Fixing an Overlooked Bug
Incidentally, this version fixes a bug we had previously in the substitution version of FLANG:
{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 §11 (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:
(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:
(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:
(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
):
but we can skip the need for an extra definition and simply return an empty mapping function:
(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:
(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:
(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 —
(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:
keeping the same type that is implied by this variant means that the new
Extend
should look like this:
(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:
(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
:
(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:
(define (Extend id val restenv)
(lambda (name)
(if (eq? name id)
val
(lookup name restenv))))
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 instead:
(define (Extend id val restenv)
(lambda (name)
(if (eq? name id)
val
(restenv name)))) ; same as (lookup name restenv)
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:
(cond [(eq? 'a id) 1]
[(eq? 'b id) 2]
[else (error ...)]))
behaves in the same way (if the type of values is numbers) as
The new code is now the same, except for the environment code:
#
The grammar:
<FLANG> ::= <num>
 { + <FLANG> <FLANG> }
 {  <FLANG> <FLANG> }
 { * <FLANG> <FLANG> }
 { / <FLANG> <FLANG> }
 { with { <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({fun {x} E},env) = <{fun {x} E}, env>
eval({call E1 E2},env1) = eval(B,extend(x,eval(E2,env1),env2))
if eval(E1,env1) = <{fun {x} B}, 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)
(define 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:
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:
[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:
(FunV boundid boundbody env)]
[(Call funexpr argexpr)
(define 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:
(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:
(FunV (lambda (argval)
(eval boundbody (Extend boundid argval env))))]
[(Call funexpr argexpr)
(define 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:
[NumV Number]
[FunV (VAL > VAL)])
And again, the rest of the code is unmodified:
(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)
(define 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:
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:
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.
This is essentially the concept of a “universal” evaluator, as in a “universal turing machine”.
(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:
is eager, and
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:
;; 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:
;; 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:
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:
And the evaluation function can now be:
;; 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
:
(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.
(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)
Recursion, Recursion, Recursion
There is one major feature that is still missing from our language: we
have no way to perform recursion (therefore no kind of loops). So far,
we could only use recursion when we had names. In FLANG, the only way
we can have names is through with
which not good enough for recursion.
To discuss the issue of recursion, we switch to a “broken” version of
(untyped) Racket — one where a define
has a different scoping rules:
the scope of the defined name does not cover the defined expression.
Specifically, in this language, this doesn’t work:
(define (fact n)
(if (zero? n) 1 (* n (fact ( n 1)))))
(fact 5)
In our language, this translation would also not work (assuming we have
if
etc):
{if {= n 0} 1 {* n {call fact { n 1}}}}}}
{call fact 5}}
And similarly, in plain Racket this won’t work if let
is the only tool
you use to create bindings:
(if (zero? n) 1 (* n (fact ( n 1)))))])
(fact 5))
In the brokenscope language, the define
form is more similar to a
mathematical definition. For example, when we write:
(define (G y) (F y))
(G F)
it is actually shorthand for
(define G (lambda (y) (F y)))
(G F)
we can then replace defined names with their definitions:
(define G (lambda (y) (F y)))
((lambda (y) (F y)) (lambda (x) x))
and this can go on, until we get to the actual code that we wrote:
This means that the above fact
definition is similar to writing:
(if (zero? n) 1 (* n (fact ( n 1)))))
(fact 5)
which is not a wellformed definition — it is meaningless (this is a
formal use of the word “meaningless”). What we’d really want, is to take
the equation (using =
instead of :=
)
(if (zero? n) 1 (* n (fact ( n 1)))))
and find a solution which will be a value for fact
that makes this
true.
If you look at the Racket evaluation rules handout on the web page, you
will see that this problem is related to the way that we introduced the
Racket define
: there is a handwavy explanation that talks about
knowing things.
The big question is: can we define recursive functions without Racket’s
magical define
form?
Note: This question is a little different than the question of implementing recursion in our language — in the Racket case we have no control over the implementation of the language. As it will eventually turn out, implementing recursion in our own language will be quite easy when we use mutation in a specific way. So the question that we’re now facing can be phrased as either “can we get recursion in Racket without Racket’s magical definition forms?” or “can we get recursion in our interpreter without mutation?”.