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[Caml-list] equality over functional value
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 Date: 2001-04-23 (07:56) From: Xavier Leroy Subject: Re: [Caml-list] equality over functional value
```> Hi,
>
> A friend of mine is just starting with ocaml. He was puzzled by the
> following result:
>
> # let a i = i;;
> val a : 'a -> 'a = <fun>
> # let b i = i;;
> val b : 'a -> 'a = <fun>
> # a=b;;
> Uncaught exception: Invalid_argument "equal: functional value".
> # a=a;;
> - : bool = true
> #
>
> That is 'a=a' does not return the expected exception.  Actually it
> first hit this "curiosity" when creating a polymorphic 'sort' function
> on lists - and by applying it to a [sort;sort..] list. It worked.
>
> Is this a bug?

It's a questionable behavior.  As Alain Frisch said, polymorphic
structural equality is implemented by first testing physical (pointer)
equality.  Besides speed, this has the advantage that x == y implies x = y.

However, this causes a = a where a is a function to return true
instead of raising an exception as you expect.

(Another weird consequence of this implementation of physical equality
is that
(nan, nan) = (nan, nan)
returns true, which is incorrect according to the IEEE specs: the NaN
float is not equal to itself!)

The special case (first test pointer equality) in structural equality
could be eliminated, although I don't know how big a performance
impact this would have.

More generally, equality between functions can be interpreted in
several ways:

1- Extensional, pessimistic: f = g iff for all x, f x = g x,
and since this is undecidable, equality always raises an exception
when passed function values.

2- Extensional, approximated: same definition as above, but we return
"true" in some cases where the two functions are guaranteed to be
extensionally equal, e.g. f and g point to the same closure, or even f
and g have the same code pointer and contain equal values in their
environments.  Otherwise, we raise an exception.

3- By representation: two functions are equal iff their closures are
structurally equal, i.e. they have the same code pointer and contain
equal values in their environment.

Interpretation 3- is useless I think, because it depends very much on
the compiler's closure representation strategy.  In other terms, while
a "true" result guarantees that the two functions are extensionally
equal, a "false" result does not mean anything.

The current interpretation 2- is semantically more sound: if it
returns "true", then the two functions are extensionally equal;
and it never returns "false", so it never claims that two functions
are extensionally different!  Still, it is rather unintuitive.
Also, the current implementation exposes too much the order of the tests,
i.e. "(0, f) = (1, g)" returns "false", but
"(f, 0) = (g, 1)" raises an exception.

Interpretation 1- is easiest to explain, although it entails a bit of
a performance penalty as I said above.

Food for thoughts...

- Xavier Leroy
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