# Assignment 2 Solution.

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1. (5 points) Consider the language used to illustrate code generation via attribute gram-mars. Suppose we extend it with do-while loops: hci1 ::= do hci2 while hbei Extend the attribute grammar discussed in class to handle such loops. Show the relevant rules for computing the necessary attributes.

Illustrate your solution using the parse tree for string i:=0; do i:=i+1 while …

Show the values of attributes temp, labin, and labout at relevant parse tree nodes. Show the value of attribute code only for the root node. In your solution, dis-play the value of hbei.code as [..be..] | that is, as some sequence of assembly instructions that is not displayed in detail in the solution.

1. (6 points) In class we brie y discussed several helper functions used in the de nition of the Lisp interptreter. These were

bound(x; z): given a literal atom x and a list z of S-expressions, does there exist an element e of list z such that car(e) is the same as x? If so, the resulting value is T; otherwise, it is NIL.

getval(x; z): given a literal atom x and a list z of S-expressions such that

bound(x; z) is T, the resulting value is cdr(e) where e is the rst element of z such that car(e) is the same as x.

addpairs(x; y; z): given a list x of literal atoms, a list y of arbitrary S-expressions, and an list z of pairs (literal atom, arbitrary S-expression), produce a new list where the i-th element of x is paired with the i-th element of y, and the resulting pairs are prepended to list z (see example in the lecture notes). Precondition: the length of x is the same as the length of y (do not check this condition, just assume it is true).

Write the Lisp functions that correspond to these three mathematical functions. That is, write (DEFUN BOUND (X Z) …) etc. and make sure that you code is a correct Lisp function that can be successfully executed by the interpreter discussed in class.

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1. (6 points) Consider the following subset of the language from our discussion of type systems:

hE i ::= const j NIL j ( PLUS hE i1 hE i2 ) j ( CAR hE i ) j ( CDR hE i ) j ( CONS hE i1 hE i2 )

Here const denotes a numeric atom representing an integer value.

Suppose we wanted to check statically (i.e., without evaluating an expression), whether a given input expression satis es the following condition: the expression and each of its subexpressions evaluate to either a numeric atom or a list of numeric atoms, but nothing else: not something that is not a list, not a list of lists, not a list containing a mix of atoms and lists, etc. Further, the following informal guidelines apply: (1) NIL is an (empty) list of numeric atoms; (2) the operands in (PLUS …) evaluate to numeric atoms; (3) the operands in (CAR …) and (CDR …) evaluate to lists of numeric atoms; (4) the rst operand in (CONS …) evaluates to a numeric atom, while the second one evaluates to a list of numeric atoms.

Part 1 (3 points): Write the inference rules for the typing relation E : T to formalize the informal guidelines from above. Show the derivation tree for

(CAR (CDR (CONS 1 (CONS (PLUS 2 3) NIL)))) : Nat

Note: do not worry about the case where the operand value of CAR or CDR at run time is an empty list. Your typing rules should not try to catch and reject this case|for example, they should determine that (CAR (CDR (CONS 8 NIL))) is well-typed.

Part 2 (3 points): Write a type checker based on these inference rules, using an attribute grammar. De ne an attribute type for hE i that represents the type of the expression, and design an arrtibute grammar based on this attribute. The grammars should reject all and only input expressions that are not well-typed. Do not add any other attributes; hExpri.type is enough to solve the problem.

4. (3 points) Suppose we extend the language from the previous question with hE i ::=

• : : j (RAND hE i1 hE i2). Assume the run-time semantics is the following: randomly, with equal probability, we choose the rst or the second subexpression. The chosen subexpression is evaluated and the result is used as the result of the entire RAND expression. Generalize your inference rules and attribute grammar from Problem 3 to ensure that the two subexpressions in a RAND have the same type (could be \numeric atom” or \list of numertic atoms”; either case is possible and allowed) and to associate a suitable type with the entire RAND expression.

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