Re: [Caml-list] Reasoning about categories at compile-time.

[Guillaume Yziquel <guillaume.yziquel@citycable.ch>]

Le Thursday 31 Mar 2011 à 08:02:30 (+0200), Gabriel Scherer a écrit :
>    First, one obvious solution : parse the equations, run a solver, then
>    output "()" if it can deduce the queries from the equations, and "1 =
>    true" otherwise. Why isn't this solution satisfying ?

Because that would mean, in Camlp4, doing a processing of the whole

module X : sig
	(* category constraints / proof goals *)
end = struct
	(* category axioms *)
end

I want to be able to deal separately with .ml and .mli files. Or with a module
structure and a module signature.

Therefore I need it to be such that it is the type checker that is in
charge of being the solver.

I'd love a 'type system extension' mechanism... this way I could include
the solver in it.

>    With the type system there is one thing that is easy and natural to do
>    : you could encode your arrows as elements with a specific type (eg.
>    for an arrow `f` of source `src` and destination `dst` you would write
>    `(src, f, dst) arrow` with `src, f, dst` abstract types) :
>
>      arrow_f : (foo, f, bar) arrow
>
>    then write both a composition function
>
>      compose : ('a, 'f, 'b) arrow -> ('b, 'g, 'c) arrow -> ('a, ('f, 'g) comp, 'c) arrow
>
>    and terms representing your equational theory:
>
>      assoc : ('a, (('f, 'g) comp, 'h) comp, 'b) arrow -> ('a, ('f, ('g, 'h) comp) comp, 'b)
>
>    and for each equation f o g = h (not that I use `g` and not `'g`)
>
>      witness_fg_h : ('a, (f, g) comp, 'b) arrow -> ('a, h, 'b) arrow
>
>    Then, for each deducible relation, there exist a typed term that is a
>    proof witness of the relation deductibility (you just unify the type of
>    the two arrows that should be equal). If you have a solver for equation
>    deductibility, you can ask him to output the proof witness in this
>    format.

I thought of that, though not as precisely as you're exposing it.

But keeping the .ml / .mli distinction and compiling them separately,
you have a problem: Essentially, the witnesses live in the .ml file, and
the proof witness also lives in the .ml file. In the .mli file, you
simply have to declare the existence of a proof witness term.

However, in order to construct the proof witness, you need to know the
goal that you wish to prove, which means providing to camlp4 some
information that lives in the .mli file.

This would break separate compilation. Or rather separate preprocessing.
You could not preprocess the .ml file before the .mli file. Worse: you
could not do:

module X with syntax extension = struct

	...

end

module type S with syntax extension = sig

	...

end

let m = (module X : S)

(which is what I want to do, specifically).

>    This does not use the inference system to solve your problem, which is
>    what you asked for. But this is a solid design that is going to still
>    work if you change your language, eg. can still be used on non
>    decidable questions. I think this is a good first step; feel free to
>    improve it by whatever advanced type hackery.

Unfortunately, this doesn't fit the bill. I had this picture in mind
intuitively but not formalised, and I saw the limitation I explained
above. That's what prompted me to ask on the list in the first place.

But thanks for clarifying my hazy thoughts.

What bugs me is that, to put it simply, the category axioms are
generated by Camlp4 in a non-deterministic fashion, dependent on the
outside world. You only have control in user code on the proof goals.

What you propose means linking module struct and module type
syntactically. However, I'd need them decoupled as I'd like in the long
run to do the following:

You have two categories, C1 and C2. You take the set union of both
categories C1 U C2. (Set union of objects, and set unions of arrows).
This gives two OCaml modules. You include both modules into a bigger
module, add some arrows and arrow composition axioms. You then want to
match them against a module signature describing the proof goals.

Do the same thing with first-class modules (OCaml 3.13 doesn't need so
much module signatures as OCaml 3.12 for first-class modules, which goes
in the sense of 'decoupling' preprocessed modules and preprocessed
signatures), and you get an idea of what I'm up to.

That's why I'm looking for a purely type-system encoding of this
problem. Somehow lifting the solver for equational deducibility you
mentionned into the type inference algo itself.

So instead of having arrows as private types, I was thinking of having
arrows as type parameters. And have unification doing the solving.
Unfortunately, if you put equational constraints in the module
signature, they leak into the module implementation (funny to have a
signature leaking in an implementation by the way). Just think of the
simple

module X : sig 
	val f : 'a -> 'a 
end = struct
	let f : 'a -> 'b = Obj.magic
end

This is somehow reversed:

module X : sig
	val f : 'a -> 'b
end = struct
	let f : 'a -> 'a = assert false
end

fails to type-check, which could perhaps be understood as 'given two
types not known to be equal, I fail to conclude that 'a = 'a and 'b =
'a, i.e. I fail to conclude that 'b = 'a. This is tautological, but it's
interesting to see that the axioms are now in the signature and the goal
in the module structure.

I was wondering perhaps naïvely how much truth there is in the above
paragraph, and if it could be used to encode the equational solver into
the unification algo in one way or another.

-- 
     Guillaume Yziquel