# Implementing `define-type` & `cases` in Schlac

Another interesting way to implement lists follows the pattern matching approach, where both pairs and the null value are represented by a function that serves as a kind of a `match` dispatcher. This function takes in two inputs — if it is the representation of null then it will return the first input, and if it is a pair, then it will apply the second input on the two parts of the pair. This is implemented as follows (with type comments to make it clear):

;; null : List
(define null
(lambda (n p)
n))

;; cons : A List -> List
(define cons
(lambda (x y)
(lambda (n p)
(p x y))))

This might seem awkward, but it follows the intended use of pairs and null as a match-like construct. Here is an example, with the equivalent Racket code on the side:

;; Sums up a list of numbers
(define/rec (sum l)
(l                    ; (match l
0                    ;  ['() 0]
(lambda (x xs)      ;  [(cons x xs)
(+ x (sum xs)))))  ;    (+ x (sum xs))])

In fact, it’s easy to implement our selectors and predicate using this:

(define null? (lambda (l) (l #t (lambda (x xs) #f))))
(define car  (lambda (l) (l #f (lambda (x xs) x))))
(define cdr  (lambda (l) (l #f (lambda (x xs) xs))))
;; in the above `#f' is really any value, since it
;; should be an error alternatively:
(define car (lambda (l)
(l ((lambda (x) (x x)) (lambda (x) (x x))) ; "error"
(lambda (x y) x))))

The same approach can be used to define any kind of new data type in a way that looks like our own `define-type` definitions. For example, consider a much-simplified definition of the AE type we’ve seen early in the semester, and a matching `eval` definition as an example for using `cases`:

(define-type AE
[Num Number]
(: eval : AE -> Number)
(define (eval expr)
(cases expr
[(Num n)  n]
[(Add l r) (+ (eval l) (eval r))]))

We can follow the above approach now to write Schlac code that more than being equivalent, is also very similar in nature. Note that the type definition is replaced by two definitions for the two constructors:

(define Num (lambda (n)  (lambda (num add) (num n  ))))
(define/rec eval
(lambda (expr) ; `expr` is always a (lambda (num add) ...), and it
; expects a unary `num` argument and a binary `add`
(expr (lambda (n)  n)
(lambda (l r) (+ (eval l) (eval r))))))
(test (->nat (eval (Add (Num 1) (Num 2)))) => '3)

We can even take this further: the translations from `define-type` and `cases` are mechanical enough that we could implement them almost exactly via rewrites (there are a subtle change in that we’re list field names rather than types):

(rewrite (define-type -ignored-Type- [Variant arg ...] ...)
=> (define Variant
(lambda (arg ...)
(lambda (Variant ...) (Variant arg ...))))
...)
(rewrite (cases value [(-ignored-Variant- arg ...) result] ...)
=> (value (lambda (arg ...) result) ...))

And using that, an evluator is simple:

(define-type AE [Num n] [Add l r] [Sub l r] [Mul l r])
(define/rec eval
(lambda (expr)
(cases expr
[(Num n)  n]
[(Add l r) (+ (eval l) (eval r))]
[(Sub l r) (- (eval l) (eval r))]
[(Mul l r) (* (eval l) (eval r))])))
(test (->nat (eval (Mul (Add (Num 1) (Num 2))
(Sub (Num 4) (Num 2)))))
=> '6)

# Recursive Environments

PLAI §11.5

What we really need for recursion, is a special kind of an environment, one that can refer to itself. So instead of doing (note: `call`s removed for readability):

{with {fact {fun {n}
{if {zero? n} 1 {* n {fact {- n 1}}}}}}
{fact 5}}

which does not work for the usual reasons, we want to use some

{rec {fact {fun {n}
{if {zero? n} 1 {* n {fact {- n 1}}}}}}
{fact 5}}

that will do the necessary magic.

One way to achieve this is using the Y combinator as we have seen — a kind of a “constructor” for recursive functions. We can do that in a similar way to the `rewrite` rule that we have seen in Schlac — translate the above expression to:

{with {fact {make-rec
{fun {fact}
{fun {n}
{if {zero? n} 1 {* n {fact {- n 1}}}}}}}}
{fact 5}}

or even:

{with {fact {{fun {f} {{fun {x} {f {x x}}} {fun {x} {f {x x}}}}}
{fun {fact}
{fun {n}
{if {zero? n} 1 {* n {fact {- n 1}}}}}}}}
{fact 5}}

Now, we will see how it can be used in our code to implement a recursive environment.

If we look at what `with` does in

{with {fact {fun {n}
{if {zero? n} 1 {* n {call fact {- n 1}}}}}}
{call fact 5}}

then we can say that to evaluate this expression, we evaluate the body expression in an extended environment that contains `fact`, even if a bogus one that is good for `0` only — the new environment is created with something like this:

extend("fact", make-fact-closure(), env)

so we can take this whole thing as an operation over `env`

This gives us the first-level fact. But `fact` itself is still undefined in `env`, so it cannot call itself. We can try this:

but that still doesn’t work, and it will never work no matter how far we go:

What we really want is infinity: a place where add-fact works and the result is the same as what we’ve started with — we want to create a “magical” environment that makes this possible:

let magic-env = ???
such that:

which basically gives us the illusion of being at the infinity point. This magic-env thing is exactly the fixed-point of the `add-fact` operation. We can use:

and following the main property of the Y combinator, we know that:

magic-env = rec(add-fact)          ; def. of magic-env
= add-fact(magic-env)    ; def. of magic-env

What does all this mean? It means that if we have a fixed-point operator at the level of the implementation of our environments, then we can use it to implement a recursive binder. In our case, this means that a fixpoint in Racket can be used to implement a recursive language. But we have that — Racket does have recursive functions, so we should be able to use that to implement our recursive binder.

There are two ways that make it possible to write recursive functions in Racket. One is to define a function, and use its name to do a recursive call — using the Racket formal rules, we can see that we said that we mark that we now know that a variable is bound to a value. This is essentially a side-effect — we modify what we know, which corresponds to modifying the global environment. The second way is a new form: `letrec`. This form is similar to `let`, except that the scope that is established includes the named expressions — it is exactly what we want `rec` to do. A third way is using recursive local definitions, but that is equivalent to using `letrec`, more on this soon.

# Recursion: Racket’s `letrec`

So we want to add recursion to our language, practically. We already know that Racket makes it possible to write recursive functions, which is possible because of the way it implements its “global environment”: our evaluator can only extend an environment, while Racket modifies its global environment. This means that whenever a function is defined in the global environment, the resulting closure will have it as its environment “pointer”, but the global environment was not extended — it stays the same, and was just modified with one additional binding.

But Racket has another, a bit more organized way of using recursion: there is a special local-binding construct that is similar to `let`, but allows a function to refer to itself. It is called `letrec`:

(letrec ([fact (lambda (n)
(if (zero? n)
1
(* n (fact (- n 1)))))])
(fact 5))

Some people may remember that there was a third way for creating recursive functions: using local definition in function bodies. For example, we have seen things like:

(define (length list)
(define (helper list len)
(if (null? list)
len
(helper (rest list) (+ len 1))))
(helper list 0))

This looks like the same kind of environment magic that happens with a global `define` — but actually, Racket defines the meaning of internal definitions using `letrec` — so the above code is exactly the same as:

(define (length list)
(letrec ([helper (lambda (list len)
(if (null? list)
len
(helper (rest list) (+ len 1))))])
(helper list 0)))

The scoping rules for a `letrec` is that the scope of the bound name covers both the body and the named expression. Furthermore, multiple names can be bound to multiple expressions, and the scope of each name covers all named expression as well as the body. This makes it easy to define mutually recursive functions, such as:

(letrec ([even? (lambda (n) (if (zero? n) #t (odd?  (- n 1))))]
[odd?  (lambda (n) (if (zero? n) #f (even? (- n 1))))])
(even? 99))

But it is not a required functionality — it could be done with a single recursive binding that contains several functions:

(letrec ([even+odd
(list (lambda (n)
(if (zero? n) #t ((second even+odd) (- n 1))))
(lambda (n)
(if (zero? n) #f ((first  even+odd) (- n 1)))))])
((first even+odd) 99))

This is basically the same problem we face if we want to use the Y combinator for mutually recursive bindings. The above solution is inconvenient, but it can be improved using more `let`s to have easier name access. For example:

(letrec ([even+odd
(list (lambda (n)
(let ([even? (first  even+odd)]
[odd?  (second even+odd)])
(if (zero? n) #t (odd? (- n 1)))))
(lambda (n)
(let ([even? (first  even+odd)]
[odd?  (second even+odd)])
(if (zero? n) #f (even? (- n 1))))))])
(let ([even? (first  even+odd)]
[odd?  (second even+odd)])
(even? 99)))

# Implementing Recursion using `letrec`

We will see how to add a similar construct to our language — for simplicity, we will add a `rec` form that handles a single binding:

{rec {fact {fun {n}
{if {= 0 n}
1
{* n {fact {- n 1}}}}}}
{fact 5}}

Using this, things can get a little tricky. What should we get if we do:

{rec {x x} x}

? Currently, it seems like there is no point in using any expression except for a function expression in a `rec` expression, so we will handle only these cases.

(BTW, under what circumstances would non-function values be useful in a letrec?)

One way to achieve this is to use the same trick that we have recently seen: instead of re-implementing language features, we can use existing features in our own language, which hopefully has the right functionality in a form that can be re-used to in our evaluator.

Previously, we have seen a way to implement environments using Racket closures:

;; Define a type for functional environments
(define-type ENV = Symbol -> VAL)

(: EmptyEnv : -> ENV)
(define (EmptyEnv)
(lambda (id) (error 'lookup "no binding for ~s" id)))

(: lookup : Symbol ENV -> VAL)
(define (lookup name env)
(env name))

(: Extend : Symbol VAL ENV -> ENV)
(define (Extend id val rest-env)
(lambda (name)
(if (eq? name id)
val
(rest-env name))))

We can use this implementation, and create circular environments using Racket’s `letrec`. The code for handling a `with` expressions is:

[(With bound-id named-expr bound-body)
(eval bound-body
(Extend bound-id (eval named-expr env) env))]

It looks like we should be able to handle `rec` in a similar way (the AST constructor name is `WRec` (“with-rec”) so it doesn’t collide with TR’s `Rec` constructor for recursive types):

[(WRec bound-id named-expr bound-body)
(eval bound-body
(Extend bound-id (eval named-expr env) env))]

but this won’t work because the named expression is evaluated prematurely, in the previous environment. Instead, we will move everything that needs to be done, including evaluation, to a separate `extend-rec` function:

[(WRec bound-id named-expr bound-body)
(eval bound-body
(extend-rec bound-id named-expr env))]

Now, the `extend-rec` function needs to provide the new, “magically circular” environment. Following what we know about the arguments to `extend-rec`, and the fact that it returns a new environment (= a lookup function), we can sketch a rough definition:

(: extend-rec : Symbol FLANG ENV -> ENV) ; FLANG, not VAL!
;; extend an environment with a new binding that is the result of
;; evaluating an expression in the same environment as the extended
;; result
(define (extend-rec id expr rest-env)
(lambda (name)
(if (eq? name id)
... something that uses expr to get a value ...
(rest-env name))))

What should the missing expression be? It can simply evaluate the object given itself:

(define (extend-rec id expr rest-env)
(lambda (name)
(if (eq? name id)
(eval expr ...this environment...)
(rest-env name))))

But how do we get this environment, before it is defined? Well, the environment is itself a Racket function, so we can use Racket’s `letrec` to make the function refer to itself recursively:

(define (extend-rec id expr rest-env)
(letrec ([rec-env (lambda (name)
(if (eq? name id)
(eval expr rec-env)
(rest-env name)))])
rec-env))

It’s a little more convenient to use an internal definition, and add a type for clarity:

(define (extend-rec id expr rest-env)
(: rec-env : Symbol -> VAL)
(define (rec-env name)
(if (eq? name id)
(eval expr rec-env)
(rest-env name)))
rec-env)

This works, but there are several problems:

1. First, we no longer do a simple lookup in the new environment. Instead, we evaluate the expression on every such lookup. This seems like a technical point, because we do not have side-effects in our language (also because we said that we want to handle only function expressions). Still, it wastes space since each evaluation will allocate a new closure.

2. Second, a related problem — what happens if we try to run this:

{rec {x x} x}

? Well, we do that stuff to extend the current environment, then evaluate the body in the new environment, this body is a single variable reference:

(eval (Id 'x) the-new-env)

so we look up the value:

(lookup 'x the-new-env)

which is:

(the-new-env 'x)

which goes into the function which implements this environment, there we see that `name` is the same as `name1`, so we return:

(eval expr rec-env)

but the `expr` here is the original named-expression which is itself `(Id 'x)`, and we’re in an infinite loop.

We can try to get over these problems using another binding. Racket allows several bindings in a single `letrec` expression or multiple internal function definitions, so we change `extend-rec` to use the newly-created environment:

(define (extend-rec id expr rest-env)
(: rec-env : Symbol -> VAL)
(define (rec-env name)
(if (eq? name id)
val
(rest-env name)))
(: val : VAL)
(define val (eval expr rec-env))
rec-env)

This runs into an interesting type error, which complains about possibly getting some `Undefined` value. It does work if we switch to the untyped language for now (using `#lang pl untyped`) — and it seems to run fine too. But it raises more questions, beginning with: what is the meaning of:

(letrec ([x ...]
[y ...x...])
...)

or equivalently, an internal block of

(define x ...)
(define y ...x...)

? Well, DrRacket seems to do the “right thing” in this case, but what about:

(letrec ([y ...x...]
[x ...])
...)

? As a hint, see what happens when we now try to evaluate the problematic

{rec {x x} x}

expression, and compare that with the result that you’d get from Racket. This also clarifies the type error that we received.

It should be clear now why we want to restrict usage to just binding recursive functions. There are no problems with such definitions because when we evaluate a `fun` expression, there is no evaluation of the body, which is the only place where there are potential references to the same function that is defined — a function’s body is delayed, and executed only when the function is applied later.

But the biggest question that is still open: we just implemented a circular environment using Racket’s own circular environment implementation, and that does not explain how they are actually implemented. The cycle of pointers that we’ve implemented depends on the cycle of pointers that Racket uses, and that is a black box we want to open up.

For reference, the complete code is below.

#lang pl

#|
The grammar:
<FLANG> ::= <num>
| { + <FLANG> <FLANG> }
| { - <FLANG> <FLANG> }
| { * <FLANG> <FLANG> }
| { / <FLANG> <FLANG> }
| { with { <id> <FLANG> } <FLANG> }
| { rec { <id> <FLANG> } <FLANG> }
| <id>
| { fun { <id> } <FLANG> }
| { call <FLANG> <FLANG> }

Evaluation rules:
eval(N,env)                = N
eval({+ E1 E2},env)        = eval(E1,env) + eval(E2,env)
eval({- E1 E2},env)        = eval(E1,env) - eval(E2,env)
eval({* E1 E2},env)        = eval(E1,env) * eval(E2,env)
eval({/ E1 E2},env)        = eval(E1,env) / eval(E2,env)
eval(x,env)                = lookup(x,env)
eval({with {x E1} E2},env) = eval(E2,extend(x,eval(E1,env),env))
eval({rec {x E1} E2},env)  = ???
eval({fun {x} E},env)      = <{fun {x} E}, env>
eval({call E1 E2},env1)
= eval(Ef,extend(x,eval(E2,env1),env2))
if eval(E1,env1) = <{fun {x} Ef}, env2>
= error!          otherwise
|#

(define-type FLANG
[Num  Number]
[Sub  FLANG FLANG]
[Mul  FLANG FLANG]
[Div  FLANG FLANG]
[Id  Symbol]
[With Symbol FLANG FLANG]
[WRec Symbol FLANG FLANG]
[Fun  Symbol FLANG]
[Call FLANG FLANG])

(: parse-sexpr : Sexpr -> FLANG)
;; parses s-expressions into FLANGs
(define (parse-sexpr sexpr)
(match sexpr
[(number: n)    (Num n)]
[(symbol: name) (Id name)]
[(cons 'with more)
(match sexpr
[(list 'with (list (symbol: name) named) body)
(With name (parse-sexpr named) (parse-sexpr body))]
[else (error 'parse-sexpr "bad `with' syntax in ~s" sexpr)])]
[(cons 'rec more)
(match sexpr
[(list 'rec (list (symbol: name) named) body)
(WRec name (parse-sexpr named) (parse-sexpr body))]
[else (error 'parse-sexpr "bad `rec' syntax in ~s" sexpr)])]
[(cons 'fun more)
(match sexpr
[(list 'fun (list (symbol: name)) body)
(Fun name (parse-sexpr body))]
[else (error 'parse-sexpr "bad `fun' syntax in ~s" sexpr)])]
[(list '+ lhs rhs) (Add (parse-sexpr lhs) (parse-sexpr rhs))]
[(list '- lhs rhs) (Sub (parse-sexpr lhs) (parse-sexpr rhs))]
[(list '* lhs rhs) (Mul (parse-sexpr lhs) (parse-sexpr rhs))]
[(list '/ lhs rhs) (Div (parse-sexpr lhs) (parse-sexpr rhs))]
[(list 'call fun arg)
(Call (parse-sexpr fun) (parse-sexpr arg))]
[else (error 'parse-sexpr "bad syntax in ~s" sexpr)]))

(: parse : String -> FLANG)
;; parses a string containing a FLANG expression to a FLANG AST
(define (parse str)
(parse-sexpr (string->sexpr str)))

;; Types for environments, values, and a lookup function

(define-type VAL
[NumV Number]
[FunV Symbol FLANG ENV])

;; Define a type for functional environments
(define-type ENV = Symbol -> VAL)

(: EmptyEnv : -> ENV)
(define (EmptyEnv)
(lambda (id) (error 'lookup "no binding for ~s" id)))

(: lookup : Symbol ENV -> VAL)
;; lookup a symbol in an environment, return its value or throw an
;; error if it isn't bound
(define (lookup name env)
(env name))

(: Extend : Symbol VAL ENV -> ENV)
;; extend a given environment cache with a new binding
(define (Extend id val rest-env)
(lambda (name)
(if (eq? name id)
val
(rest-env name))))

(: extend-rec : Symbol FLANG ENV -> ENV)
;; extend an environment with a new binding that is the result of
;; evaluating an expression in the same environment as the extended
;; result
(define (extend-rec id expr rest-env)
(: rec-env : Symbol -> VAL)
(define (rec-env name)
(if (eq? name id)
val
(rest-env name)))
(: val : VAL)
(define val (eval expr rec-env))
rec-env)

(: NumV->number : VAL -> Number)
;; convert a FLANG runtime numeric value to a Racket one
(define (NumV->number val)
(cases val
[(NumV n) n]
[else (error 'arith-op "expected a number, got: ~s" val)]))

(: arith-op : (Number Number -> Number) VAL VAL -> VAL)
;; gets a Racket numeric binary operator, and uses it within a NumV
;; wrapper
(define (arith-op op val1 val2)
(NumV (op (NumV->number val1) (NumV->number val2))))

(: eval : FLANG ENV -> VAL)
;; evaluates FLANG expressions by reducing them to values
(define (eval expr env)
(cases expr
[(Num n) (NumV n)]
[(Add l r) (arith-op + (eval l env) (eval r env))]
[(Sub l r) (arith-op - (eval l env) (eval r env))]
[(Mul l r) (arith-op * (eval l env) (eval r env))]
[(Div l r) (arith-op / (eval l env) (eval r env))]
[(With bound-id named-expr bound-body)
(eval bound-body
(Extend bound-id (eval named-expr env) env))]
[(WRec bound-id named-expr bound-body)
(eval bound-body
(extend-rec bound-id named-expr env))]
[(Id name) (lookup name env)]
[(Fun bound-id bound-body)
(FunV bound-id bound-body env)]
[(Call fun-expr arg-expr)
(let ([fval (eval fun-expr env)])
(cases fval
[(FunV bound-id bound-body f-env)
(eval bound-body
(Extend bound-id (eval arg-expr env) f-env))]
[else (error 'eval "`call' expects a function, got: ~s"
fval)]))]))

(: run : String -> Number)
;; evaluate a FLANG program contained in a string
(define (run str)
(let ([result (eval (parse str) (EmptyEnv))])
(cases result
[(NumV n) n]
[else (error 'run "evaluation returned a non-number: ~s"
result)])))

;; tests
(test (run "{call {fun {x} {+ x 1}} 4}")
=> 5)
(test (run "{with {add3 {fun {x} {+ x 3}}}
=> 4)
(test (run "{with {add3 {fun {x} {+ x 3}}}
{with {add1 {fun {x} {+ x 1}}}
{with {x 3}
=> 7)
(test (run "{with {identity {fun {x} x}}
{with {foo {fun {x} {+ x 1}}}
{call {call identity foo} 123}}}")
=> 124)
(test (run "{with {x 3}
{with {f {fun {y} {+ x y}}}
{with {x 5}
{call f 4}}}}")
=> 7)
(test (run "{call {with {x 3}
{fun {y} {+ x y}}}
4}")
=> 7)
(test (run "{with {f {with {x 3} {fun {y} {+ x y}}}}
{with {x 100}
{call f 4}}}")
=> 7)
(test (run "{call {call {fun {x} {call x 1}}
{fun {x} {fun {y} {+ x y}}}}
123}")
=> 124)

# Implementing `rec` Using Cyclic Structures

PLAI §10

Looking at the arrows in the environment diagrams, what we’re really looking for is a closure that has an environment pointer which is the same environment in which it was defined. This will make it possible for `fact` to be bound to a closure that can refer to itself since its environment is the same one in which it is defined. However, so far we have no tools that makes it possible to do this.

What we need is to create a “cycle of pointers”, and so far we do not have a way of achieving that: when we create a closure, we begin with an environment which is saved in the slot’s environment slot, but we want that closure to be the value of a binding in that same environment.

# Boxes and Mutation

To actually implement a circular structure, we will now use side-effects, using a new kind of Racket value which supports mutation: a box. A box value is built with the `box` constructor:

(define my-thing (box 7))

the value is retrieved with the `unbox’ function,

(* 6 (unbox my-thing))

and finally, the value can be changed with the `set-box!` function.

(set-box! my-thing 17)
(* 6 (unbox my-thing))

An important thing to note is that `set-box!` is much like `display` etc, it returns a value that is not printed in the Racket REPL, because there is no point in using the result of a `set-box!`, it is called for the side-effect it generates. (Languages like C blur this distinction between returning a value and a side-effect with its assignment statement.)

As a side note, we now have side effects of two kinds: mutation of state, and I/O (at least the O part). (Actually, there is also infinite looping that can be viewed as another form of a side effect.) This means that we’re now in a completely different world, and lots of new things can make sense now. A few things that you should know about:

• We never used more than one expression in a function body because there was no point in it, but now there is. To evaluate a sequence of Racket expressions, you wrap them in a `begin` expression.

• In most places you don’t actually need to use `begin` — these are places that are said to have an implicit `begin`: the body of a function (or any lambda expression), the body of a `let` (and `let`-relatives), the consequence positions in `cond`, `match`, and `cases` clauses and more. One of the common places where a `begin` is used is in an `if` expression (and some people prefer using `cond` instead when there is more than a single expression).

• `cond` without an `else` in the end can make sense, if all you’re using it it for is side-effects.

• `if` could get a single expression which is executed when the condition is true (and an unspecified value is used otherwise), but our language (as well as the default Racket language) always forbids this — there are convenient special forms for a one-sided `if`s: `when` & `unless`, and they can have any number of expressions (they have an implicit `begin`). They have an advantage of saying “this code does some side-effects here” more explicit.

• There is a function called `for-each` which is just like `map`, except that it doesn’t collect the list of results, it is used only for performing side effects.

• Aliasing and the concept of “object equality”: `equal?` vs `eq?`. For example:

(: foo : (Boxof ...) (Boxof ...) -> ...)
(define (foo a b)
(set-box! a 1)) ;*** this might change b, can check `eq?`

When any one of these things is used (in Racket or other languages), you can tell that side-effects are involved, because there is no point in any of them otherwise. In addition, any name that ends with a `!` (“bang”) is used to mark a function that changes state (usually a function that only changes state).

So how do we create a cycle? Simple, boxes can have any value, and they can be put in other values like lists, so we can do this:

#lang pl untyped
(define foo (list 1 (box 3)))
(set-box! (second foo) foo)

and we get a circular value. (Note how it is printed.) And with types:

#lang pl
(: foo : (List Number (Boxof Any)))
(define foo (list 1 (box 3)))
(set-box! (second foo) foo)