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E=Sophos;i="5.93,275,1654552800"; d="scan'208";a="22444603" X-MGA-submission: =?us-ascii?q?MDEruZh6Qlqj/nqHjRMLhO8G2l7n4VywipPffJ?= =?us-ascii?q?ccUy5QGSS9wNZGm2yhf86qroy6N7XS5Kp2X/1ge8vpNi7mrkZcZNq5uz?= =?us-ascii?q?T78bnTAPqpworrIodFbOqPtNcrAeVUGEqfsvuGXm20VDBk+5xtcgjJW2?= =?us-ascii?q?bM2gcK6UNDzVww29S1FOBmiA=3D=3D?= Received: from juno.mpi-klsb.mpg.de ([139.19.86.40]) by mail3-smtp-sop.national.inria.fr with ESMTP/TLS/ECDHE-RSA-AES256-GCM-SHA384; 30 Aug 2022 18:45:39 +0200 DKIM-Signature: v=1; a=rsa-sha256; q=dns/txt; c=relaxed/relaxed; d=mpi-sws.org ; s=mail201904; h=To:References:Message-Id:Content-Transfer-Encoding:Cc:Date: In-Reply-To:From:Subject:Mime-Version:Content-Type:sender:reply-to:content-id :content-description:resent-date:resent-from:resent-sender:resent-to: resent-cc:resent-message-id:list-id:list-help:list-unsubscribe:list-subscribe :list-post:list-owner:list-archive; bh=WDqG83f0yfu/BEz0lmWlggtdMlIfHdLMu5BjkL9lWms=; b=PjSZWbwWQyKzS8V1SYSH96VBzZ 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<25e68e2c-04d2-7764-e189-00812c08a34a@inria.fr> <8078ad33-f220-b233-4863-06ce30cc8cff@inria.fr> To: =?utf-8?Q?Fran=C3=A7ois_Pottier?= X-Mailer: Apple Mail (2.3696.120.41.1.1) X-RSPAMD-Score: 0.4 (/) X-RSPAMD-Report: Action: no action Symbol: ARC_NA(0.00) Symbol: RCVD_VIA_SMTP_AUTH(0.00) Symbol: MID_RHS_MATCH_FROM(0.00) Symbol: R_SPF_NEUTRAL(0.00) Symbol: FROM_HAS_DN(0.00) Symbol: MV_CASE(0.50) Symbol: TO_MATCH_ENVRCPT_ALL(0.00) Symbol: MIME_GOOD(-0.10) Symbol: DMARC_NA(0.00) Symbol: TO_DN_SOME(0.00) Symbol: RCPT_COUNT_TWO(0.00) Symbol: NEURAL_HAM(-0.00) Symbol: FROM_EQ_ENVFROM(0.00) Symbol: R_DKIM_NA(0.00) Symbol: MIME_TRACE(0.00) Symbol: ASN(0.00) Symbol: RCVD_COUNT_TWO(0.00) Symbol: RCVD_TLS_ALL(0.00) Message-ID: 2C890C66-F8F7-402C-B88E-587C3E21DE89@mpi-sws.org X-RSPAMD-Bar: / Subject: Re: [Caml-list] coinductive data types Hi Fran=C3=A7ois, I=E2=80=99m curious why you would categorise iso-recursive types as = nominal. I have always considered them structural as well, since two = structurally matching iso-recursive type expressions are still deemed = equivalent. That matters in so far as it makes them modular in a way = that true nominal types are not. Iso-recursive types would only behave like nominal in the degenerate = case where all type definitions occurring in the entire program (across = module boundaries) are tied into a single grand iso-recursive knot, I = think. /Andreas > On 30. 8. 2022, at 17:47, Fran=C3=A7ois Pottier = wrote: >=20 >=20 > Hi, >=20 > On 30/08/2022 14:37, Aaron Gray wrote: > > 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. >=20 > Do you mean implementing an algorithm that decides whether two types = are > related by subtyping, in the presence of mutually recursive algebraic > data types? >=20 > I guess the answer depends on exactly how the subtying relation is = defined. > E.g., if "=CE=B1 foo" and "=CE=B2 bar" are (parameterized) algebraic = data types, does > the subtyping relation "=CE=B1 foo =E2=89=A4 =CE=B2 bar" necessarily = imply that "foo" and "bar" > are the same algebraic data type? >=20 > - If the answer is positive, then the problem boils down to deciding > "=CE=B1 =E2=89=A4 =CE=B2" or "=CE=B2 =E2=89=A4 =CE=B1" or neither or = both, depending on whether this > algebraic data type is covariant or contravariant or neither or both. > In that case, you are working with "isorecursive" types, > also known as "nominal" types. >=20 > - If the answer is negative, then (I imagine) you wish to unfold the > definitions of "foo" and "bar" and decide whether the two unfoldings > are related by subtyping. > In that case, you are working with "equirecursive" types, > also known as "structural" types. >=20 > Brandt and Henglein ("Coinductive axiomatization of recursive type = equality > and subtyping", 1998) study the second problem, if I remember = correctly. >=20 > > Does OCaML use/handle subtyping of mutually recursive algebraic data > > types ? And if so, is its implementation easily accessible ? >=20 > OCaml has explicit subtyping coercions: a user can write `(e : t1 :> = t2)` > and the compiler will check that `t1` is a subtype of `t2`. >=20 > As far as algebraic data types are concerned, OCaml's notion of = subtyping is > nominal, I believe. An algebraic data type can be decorated with = variance > annotations, so, for instance, "list" can be declared covariant and = "=CE=B1 list" > is considered a subtype of "=CE=B2 list" if =CE=B1 is a subtype of =CE=B2= . However, "=CE=B1 list" > can never be considered a subtype of an algebraic data type other than = "list". >=20 > The variance annotations provided by the programmer are checked when = an > algebraic data type is defined, and are later used when deciding = whether a > subtyping relation holds. >=20 > Some more information here: >=20 > https://blog.janestreet.com/a-and-a/ > https://v2.ocaml.org/manual/expr.html#ss%3Aexpr-coercions >=20 > -- > Fran=C3=A7ois Pottier > Francois.Pottier@inria.fr > http://cambium.inria.fr/~fpottier/