PL: Lecture #16  Tuesday, November 1st
(text)

Boxof’s Lack of Subtyping

The lack of any subtype relations between (Boxof T) and (Boxof S) regardless of S and T can roughly be explained as follows.

First, a box is a container that you can pull a value out of — which makes it similar to lists. In the case of lists, we have:

if:          S  subtype-of          T
then: (Listof S)  subtype-of  (Listof T)

This is true for all such containers that you can pull a value out of: if you expect to pull a T but you’re given a container of a subtype S, then things are still fine (you’ll get an S which is also a T). Such “containers” include functions that produce a value — for example:

if:        S  subtype-of      T
then:  Q -> S  subtype-of  Q -> T

However, functions also have the other side, where things are different — instead of a side of some produced value, it’s the side of the consumed value. We get the opposite rule there:

if:    T      subtype-of  S
then:  S -> Q  subtype-of  T -> Q

To see why this is right, use Number and Integer for S and T:

if:    Integer      subtype-of  Number
then:  Number -> Q  subtype-of  Integer -> Q

so — if you expect a function that takes an integer, a valid subtype value that I can give you is a function that takes a number. In other words, every function that takes a number is also a function that takes an integer, but not the other way.

To summarize all of this, when you make the output type of a function “smaller” (more constrained), the resulting type is smaller (a subset), but on the input side things are flipped — a bigger input type means a more constrained function.

The technical names for these properties are: a “covariant” type is one that preserves the subtype relationship, and a “contravairant” type is one that reverses it. (Which is similar to how these terms are used in math.)

(Side note: this is related to the fact that in logic, P => Q is roughly equivalent to not(P) or Q — the left side, P, is inside negation. It also explains why in ((S -> T) -> Q) the S obeys the first rule, as if it was on the right side — because it’s negated twice.)

Now, a (Boxof T) is a producer of T when you pull a value out of the box, but it’s also a consumer of T when you put such a value in it. This means that — using the above analogy — the T is on both sides of the arrow. This means that

if:    S subtype-of T  *and*  T subtype-of S
then:  (Boxof S) subtype-of (Boxof T)

which is actually:

if:          S  is-the-same-type-as        T
then:  (Boxof S)  is-the-same-type-as  (Boxof T)

A different way to look at this conclusion is to consider the function type of (A -> A): when is it a subtype of some other (B -> B)? Only when A is a subtype of B and B is a subtype of A, which means that this happens only when A and B are the same type.

The term for this is “nonvariant” (or “invariant”): (A -> A) is unrelated to (B -> B) regardless of how A and B are related. The only exception is, of course, when they are the same type. The Wikipedia entry about these puts the terms together nicely in the face of mutation:

Read-only data types (sources) can be covariant; write-only data types (sinks) can be contravariant. Mutable data types which act as both sources and sinks should be invariant.

The following piece of code makes the analogy to function types more formally. Boxes behave as if their contents is on both sides of a function arrow — on the right because they’re readable, and on the left because they’re writable, which the conclusion that a (Boxof A) type is a subtype of itself and no other (Boxof B).

#lang pl

;; a type for a "read-only" box
(define-type (Boxof/R A) = (-> A))
;; Boxof/R constructor
(: box/r : (All (A) A -> (Boxof/R A)))
(define (box/r x) (lambda () x))
;; we can see that (Boxof/R T1) is a subtype of (Boxof/R T2)
;; if T1 is a subtype of T2 (this is not surprising, since
;; these boxes are similar to any other container, like lists):
(: foo1 : Integer -> (Boxof/R Integer))
(define (foo1 b) (box/r b))
(: bar1 : (Boxof/R Number) -> Number)
(define (bar1 b) (b))
(test (bar1 (foo1 123)) => 123)

;; a type for a "write-only" box
(define-type (Boxof/W A) = (A -> Void))
;; Boxof/W constructor
(: box/w : (All (A) A -> (Boxof/W A)))
(define (box/w x) (lambda (new) (set! x new)))
;; in contrast to the above, (Boxof/W T1) is a subtype of
;; (Boxof/W T2) if T2 is a subtype of T1, *not* the other way
;; (and note how this is related to A being on the *left* side
;; of the arrow in the `Boxof/W' type):
(: foo2 : Number -> (Boxof/W Number))
(define (foo2 b) (box/w b))
(: bar2 : (Boxof/W Integer) Integer -> Void)
(define (bar2 b new) (b new))
(test (bar2 (foo2 123) 456))

;; combining the above two into a type for a "read/write" box
(define-type (Boxof/RW A) = (A -> A))
;; Boxof/RW constructor
(: box/rw : (All (A) A -> (Boxof/RW A)))
(define (box/rw x) (lambda (new) (let ([old x]) (set! x new) old)))
;; this combines the above two: `A' appears on both sides of the
;; arrow, so (Boxof/RW T1) is a subtype of (Boxof/RW T2) if T1
;; is a subtype of T2 (because there's an A on the right) *and*
;; if T2 is a subtype of T1 (because there's another A on the
;; left) -- and that can happen only when T1 and T2 are the same
;; type. So this is a type error:
;;  (: foo3 : Integer -> (Boxof/RW Integer))
;;  (define (foo3 b) (box/rw b))
;;  (: bar3 : (Boxof/RW Number) Number -> Number)
;;  (define (bar3 b new) (b new))
;;  (test (bar3 (foo3 123) 456) => 123)
;;  ** Expected (Number -> Number), but got (Integer -> Integer)
;; And this a type error too:
;;  (: foo3 : Number -> (Boxof/RW Number))
;;  (define (foo3 b) (box/rw b))
;;  (: bar3 : (Boxof/RW Integer) Integer -> Integer)
;;  (define (bar3 b new) (b new))
;;  (test (bar3 (foo3 123) 456) => 123)
;;  ** Expected (Integer -> Integer), but got (Number -> Number)
;; The two types must be the same for this to work:
(: foo3 : Integer -> (Boxof/RW Integer))
(define (foo3 b) (box/rw b))
(: bar3 : (Boxof/RW Integer) Integer -> Integer)
(define (bar3 b new) (b new))
(test (bar3 (foo3 123) 456) => 123)

Implementing a Circular Environment

We now use this to implement rec in the following way:

  1. Change environments so that instead of values they hold boxes of values: (Boxof VAL) instead of VAL, and whenever lookup is used, the resulting boxed value is unboxed,

  2. In the WRec case, create the new environment with some temporary binding for the identifier — any value will do since it should not be used (when named expressions are always fun expressions),

  3. Evaluate the expression in the new environment,

  4. Change the binding of the identifier (the box) to the result of this evaluation.

The resulting definition is:

(: 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)
  (let ([new-cell (box (NumV 42))])
    (let ([new-env (Extend id new-cell rest-env)])
      (let ([value (eval expr new-env)])
        (set-box! new-cell value)
        new-env))))

Racket has another let relative for such cases of multiple-nested lets — let*. This form is a derived form — it is defined as a shorthand for using nested lets. The above is therefore exactly the same as this code:

(: 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)
  (let* ([new-cell (box (NumV 42))]
        [new-env  (Extend id new-cell rest-env)]
        [value    (eval expr new-env)])
    (set-box! new-cell value)
    new-env))

This let* form can be read almost as a C/Java-ish kind of code:

fun extend_rec(id, expr, rest_env) {
  new_cell  = new NumV(42);
  new_env  = Extend(id, new_cell, rest_env);
  value    = eval(expr, new_env);
  *new_cell = value;
  return new_env;
}

The code can be simpler if we fold the evaluation into the set-box! (since value is used just there), and if use lookup to do the mutation — since this way there is no need to hold onto the box. This is a bit more expensive, but since the binding is guaranteed to be the first one in the environment, the addition is just one quick step. The only binding that we need is the one for the new environment, which we can do as an internal definition, leaving us with:

(: extend-rec : Symbol FLANG ENV -> ENV)
(define (extend-rec id expr rest-env)
  (define new-env (Extend id (box (NumV 42)) rest-env))
  (set-box! (lookup id new-env) (eval expr new-env))
  new-env)

A complete rehacked version of FLANG with a rec binding follows. We can’t test rec easily since we have no conditionals, but you can at least verify that

(run "{rec {f {fun {x} {call f x}}} {call f 0}}")

is an infinite loop.

#lang pl

(define-type FLANG
  [Num  Number]
  [Add  FLANG FLANG]
  [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 (or 'with 'rec) more)
    (match sexpr
      [(list 'with (list (symbol: name) named) body)
        (With name (parse-sexpr named) (parse-sexpr body))]
      [(list 'rec (list (symbol: name) named) body)
        (WRec name (parse-sexpr named) (parse-sexpr body))]
      [(cons x more)
        (error 'parse-sexpr "bad `~s' syntax in ~s" x 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 ENV
  [EmptyEnv]
  [Extend Symbol (Boxof VAL) ENV])

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

(: lookup : Symbol ENV -> (Boxof VAL))
;; lookup a symbol in an environment, return its value or throw an
;; error if it isn't bound
(define (lookup name env)
  (cases env
    [(EmptyEnv) (error 'lookup "no binding for ~s" name)]
    [(Extend id boxed-val rest-env)
    (if (eq? id name) boxed-val (lookup name rest-env))]))

(: 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)
  (define new-env (Extend id (box (NumV 42)) rest-env))
  (set-box! (lookup id new-env) (eval expr new-env))
  new-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 (box (eval named-expr env)) env))]
    [(WRec bound-id named-expr bound-body)
    (eval bound-body
          (extend-rec bound-id named-expr env))]
    [(Id name) (unbox (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 (box (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}}}
              {call add3 1}}")
      => 4)
(test (run "{with {add3 {fun {x} {+ x 3}}}
              {with {add1 {fun {x} {+ x 1}}}
                {with {x 3}
                  {call add1 {call add3 x}}}}}")
      => 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)

Variable Mutation

PLAI §12 and PLAI §13 (different: adds boxes to the language)

PLAI §14 (that’s what we do)

The code that we now have implements recursion by changing bindings, and to make that possible we made environments hold boxes for all bindings, therefore bindings are all mutable now. We can use this to add more functionality to our evaluator, by allowing changing any variable — we can add a set! form:

{set! <id> <FLANG>}

to the evaluator that will modify the value of a variable. To implement this functionality, all we need to do is to use lookup to retrieve some box, then evaluate the expression and put the result in that box. The actual implementation is left as a homework exercise.

One thing that should be considered here is — all of the expressions in our language evaluate to some value, the question is what should be the value of a set! expression? There are three obvious choices:

  1. return some bogus value,

  2. return the value that was assigned,

  3. return the value that was previously in the box.

Each one of these has its own advantage — for example, C uses the second option to chain assignments (eg, x = y = 0) and to allow side effects where an expression is expected (eg, while (x = x-1) ...).

The third one is useful in cases where you might use the old value that is overwritten — for example, if C had this behavior, you could pop a value from a linked list using something like:

first(stack = rest(stack));

because the argument to first will be the old value of stack, before it changed to be its rest. You could also swap two variables in a single expression: x = y = x.

(Note that the expression x = x + 1 has the meaning of C’s ++x when option (2) is used, and x++ when option (3) is used.)

Racket chooses the first option, and we will do the same in our language. The advantage here is that you get no discounts, therefore you must be explicit about what values you want to return in situations where there is no obvious choice. This leads to more robust programs since you do not get other programmers that will rely on a feature of your code that you did not plan on.

In any case, the modification that introduces mutation is small, but it has a tremendous effect on our language: it was true for Racket, and it is true for FLANG. We have seen how mutation affects the language subset that we use, and in the extension of our FLANG the effect is even stronger: since any variable can change (no need for explicit boxes). In other words, a binding is not always the same — in can change as a result of a set! expression. Of course, we could extend our language with boxes (using Racket boxes to implement FLANG boxes), but that will be a little more verbose.

Note that Racket does have a set! form, and in addition, fields in structs can be made modifiable. However, we do not use any of these. At least not for now.