Lecture #10, Tuesday, February 4th
==================================
 Fixing an Overlooked Bug
 Lexical Scope using Racket Closures
 More Closures (on both levels)
 Types of Evaluators
 Feature Embedding

# Fixing an Overlooked Bug
Incidentally, this version fixes a bug we had previously in the
substitution version of FLANG:
(run "{with {f {fun {y} {+ x y}}}
{with {x 7}
{call f 1}}}")
This bug was due to our naive `subst`, which doesn't avoid capturing
renames. But note that since that version of the evaluator makes its
way from the outside in, there is no difference in semantics for *valid*
programs  ones that don't have free identifiers.
(Reminder: This was *not* a dynamically scoped language, just a bug that
happened when `x` wasn't substituted away before `f` was replaced with
something that refers to `x`.)

# Lexical Scope using Racket Closures
> [PLAI ยง11] (without the last part about recursion)
An alternative representation for an environment.
We've already seen how firstclass functions can be used to implement
"objects" that contain some information. We can use the same idea to
represent an environment. The basic intuition is  an environment is
a *mapping* (a function) between an identifier and some value. For
example, we can represent the environment that maps `'a` to `1` and `'b`
to `2` (using just numbers for simplicity) using this function:
(: mymap : Symbol > Number)
(define (mymap id)
(cond [(eq? 'a id) 1]
[(eq? 'b id) 2]
[else (error ...)]))
An empty mapping that is implemented in this way has the same type:
(: emptymapping : Symbol > Number)
(define (emptymapping id)
(error ...))
We can use this idea to implement our environments: we only need to
define three things  `EmptyEnv`, `Extend`, and `lookup`. If we
manage to keep the contract to these functions intact, we will be able
to simply plug it into the same evaluator code with no other changes.
It will also be more convenient to define `ENV` as the appropriate
function type for use in the `VAL` type definition instead of using the
actual type:
;; Define a type for functional environments
(definetype ENV = Symbol > VAL)
Now we get to `EmptyEnv`  this is expected to be a function that
expects no arguments and creates an empty environment, one that behaves
like the `emptymapping` function defined above. We could define it
like this (changing the `emptymapping` type to return a `VAL`):
(define (EmptyEnv) emptymapping)
but we can skip the need for an extra definition and simply return an
empty mapping function:
(: EmptyEnv : > ENV)
(define (EmptyEnv)
(lambda (id) (error ...)))
(The unRackety name is to avoid replacing previous code that used the
`EmptyEnv` name for the constructor that was created by the type
definition.)
The next thing we tackle is `lookup`. The previous definition that was
used is:
(: lookup : Symbol ENV > VAL)
(define (lookup name env)
(cases env
[(EmptyEnv) (error 'lookup "no binding for ~s" name)]
[(Extend id val restenv)
(if (eq? id name) val (lookup name restenv))]))
How should it be modified now? Easy  an environment is a mapping: a
Racket function that will do the searching job itself. We don't need to
modify the contract since we're still using `ENV`, except a different
implementation for it. The new definition is:
(: lookup : Symbol ENV > VAL)
(define (lookup name env)
(env name))
Note that `lookup` does almost nothing  it simply delegates the real
work to the `env` argument. This is a good hint for the error message
that empty mappings should throw 
(: EmptyEnv : > ENV)
(define (EmptyEnv)
(lambda (id) (error 'lookup "no binding for ~s" id)))
Finally, `Extend`  this was previously created by the variant case of
the ENV type definition:
[Extend Symbol VAL ENV]
keeping the same type that is implied by this variant means that the new
`Extend` should look like this:
(: Extend : Symbol VAL ENV > ENV)
(define (Extend id val restenv)
...)
The question is  how do we extend a given environment? Well, first,
we know that the result should be mapping  a `symbol > VAL` function
that expects an identifier to look for:
(: Extend : Symbol VAL ENV > ENV)
(define (Extend id val restenv)
(lambda (name)
...))
Next, we know that in the generated mapping, if we look for `id` then
the result should be `val`:
(: Extend : Symbol VAL ENV > ENV)
(define (Extend id val restenv)
(lambda (name)
(if (eq? name id)
val
...)))
If the `name` that we're looking for is not the same as `id`, then we
need to search through the previous environment, eg: (lookup name rest).
But we know what `lookup` does  it simply delegates back to the
mapping function (which is our `rest` argument), so we can take a direct
route:
(: Extend : Symbol VAL ENV > ENV)
(define (Extend id val restenv)
(lambda (name)
(if (eq? name id)
val
(restenv name))))
(Note that the last line is simply `(lookup name restenv)`, but we know
that we have a functional implementation.)
To see how all this works, try out extending an empty environment a few
times and examine the result. For example, the environment that we
began with:
(define (mymap id)
(cond [(eq? 'a id) 1]
[(eq? 'b id) 2]
[else (error ...)]))
behaves in the same way (if the type of values is numbers) as
(Extend 'a 1 (Extend 'b 2 (EmptyEnv)))
The new code is now the same, except for the environment code:
#lang pl
#
The grammar:
::=
 { + }
 {  }
 { * }
 { / }
 { with { } }

 { fun { } }
 { call }
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({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
#
(definetype FLANG
[Num Number]
[Add FLANG FLANG]
[Sub FLANG FLANG]
[Mul FLANG FLANG]
[Div FLANG FLANG]
[Id Symbol]
[With Symbol FLANG FLANG]
[Fun Symbol FLANG]
[Call FLANG FLANG])
(: parsesexpr : Sexpr > FLANG)
;; parses sexpressions into FLANGs
(define (parsesexpr 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 (parsesexpr named) (parsesexpr body))]
[else (error 'parsesexpr "bad `with' syntax in ~s" sexpr)])]
[(cons 'fun more)
(match sexpr
[(list 'fun (list (symbol: name)) body)
(Fun name (parsesexpr body))]
[else (error 'parsesexpr "bad `fun' syntax in ~s" sexpr)])]
[(list '+ lhs rhs) (Add (parsesexpr lhs) (parsesexpr rhs))]
[(list ' lhs rhs) (Sub (parsesexpr lhs) (parsesexpr rhs))]
[(list '* lhs rhs) (Mul (parsesexpr lhs) (parsesexpr rhs))]
[(list '/ lhs rhs) (Div (parsesexpr lhs) (parsesexpr rhs))]
[(list 'call fun arg)
(Call (parsesexpr fun) (parsesexpr arg))]
[else (error 'parsesexpr "bad syntax in ~s" sexpr)]))
(: parse : String > FLANG)
;; parses a string containing a FLANG expression to a FLANG AST
(define (parse str)
(parsesexpr (string>sexpr str)))
;; Types for environments, values, and a lookup function
(definetype VAL
[NumV Number]
[FunV Symbol FLANG ENV])
;; Define a type for functional environments
(definetype ENV = Symbol > VAL)
(: EmptyEnv : > ENV)
(define (EmptyEnv)
(lambda (id) (error 'lookup "no binding for ~s" id)))
(: Extend : Symbol VAL ENV > ENV)
;; extend a given environment cache with a new binding
(define (Extend id val restenv)
(lambda (name)
(if (eq? name id)
val
(restenv name))))
(: 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))
(: 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 'arithop "expected a number, got: ~s" val)]))
(: arithop : (Number Number > Number) VAL VAL > VAL)
;; gets a Racket numeric binary operator, and uses it within a NumV
;; wrapper
(define (arithop 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) (arithop + (eval l env) (eval r env))]
[(Sub l r) (arithop  (eval l env) (eval r env))]
[(Mul l r) (arithop * (eval l env) (eval r env))]
[(Div l r) (arithop / (eval l env) (eval r env))]
[(With boundid namedexpr boundbody)
(eval boundbody
(Extend boundid (eval namedexpr env) env))]
[(Id name) (lookup name env)]
[(Fun boundid boundbody)
(FunV boundid boundbody env)]
[(Call funexpr argexpr)
(let ([fval (eval funexpr env)])
(cases fval
[(FunV boundid boundbody fenv)
(eval boundbody
(Extend boundid (eval argexpr env) fenv))]
[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 nonnumber: ~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)

# More Closures (on both levels)
Racket closures (= functions) can be used in other places too, and as we
have seen, they can do more than encapsulate various values  they can
also hold the behavior that is expected of these values.
To demonstrate this we will deal with closures in our language. We
currently use a variant that holds the three pieces of relevant
information:
[FunV Symbol FLANG ENV]
We can replace this by a functional object, which will hold the three
values. First, change the `VAL` type to hold functions for `FunV`
values:
(definetype VAL
[NumV Number]
[FunV (? > ?)])
And note that the function should somehow encapsulate the same
information that was there previously, the question is *how* this
information is going to be done, and this will determine the actual
type. This information plays a role in two places in our evaluator 
generating a closure in the `Fun` case, and using it in the `Call` case:
[(Fun boundid boundbody)
(FunV boundid boundbody env)]
[(Call funexpr argexpr)
(let ([fval (eval funexpr env)])
(cases fval
[(FunV boundid boundbody fenv)
(eval boundbody ;***
(Extend boundid ;***
(eval argexpr env) ;***
fenv))] ;***
[else (error 'eval "`call' expects a function, got: ~s"
fval)]))]
we can simply fold the marked functionality bit of `Call` into a Racket
function that will be stored in a `FunV` object  this piece of
functionality takes an argument value, extends the closure's environment
with its value and the function's name, and continues to evaluate the
function body. Folding all of this into a function gives us:
(lambda (argval)
(eval boundbody (Extend boundid argval env)))
where the values of `boundbody`, `boundid`, and `val` are known at the
time that the `FunV` is *constructed*. Doing this gives us the
following code for the two cases:
[(Fun boundid boundbody)
(FunV (lambda (argval)
(eval boundbody (Extend boundid argval env))))]
[(Call funexpr argexpr)
(let ([fval (eval funexpr env)])
(cases fval
[(FunV proc) (proc (eval argexpr env))]
[else (error 'eval "`call' expects a function, got: ~s"
fval)]))]
And now the type of the function is clear:
(definetype VAL
[NumV Number]
[FunV (VAL > VAL)])
And again, the rest of the code is unmodified:
#lang pl
(definetype FLANG
[Num Number]
[Add FLANG FLANG]
[Sub FLANG FLANG]
[Mul FLANG FLANG]
[Div FLANG FLANG]
[Id Symbol]
[With Symbol FLANG FLANG]
[Fun Symbol FLANG]
[Call FLANG FLANG])
(: parsesexpr : Sexpr > FLANG)
;; parses sexpressions into FLANGs
(define (parsesexpr 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 (parsesexpr named) (parsesexpr body))]
[else (error 'parsesexpr "bad `with' syntax in ~s" sexpr)])]
[(cons 'fun more)
(match sexpr
[(list 'fun (list (symbol: name)) body)
(Fun name (parsesexpr body))]
[else (error 'parsesexpr "bad `fun' syntax in ~s" sexpr)])]
[(list '+ lhs rhs) (Add (parsesexpr lhs) (parsesexpr rhs))]
[(list ' lhs rhs) (Sub (parsesexpr lhs) (parsesexpr rhs))]
[(list '* lhs rhs) (Mul (parsesexpr lhs) (parsesexpr rhs))]
[(list '/ lhs rhs) (Div (parsesexpr lhs) (parsesexpr rhs))]
[(list 'call fun arg)
(Call (parsesexpr fun) (parsesexpr arg))]
[else (error 'parsesexpr "bad syntax in ~s" sexpr)]))
(: parse : String > FLANG)
;; parses a string containing a FLANG expression to a FLANG AST
(define (parse str)
(parsesexpr (string>sexpr str)))
;; Types for environments, values, and a lookup function
(definetype VAL
[NumV Number]
[FunV (VAL > VAL)])
;; Define a type for functional environments
(definetype ENV = Symbol > VAL)
(: EmptyEnv : > ENV)
(define (EmptyEnv)
(lambda (id) (error 'lookup "no binding for ~s" id)))
(: Extend : Symbol VAL ENV > ENV)
;; extend a given environment cache with a new binding
(define (Extend id val restenv)
(lambda (name)
(if (eq? name id)
val
(restenv name))))
(: 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))
(: 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 'arithop "expected a number, got: ~s" val)]))
(: arithop : (Number Number > Number) VAL VAL > VAL)
;; gets a Racket numeric binary operator, and uses it within a NumV
;; wrapper
(define (arithop 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) (arithop + (eval l env) (eval r env))]
[(Sub l r) (arithop  (eval l env) (eval r env))]
[(Mul l r) (arithop * (eval l env) (eval r env))]
[(Div l r) (arithop / (eval l env) (eval r env))]
[(With boundid namedexpr boundbody)
(eval boundbody
(Extend boundid (eval namedexpr env) env))]
[(Id name) (lookup name env)]
[(Fun boundid boundbody)
(FunV (lambda (argval)
(eval boundbody (Extend boundid argval env))))]
[(Call funexpr argexpr)
(let ([fval (eval funexpr env)])
(cases fval
[(FunV proc) (proc (eval argexpr 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 nonnumber: ~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)

# Types of Evaluators
What we did just now is implement lexical environments and closures in
the language we implement using lexical environments and closures in our
own language (Racket)!
This is another example of embedding a feature of the host language in
the implemented language, an issue that we have already discussed.
There are many examples of this, even when the two languages involved
are different. For example, if we have this bit in the C implementation
of Racket:
// Disclaimer: not real Racket code
Racket_Object *eval_and(int argc, Racket_Object *argv[]) {
Racket_Object *tmp;
if ( argc != 2 )
signal_racket_error("bad number of arguments");
else if ( racket_eval(argv[0]) != racket_false &&
(tmp = racket_eval(argv[1])) != racket_false )
return tmp;
else
return racket_false;
}
then the special semantics of evaluating a Racket `and` form is being
inherited from C's special treatment of `&&`. You can see this by the
fact that if there is a bug in the C compiler, then it will propagate to
the resulting Racket implementation too. A different solution is to not
use `&&` at all:
// Disclaimer: not real Racket code
Racket_Object *eval_and(int argc, Racket_Object *argv[]) {
Racket_Object *tmp;
if ( argc != 2 )
signal_racket_error("bad number of arguments");
else if ( racket_eval(argv[0]) != racket_false )
return racket_eval(argv[1]);
else
return racket_false;
}
and we can say that this is even better since it evaluates the second
expression in tail position. But in this case we don't really get that
benefit, since C itself is not doing tailcall optimization as a
standard feature (though some compilers do so under some circumstances).
We have seen a few different implementations of evaluators that are
quite different in flavor. They suggest the following taxonomy.
* A ___syntactic evaluator___ is one that uses its own language to
represent expressions and semantic runtime values of the evaluated
language, implementing all the corresponding behavior explicitly.
* A ___meta evaluator___ is an evaluator that uses language features of
its own language to directly implement behavior of the evaluated
language.
While our substitutionbased FLANG evaluator was close to being a
syntactic evaluator, we haven't written any purely syntactic evaluators
so far: we still relied on things like Racket arithmetics etc. The most
recent evaluator that we have studied, is even more of a *meta*
evaluator than the preceding ones: it doesn't even implement closures
and lexical scope, and instead, it uses the fact that Racket itself has
them.
With a good match between the evaluated language and the implementation
language, writing a meta evaluator can be very easy. With a bad match,
though, it can be very hard. With a syntactic evaluator, implementing
each semantic feature will be somewhat hard, but in return you don't
have to worry as much about how well the implementation and the
evaluated languages match up. In particular, if there is a particularly
strong mismatch between the implementation and the evaluated language,
it may take less effort to write a syntactic evaluator than a meta
evaluator. As an exercise, we can build upon our latest evaluator to
remove the encapsulation of the evaluator's response in the VAL type.
The resulting evaluator is shown below. This is a true meta evaluator:
it uses Racket closures to implement FLANG closures, Racket function
application for FLANG function application, Racket numbers for FLANG
numbers, and Racket arithmetic for FLANG arithmetic. In fact, ignoring
some small syntactic differences between Racket and FLANG, this latest
evaluator can be classified as something more specific than a meta
evaluator:
* A ___metacircular evaluator___ is a meta evaluator in which the
implementation and the evaluated languages are the same.
(Put differently, the trivial nature of the evaluator clues us in to the
deep connection between the two languages, whatever their syntactic
differences may be.)

# Feature Embedding
We saw that the difference between lazy evaluation and eager evaluation
is in the evaluation rules for `with` forms, function applications, etc:
eval({with {x E1} E2}) = eval(E2[eval(E1)/x])
is eager, and
eval({with {x E1} E2}) = eval(E2[E1/x])
is lazy. But is the first rule *really* eager? The fact is that the
only thing that makes it eager is the fact that our understanding of the
mathematical notation is eager  if we were to take math as lazy, then
the description of the rule becomes a description of lazy evaluation.
Another way to look at this is  take the piece of code that
implements this evaluation:
(: eval : FLANG > Number)
;; evaluates FLANG expressions by reducing them to numbers
(define (eval expr)
(cases expr
...
[(With boundid namedexpr boundbody)
(eval (subst boundbody
boundid
(Num (eval namedexpr))))]
...))
and the same question applies: is this really implementing eager
evaluation? We know that this is indeed eager  we can simply try it
and check that it is, but it is only eager because we are using an eager
language for the implementation! If our own language was lazy, then the
evaluator's implementation would run lazily, which means that the above
applications of the `eval` and the `subst` functions would also be lazy,
making our evaluator lazy as well.
This is a general phenomena where some of the semantic features of the
language we use (math in the formal description, Racket in our code)
gets *embedded* into the language we implement.
Here's another example  consider the code that implements
arithmetics:
(: eval : FLANG > Number)
;; evaluates FLANG expressions by reducing them to numbers
(define (eval expr)
(cases expr
[(Num n) n]
[(Add l r) (+ (eval l) (eval r))]
...))
what if it was written like this:
FLANG eval(FLANG expr) {
if (is_Num(expr))
return num_of_Num(expr);
else if (is_Add(expr))
return eval(lhs_of_Add(expr)) + eval(rhs_of_Add(expr));
else if ...
...
}
Would it still implement unlimited integers and exact fractions? That
depends on the language that was used to implement it: the above syntax
suggests C, C++, Java, or some other relative, which usually come with
limited integers and no exact fractions. But this really depends on the
language  even our own code has unlimited integers and exact
rationals only because Racket has them. If we were using a language
that didn't have such features (there are such Scheme implementations),
then our implemented language would absorb these (lack of) features too,
and its own numbers would be limited in just the same way. (And this
includes the syntax for numbers, which we embedded intentionally, like
the syntax for identifiers).
The bottom line is that we should be aware of such issues, and be very
careful when we talk about semantics. Even the language that we use to
communicate (semiformal logic) can mean different things.

Aside: read "Reflections on Trusting Trust" by Ken Thompson
(You can skip to the "Stage II" part to get to the interesting stuff.)
(And when you're done, look for "XcodeGhost" to see a relevant example,
and don't miss the leaked document on the wikipedia page...)

Here is yet another variation of our evaluator that is even closer to a
metacircular evaluator. It uses Racket values directly to implement
values, so arithmetic operations become straightforward. Note
especially how the case for function application is similar to
arithmetics: a FLANG function application translates to a Racket
function application. In both cases (applications and arithmetics) we
don't even check the objects since they are simple Racket objects  if
our language happens to have some meaning for arithmetics with
functions, or for applying numbers, then we will inherit the same
semantics in our language. This means that we now specify less behavior
and fall back more often on what Racket does.
We use Racket values with this type definition:
(definetype VAL = (U Number (VAL > VAL)))
And the evaluation function can now be:
(: eval : FLANG ENV > VAL)
;; evaluates FLANG expressions by reducing them to values
(define (eval expr env)
(cases expr
[(Num n) n] ;*** return the actual number
[(Add l r) (+ (eval l env) (eval r env))]
[(Sub l r) ( (eval l env) (eval r env))]
[(Mul l r) (* (eval l env) (eval r env))]
[(Div l r) (/ (eval l env) (eval r env))]
[(With boundid namedexpr boundbody)
(eval boundbody
(Extend boundid (eval namedexpr env) env))]
[(Id name) (lookup name env)]
[(Fun boundid boundbody)
(lambda ([argval : VAL]) ;*** return the racket function
;; note that this requires input type specifications since
;; typed racket can't guess the right one
(eval boundbody (Extend boundid argval env)))]
[(Call funexpr argexpr)
((eval funexpr env) ;*** trivial like the arithmetics!
(eval argexpr env))]))
Note how the arithmetics implementation is simple  it's a direct
translation of the FLANG syntax to Racket operations, and since we don't
check the inputs to the Racket operations, we let Racket throw type
errors for us. Note also how function application is just like the
arithmetic operations: a FLANG application is directly translated to a
Racket application.
However, this does not work quite as simply in Typed Racket. The whole
point of typechecking is that we never run into type errors  so we
cannot throw back on Racket errors since code that might produce them is
forbidden! A way around this is to perform explicit checks that
guarantee that Racket cannot run into type errors. We do this with the
following two helpers that are defined inside `eval`:
(: evalN : FLANG > Number)
(define (evalN e)
(let ([n (eval e env)])
(if (number? n)
n
(error 'eval "got a nonnumber: ~s" n))))
(: evalF : FLANG > (VAL > VAL))
(define (evalF e)
(let ([f (eval e env)])
(if (function? f)
f
(error 'eval "got a nonfunction: ~s" f))))
Note that Typed Racket is "smart enough" to figure out that in `evalF`
the result of the recursive evaluation has to be either `Number` or
`(VAL > VAL)`; and since the `if` throws out on numbers, we're left
with `(VAL > VAL)` functions, not just any function.
#lang pl
(definetype FLANG
[Num Number]
[Add FLANG FLANG]
[Sub FLANG FLANG]
[Mul FLANG FLANG]
[Div FLANG FLANG]
[Id Symbol]
[With Symbol FLANG FLANG]
[Fun Symbol FLANG]
[Call FLANG FLANG])
(: parsesexpr : Sexpr > FLANG)
;; parses sexpressions into FLANGs
(define (parsesexpr 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 (parsesexpr named) (parsesexpr body))]
[else (error 'parsesexpr "bad `with' syntax in ~s" sexpr)])]
[(cons 'fun more)
(match sexpr
[(list 'fun (list (symbol: name)) body)
(Fun name (parsesexpr body))]
[else (error 'parsesexpr "bad `fun' syntax in ~s" sexpr)])]
[(list '+ lhs rhs) (Add (parsesexpr lhs) (parsesexpr rhs))]
[(list ' lhs rhs) (Sub (parsesexpr lhs) (parsesexpr rhs))]
[(list '* lhs rhs) (Mul (parsesexpr lhs) (parsesexpr rhs))]
[(list '/ lhs rhs) (Div (parsesexpr lhs) (parsesexpr rhs))]
[(list 'call fun arg)
(Call (parsesexpr fun) (parsesexpr arg))]
[else (error 'parsesexpr "bad syntax in ~s" sexpr)]))
(: parse : String > FLANG)
;; parses a string containing a FLANG expression to a FLANG AST
(define (parse str)
(parsesexpr (string>sexpr str)))
;; Types for environments, values, and a lookup function
;; Values are plain Racket values, no new VAL wrapper;
;; (but note that this is a recursive definition)
(definetype VAL = (U Number (VAL > VAL)))
;; Define a type for functional environments
(definetype ENV = (Symbol > VAL))
(: EmptyEnv : > ENV)
(define (EmptyEnv)
(lambda (id) (error 'lookup "no binding for ~s" id)))
(: Extend : Symbol VAL ENV > ENV)
;; extend a given environment cache with a new binding
(define (Extend id val restenv)
(lambda (name)
(if (eq? name id)
val
(restenv name))))
(: 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))
(: eval : FLANG ENV > VAL)
;; evaluates FLANG expressions by reducing them to values
(define (eval expr env)
(: evalN : FLANG > Number)
(define (evalN e)
(let ([n (eval e env)])
(if (number? n)
n
(error 'eval "got a nonnumber: ~s" n))))
(: evalF : FLANG > (VAL > VAL))
(define (evalF e)
(let ([f (eval e env)])
(if (function? f)
f
(error 'eval "got a nonfunction: ~s" f))))
(cases expr
[(Num n) n]
[(Add l r) (+ (evalN l) (evalN r))]
[(Sub l r) ( (evalN l) (evalN r))]
[(Mul l r) (* (evalN l) (evalN r))]
[(Div l r) (/ (evalN l) (evalN r))]
[(With boundid namedexpr boundbody)
(eval boundbody
(Extend boundid (eval namedexpr env) env))]
[(Id name) (lookup name env)]
[(Fun boundid boundbody)
(lambda ([argval : VAL])
(eval boundbody (Extend boundid argval env)))]
[(Call funexpr argexpr)
((evalF funexpr)
(eval argexpr env))]))
(: run : String > VAL) ; no need to convert VALs to numbers
;; evaluate a FLANG program contained in a string
(define (run str)
(eval (parse str) (EmptyEnv)))
;; 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)