Do Not Reuse Blindly!
It may be tempting to base your decision to reuse an existing semantics of a language feature solely on syntactic considerations; for example, to reuse whenever the parser does not complain. As seen in this lesson, this could be quite risky.
Let's try (and fail) to reuse the definition of
callcc from Lesson 1:
syntax Exp ::= "callcc" Exp [strict] syntax Val ::= cc(K) rule <k> (callcc V:Val => V cc(K)) ~> K </k> rule <k> cc(K) V ~> _ => V ~> K </k>
callcc examples that we tried in Lesson 1 work, so it may look it works.
However, the problem is that
cc(K) should also include an environment,
and that environment should also be restored when
cc(K) is invoked.
Let's try to illustrate this bug with
let x = 1 in ((callcc lambda k . (let x = 2 in (k x))) + x)
where the second argument of
x, should be bound to the top
is 1. However, since
callcc does not restore the environment, that
should be looked up in the wrong, callcc-inner environment, so we should see
the overall result 4.
Hm, we get the right result, 3 ... (Note: you may get 4, depending on
your version of K and platform; but both 3 and 4 are possible results, as
explained below and seen in the tests). How can we get 3? Well, recall that
+ is strict, which means that it can evaluate its arguments in any order.
It just happened that in the execution that took place above its second
argument was evaluated first, to 1, and then the
callcc was evaluated, but
cc value K had already included the 1 instead of
x ... In Part 4 of
the tutorial we will see how to explore all the non-deterministic behaviors of
a program; we could use that feature of K to debug semantics, too.
For example, in this case, we could search for all behaviors of this program
and we would indeed get two possible value results: 3 and 4.
One may think that the problem is the non-deterministic evaluation order
+, and thus that all we need to do is to enforce a deterministic order
in which the arguments of + are evaluated. Let us follow this path to
see what happens. There are two simple ways to make the evaluation order
+'s arguments deterministic. One is to make
seqstrict in the
semantics, to enforce its evaluation from left-to-right. Do it and then
run the program above again; you should get only one behavior for the
program above, 4, which therefore shows that copying-and-pasting our old
callcc was incorrect. However, as seen shortly, that only
fixed the problem for the particular example above, but not in general.
Another conventional approach to enforce the desired evaluation order is to
modify the program to enforce the left-to-right evaluation order using let
binders, as we do in
let x = 1 in let a = callcc lambda k . (let x = 2 in (k x)) in let b = x in (a + b)
With your installation of K you may get the "expected" result 4 when you
execute this program, so it may look like our non-deterministic problem is
fixed. Unfortunately, it is not. Using the K tool to search for all the
behaviors in the program above reveals that the final result 3 is still
possible. Moreover, both the 3 and the 4 behaviors are possible regardless
+ is declared to be
seqstrict or just
strict. How is that
possible? The problem is now the non-deterministic evaluation strategy of
the function application construct. Indeed, recall that the semantics of
the let-in construct is defined by desugaring to lambda application:
rule let X = E in E' => (lambda X . E') E
With this, the program above eventually reduces to
(lambda a . ((lambda b . a + b) x)) (callcc lambda k . (let x = 2 in (k x)))
in an environment where
x is 1. If the first expression evaluates first,
then it does so to a closure in which
x is bound to a location holding 1,
so when applied later on to the
x inside the argument of
callcc (which is
2), it will correctly lookup
x in its enclosed environment and thus the
program will evaluate to 3. On the other hand, if the second expression
evaluates first, then the
cc value will freeze the first expression as is,
breaking the relationship between its
x and the current environment in which
it is bound to 1, being inadvertently captured by the environment of the
let-in construct inside the
callcc and thus making the entire expression
evaluate to 4.
So the morale is: Do not reuse blindly. Think!
In the next lesson we fix the environment-based semantics of
callcc by having
cc also wrap an environment, besides a computation. We will also give a more
direct semantics to recursion, based on environments instead of fixed-point
Go to Lesson 5, LAMBDA++: More Semantic Computation Items.