## CS275

**This is due on Monday, November 14.**

No language would be complete without the ability to create new procedures. Our new language will implement the lambda expression. A lambda expression should evaluate to a package containing the formal parameters, the body, and the environment that was current when the procedure was created (i.e. when the lambda expression was evaluated. This package is known as a **closure**. So start by making a datatype for closures that holds 3 parts: parameters, body, and environment.

We parse a lambda expression such as (lambda (x y) (+ x y) ) into a **lambda-exp** tree. This is a new kind of tree node with 2 parts: the parameter list (x y) and the tree that results from parsing the body. The parse function doesn't track the environment, so it can't build a full closure. If **exp** is the tree we get from parsing such a lambda expression, **(eval-exp exp env) **builds a closure with exp's parameter list and body combined with the environment **env**.

We are ready for MiniSchemeF. The syntax is extended once more, this time to include lambda expressions. Here is the grammar for MiniSchemeF.

EXP ::= number parse into lit-exp

| symbol parse into var-ref

| (if EXP EXP EXP) parse into if-exp

| (let (LET-BINDINGS) EXP) parse into let-exp

| (lambda (PARAMS) EXP) parse into lambda-exp

| ( EXP EXP*) parse into app-exp

LET-BINDINGS ::= LET-BINDING*

LET-BINDING ::= ( symbol EXP)

PARAMS ::= symbol*

So we parse a lambda expresion into a lambda-exp node that stores the parameter list and the parsed body. In eval-exp, we evaluate a lambda-exp node as a closure that has the lambda-exp's parameter list and parsed body and also the current enviornment.

In Section 3 of Lab 7 (Minischeme C) we defined **apply-proc** like this:

(define apply-proc (lambda (p arg-values)

(cond

[ (prim-proc? p) (apply-primitive-op (Operator p) arg-values)] ; Operator is my name for the prim-proc getter

(else (error 'apply-proc "Bad procedure: ~s" p)))))

We now extend this with a case for p being a closure. To evaluate the application of a closure to some argument values we start with the environment from the closure, extend that environment with bindings of the parameters to the arg-values, and call eval-exp on the closure's body with this extended environment. After implementing this you should be able to do the following:

> (read-eval-print)

MS> ((lambda (x) x) 1)

1

MS> ((lambda (x y) (* x y)) 2 4)

8

MS> (let ((sqr (lambda (x) (* x x)))) (sqr 64))

4096

MS> (let ((sqr (lambda (x) (* x x)))) (let ((cube (lambda (x) (* x (sqr x))))) (cube 3)))

27

Our next feature will be variable assignment, with set!. Unfortunately, our implementation of environments does not provide a way to change the value bound to a variable. We will modify our old implementation so that variable names are bound to a mutable datatype called a box, which is provided by Dr. Racket.

Take a moment to familiarize yourself with boxes in Scheme:

> (define abox (box 17))

> (box? abox)

#t

> (unbox abox)

17

> (set-box! abox 32)

> (unbox abox)

32

> ...

When variables are created we will bind them to boxes. When they are referenced we will unbox their bindings. We will take these tasks sequentially.

First, eval-exp currently evaluates a varref-exp, it gets the symbol from the expression and looks it up in the environment. When our variables all refer to boxes, eval-exp needs to do an extra step -- it gets the symbol from the expression, looks it up in the environment, and unboxes the result.

Secondly, whenever the environment is extended, the new bindings will be boxes that contain values. This occurs in two places. One is when we evaluate a let-expression in eval-exp, the other is when we apply a closure in apply-proc. For the latter our code used to be a recursive call to eval-exp on the body from the closure, using the environment **(extended-env params arg-values env)**. After we introduce boxes we will do this with a recursive call to eval-exp on the body using the environment **(extended-env params (map box arg-values) env)**

We handle let-expressions in the same way..

At this point your interpreter should be running exactly as it did for MiniSchemeF -- let expressions, lambda expressions and applications should all work correctly. Make sure this is the case before you procede. We will now take advantage of our boxed bindings to implement set!

MiniSchemeG will implement variable assignment in the form of set! expressions. Note that we will not be implementing set! as a primitive function, but as an expression -- in (set! x 5) we don't want to evaluate varible x to its previous value, as a call would, but rather to store value 5 in its box. . The grammar for MiniSchemeH will be:EXP ::= number parse into lit-exp

| symbol parse into var-ref

| (if EXP EXP EXP) parse into if-exp

| (let (LET-BINDINGS) EXP) parse into let-exp

| (lambda (PARAMS) EXP) parse into lambda-exp

| (set! symbol EXP) parse into assign-exp

| ( EXP EXP*) parse into app-exp

LET-BINDINGS ::= LET-BINDING*

LET-BINDING ::= ( symbol EXP)

PARAMS ::= symbol*

We need to extend eval-exp to handle assing-exp tree nodes. This is just a matter of putting all of the pieces together: we lookup the symbol from the expression (the variablel being assigned to) in the current environment; this should give us a box. We call set-box! on this box with the value we get from recursively calling eval-exp on the expression part of the assign-exp.

Here is what we can do when this is implemented:

> (read-eval-print)

MS> (set! + -)

#

MS> (+ 2 2)

0

MS> (set! + (lambda (x y) (- x (minus y))))

#

MS> (+ 2 2)

4

MS> (+ 2 5)

7

MS> exit

returning to Scheme proper

>

Now that we have introduced side effects, it seems a natural next step to implement sequencing. We add a begin expression to the grammar::

EXP ::= number parse into lit-exp

| symbol parse into var-ref

| (if EXP EXP EXP) parse into if-exp

| (let (LET-BINDINGS) EXP) parse into let-exp

| (lambda (PARAMS) EXP) parse into lambda-exp

| (set! symbol EXP) parse into assign-exp

| (begin EXP* ) parse into begin-exp

| ( EXP EXP*) parse into app-exp

LET-BINDINGS ::= LET-BINDING*

LET-BINDING ::= ( symbol EXP)

PARAMS ::= symbol*

Evaluating (begin e1 e2 ... en) results in the evaluation of e1, e2, .. en in that order. The returned result is the last expression, en.

A begin-exp holds a list of parsed expression. You will need to think about how eval-exp. You need to iterate through the list of expressions in such a way that

**(let ([x 1] [y 2]) (begin (set! x 23) (+ x y))) **returns 25; the whole point of** begin** is that the subexpressions might have side effects that alter the environment. Perhaps this will encourage you to be more appreciative of functional programming ....

It looks like we're about done; we've implemented just about everything except for global definitions (i.e. something like define). But let's take a closer look.

What happens if we try to define a recursive procedure in MiniSchemeG? Let's try the ever-familiar factorial function:

> (read-eval-print)

MS> (let

([fac (lambda (n)

(if (equals? n 0)

1

[ (* n (fac (- n 1)))))])

(fac 4))

MS>

This gets an error message saying there is no binding for fac. But we bound fac using let. Why is MiniScheme reporting that fac is unbound? The problem is in the recursive call to fac in (* n (fac (sub1 n))). When we evaluated the lambda expression to create the closure, we did so in an environment in which fac was not bound. Because procedures use static environments when they are executed, the recursive call failed. The same thing would happen in Scheme itself; this is why we have **letrec**.

Try this

MS> (let ([fac (lambda (x) (+ x 150))])

(let ([fac

(lambda (n)

(if (equals? n 0)

1

(* n (fac (- n 1)))))])

(fac 4)))

This time the program returns 612, which is 153*4; it is the first binding of function fac that is seen in the call to (fac (- n 1)

Recall what happens when a function is created. A closure is created that contains the environment at the time the function was created, along with the body of the function and the formal parameters. MiniScheme had no problems with this, and shouldn't have.

When a function is *called*, the formal parameters are bound to the actual parameters, and then all of the remaining free variables in the body are looked up in the environment that was present at the time of the creation of the function. This is where MiniScheme ran into problems. In the first example the fac in the line

(* n (fac (- n 1))))))

was not bound to anything at the time the function was created, and so we got an error. In the second example fac was bound to the earlier procedure produced by evaluating (lambda (x) (+ x 150)), which fortunately wasn't recursive. But neither case is what we want.

There is a clever way to get around this problem. Try running the following code:

MS> (let ([fac 0])

(let ([f (lambda (n)

(if (equals? n 0)

1

(* n (fac (- n 1)))))])

(begin

(set! fac f)

(fac 4))))

This works correctly. You can use this pattern for all recursive functions.

So then, it appears that recursive procedures are really "syntactic sugar". Here is the grammar for our final language, Mini-SchemeH:

EXP ::= number parse into lit-exp

| symbol parse into var-ref

| (if EXP EXP EXP) parse into if-exp

| (let (LET-BINDINGS) EXP) parse into let-exp

| (letrec (LET-BINDINGS) EXP) translate into equivalent let expression and parse that

| (lambda (PARAMS) EXP) parse into lambda-exp

| (set! symbol EXP) parse into assign-exp

| (begin EXP* ) parse into begin-exp

| ( EXP EXP*) parse into app-exp

LET-BINDINGS ::= LET-BINDING*

LET-BINDING ::= ( symbol EXP)

PARAMS ::= symbol*

The way we are handling letrecs is what is known as a *syntactic transformation*. When the parser sees a letrec expression it can either product an equivalent let expression LE and return (parse LE), or it can directly create the appropriate let-exp tree. The latter is what I do, but either approach works.

Implement MiniSchemeH, which implements letrec. You should only have to modify the parser, which means that InterpH is the same as InterpG. Use a help function (make-letrec ids vals body) to do the work so that you don't clutter your parser.

You will need some fresh variables to play the role of placeholders. The procedure (gensym) always returns a fresh, unused variable.

> (gensym)In the end the following examples should work.

g62

> (read-eval-print)

MS> (letrec ([fac (lambda (x) (if (equals? x 0) 1 (* x (fac (sub1 x)))))]) (fac 4))

24

MS> (letrec ([fac (lambda (x) (if (equals? x 0) 1 (* x (fac (sub1 x)))))]) (fac 10))

3628800

MS> (letrec

([even? (lambda (n)

(if (equals? 0 n)

True

(odd? (sub1 n))))]

[odd? (lambda (n)

(if (equals? 0 n)

False

(even? (sub1 n))))])

(even? 5))

False

MS> exit

returning to Scheme proper

You are done! Make the final versions of your modules be env.rkt, parse.rkt, and interp.rkt so the grader doesn't have to look through all of your code to figure out where you stopped.