# Homework 5 (Due Monday 2/21/2011 at 10:00am)

Submit this assignment via Owl-Space In contrast to the previous assignments, submit each problem in a separate `.ss`

file: `1.ss`

, `2.ss`

, `3.ss`

, and `4.ss`

(if you do the extra credit problem). Unfortunately, none of the languages supported by DrScheme will allow these files to be combined. The *Pretty Big* Scheme language allows top-level indentifiers (functions and variables) to be redefined, but it does *not* support `check-expect`

. All of the student languages--the only ones that support `check-expect`

--prohibit redefinition.

Embed answers that are not program text in a Scheme block comment or block commenting brackets (#| and |#).

Use the `Intermediate Student with lambda`

language.

Given the Scheme structure definitions:

(define-struct sum (left right)) (define-struct prod (left right)) (define-struct diff (left right)) (define-struct quot (left right))

an `arith-expr`

is either:

- a number
`n`

, - a sum
`(make-sum ae1 ae2)`

, - a product
`(make-prod ae1 ae2)`

, - a difference
`(make-diff ae1 ae2)`

, or - a quotient
`(make-quot ae1 ae2)`

where `n`

is a Scheme number, and `ae1`

and `ae2`

are `arith-exprs`

.

The following 4 exercises involve the data type `arith-expr`

. If you are asked to write a function(s), follow the design recipe: contract, purpose, examples/tests, template instantiation, code, testing (which happens automatically when the examples are given in `(check-expect ...)`

form). Follow the same recipe for any help function that you introduce.

- (40 pts.) Write an evaluator for arithmetic expressions as follows:
- Write the (function) template for
`arith-expr`

- Write a function
`to-list`

that maps an`arith-expr`

to the corresponding "list" representation in Scheme. Numbers are unchanged. Some other examples include:Note: you need to define the output type (named(to-list (make-sum (make-prod 4 7) 25)) => '(+ (* 4 7) 25) (to-list (make-quot (make-diff 4 7) 25)) => '(/ (- 4 7) 25)

`scheme-expr`

) for this function, but you can omit the template because this assignment does not include any functions that process this type. - Write a function
`eval: arith-expr -> number`

that evaluates an`arith-expr`

. Your evaluator should produce exactly the same result for an`arith-expr E`

that Scheme evaluation would produce for the list representation`(to-list E)`

.

- Write the (function) template for
- (40 pts.) Extend the definition of
`<arith-expr>`

} as follows:- Add a clause for variables represented as Scheme symbols.
- Write the (function) template for this definition.
- Modify your definition of
`to-list`

to support the new definition of arith-expr. - Given the Scheme structure definition:
a
(define-structure binding (var val))

`binding`

is`(make-binding s n)`

where`s`

is a symbol and`n`

is a number and an`environment`

is a`(list-of binding)`

. Write a (function) template for processing an`environment`

. - Define a top-level variable (constant)
`empty-env`

that is bound to the empty environment containing no bindings (*i.e.*, the empty list). - Write a function
`extend`

that takes environment`env`

, a symbol`s`

, and a number`n`

, and returns an extended environment identical to`env`

except that it adds the additional binding of`s`

to`n`

.

The definition of`extend`

is trivial; it requires no recursion. As a result,`extend`

satisfies the invariantand(check-expect (extend empty-env s n) (list (make-binding s n)))

In the remainder of the problem, use(extend empty-env 'a 4) => (list (make-binding 'a 4))

`empty-env`

and`extend`

to define example environments for test cases. - Write a function
`lookup`

that takes a symbol`s`

and an environment`env`

and returns the first binding in`env`

with a`var`

component that equals`s`

. If no match is found,`lookup`

returns empty. Note that the return type of`lookup`

is not simply`binding`

because it can return`empty`

. Define the a new union type called`option-binding`

for the the return type. - Write a new
`eval`

function for the new definition of`arith-expr`

. The new`eval`

takes*two*arguments: an`arith-expr E`

to evaluate and an`environment env`

specifying the values of free variables in`E`

. For example,If an(eval 'x (extend empty-env 'x 17)) => 17 (eval (make-prod 4 7) (extend empty-env 'x 17)) = 28 (eval 'y (extend empty-env 'x 17)) => some form of run-time error

`arith-expr E`

contains a free variable that is not bound in the`environment env`

, then`(eval E env)`

will naturally produce some form of run-time error if you have correctly coded`eval`

. Do*not*explicitly test for this form of error.

- (20 pts.) An
`environment`

is really a finite function (a finite set of ordered pairs). It is*finite*in the sense that it can be completely defined by a finite table, which is not true of nearly all the primitive and library functions in Scheme (and other programming languages). Even the identity function is*not*finite. For the purpose of this exercise, we redefine the type`environment`

as`(symbol -> option-binding)`

.- Rewrite
`eval`

to use`environment`

defined as a finite function in`(symbol -> option-binding)`

instead of as a`(list-of option-binding)`

. If you cleanly coded your definition of`eval`

in the preceding problem using`lookup`

,`make-binding`

, and`extend`

, all that you have to do to your solution to the previous problem is redefine the bindings of`lookup`

,`empty-env`

, and`extend`

, and revise your test cases for`extend`

. You can literally copy the entire text of your solution to problem 2; change the definitions of`lookup`

,`empty-env`

, and`extend`

; update your documentation (annotations) concerning the`environment`

type; and revise your tests for`extend`

. Note that`extend`

cannot be tested (since the result is a function!) without using`lookup`

to examine it. (If you wrote a correct solution to problem 2, you can do this problem is less than 15 minutes!)

**Hint:**you can use`lambda`

-notation to define a constant function for`empty-env`

, and`extend`

can be defined as a functional that takes a function (representing an environment) and adds a new pair to the function--using a`if`

embedded inside a`lambda`

-expression.

- Rewrite
- Extra Credit (50 pts.) Add support for
`lambda`

-expressions in your evaluator as follows:- Extend the definition of
`<arith-expr>`

by adding a clause for unary`lambda`

-expressions and a clause for unary applications of an`arith-expr`

to an`arith-expr`

. Use the name`lam`

for the structure representing a`lambda`

-expression and the names`var`

and`body`

for the accessors of this structure. Use the name`app`

for the structure representing an application and the names`head`

and`arg`

for the accessors of this structure. Note that the head of an`app`

is an`arith-expr`

not a`lam`

. - Write a (function) template for the newest definition of
`arith-expr`

. - Extend the definition of
`to-list`

to support the newest definition of`arith-expr`

. - Extend the definition of
`eval`

to support the newest definition of`arith-expr`

. Note that`eval`

can now return functions as well as numbers. Your biggest challenge is determining a good representation for function values. What does`eval`

return for a`lam`

input? That input may contain free variables. In principle, you could represent the value of the`lam`

input by a revised`lam`

(with no free variables) obtained by substituting the values for free variables from the environment input (just like we do in hand-evaluation). But this approach is tedious and computationally expensive. A better strategy is to define a structure type (called a*closure*) to represent a function value. The structure type must contain the original`lam`

and a description of what substitution would have been made, deferring the actual substitution just as`eval`

defers substitutions by maintaining an environment.

- Extend the definition of