterminal.trans` to mean the translation
of a nonterminal, and `terminal.value` to mean the
value of a terminal.
Assume that `ID.value` is a `String` (the name of the
identifier).
Use subscripts for translation rules that include the same nonterminal
or the same terminal more than once.