Re: [Caml-list] coinductive data types

[Aaron Gray <aaronngray.lists@gmail.com>]

Jacques,

I will get a raw OCaML Docker image, and have a play with this.

What is going on there, does ':>' return a conversion function as well
as being 'true' and secondly a predicate ?

Thanks,

Aaron



On Wed, 31 Aug 2022 at 02:15, Jacques Garrigue
<jacques.garrigue@gmail.com> wrote:
>
> Hello Aaron,
>
> If you are interested in the subtyping already available inside OCaml,
> it provides width subtyping for both objects and polymorphic variants.
> Polymorphic variants are algebraic datatypes, but their equality is
> structural rather than nominal, and they are limited to regular type
> definitions.
>
> For instance, the following subtyping on two variants of potentially infinite
> streams is available:
>
> type 'a seq = 'a seqc lazy_t and 'a seqc = [`Nil | `Cons of 'a * 'a seq]
> type 'a aseq = 'a aseqc lazy_t
>  and 'a aseqc = ['a seqc | `App of 'a aseq * 'a aseq]
> let sub x = (x : 'a seq :> 'a aseq)
> (* val sub : 'a seq -> 'a aseq *)
>
> Jacques Garrigue
>
> 2022/08/30 21:33、Aaron Gray <aaronngray.lists@gmail.com>のメール:
>
> On Tue, 30 Aug 2022 at 12:12, Xavier Leroy
> <xavier.leroy@college-de-france.fr> wrote:
>
>
> On Tue, Aug 30, 2022 at 9:24 AM François Pottier <francois.pottier@inria.fr> wrote:
>
>
>
> Hello,
>
> Le 29/08/2022 à 17:43, Aaron Gray a écrit :
>
> Does either ML or OCaML have coinductive data types ? And if so could
> you please point me at the/some appropriate documentation on them.
>
>
> ML and OCaml have algebraic data types, which are recursive (that is,
> more general than inductive and co-inductive types). Algebraic data
> type definitions are not subject to a positivity restriction, and
> algebraic data types can be constructed and deconstructed by recursive
> functions, which are not subject to a termination check.
>
>
> Hello Xavier,
>
> Thanks for putting me straight on that.
>
> My original path of inquiry which I should have actually stated was
> regarding how to go about implementing subtyping of mutually recursive
> algebraic data types.
>
> I am looking on how to go about this and using coinduction and
> bisimulation seemed like the best fit or correct way to go about this.
>
> Does OCaML use/handle subtyping of mutually recursive algebraic data
> types ? And if so, is its implementation easily accessible ?
>
> If you want to see a typical example of a "co-inductive" data structure
> encoded in OCaml, I suggest to have a look at "sequences", which can be
> described as potentially infinite lists:
>
>   https://v2.ocaml.org/api/Seq.html
>
> Lazy evaluation (type Lazy.t) can also be used to define coinductive data structures.
>
> For concreteness, here are two definitions of the type of streams (infinite lists) :
>
> type 'a stream = unit -> 'a streamcell and 'a streamcell = { head: 'a; tail: 'a stream }
> type 'a stream = 'a streamcell Lazy.t and 'a streamcell = { head: 'a; tail: 'a stream }
>
> and of the stream of integers starting from n :
>
> let rec ints n = fun () -> { head = n; tail = ints (n + 1) }
> let rec ints n = lazy { head = n; tail = ints (n + 1) }
>
>
> Thank you, yes I am familiar with this type of usage.
>
> Regards,
>
> Aaron
>
>
>
> Hope this helps,
>
> - Xavier Leroy
>
>
>
> --
> François Pottier
> francois.pottier@inria.fr
> http://cambium.inria.fr/~fpottier/
>
>
>
>
> --
> Aaron Gray
>
> Independent Open Source Software Engineer, Computer Language
> Researcher, Information Theorist, and amateur computer scientist.
>
>


-- 
Aaron Gray

Independent Open Source Software Engineer, Computer Language
Researcher, Information Theorist, and amateur computer scientist.