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Re: [Caml-list] Weird types
• Tyng-Ruey Chuang
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 Date: 2001-06-18 (08:06) From: Tyng-Ruey Chuang Subject: Re: [Caml-list] Weird types
```Jean-Christophe FILLIATRE wrote:
> Actually, there  is a  type-able way of  writing this  function, which
> consists in duplicating it into two functions, like this:
>
> ======================================================================
> type ('a,'b,'c) t =
>   | A of 'a * 'b * 'c
>   | B of ('b, 'a, 'c) t
>
> let rec gamma = function
>   | A _ -> 0
>   | B x -> 1 + gamma' x
>
> and gamma' = function
>   | A _ -> 0
>   | B x -> 1 + gamma x
> ======================================================================
>
> which gives the expected types:
>
> ======================================================================
> val gamma : ('a, 'b, 'c) t -> int = <fun>
> val gamma' : ('a, 'b, 'c) t -> int = <fun>
> ======================================================================

Interesting! But then size of the duplicated code grows exponentially.
For example, for a 3-ary type constructor sigma

type ('a, 'b, 'c) sigma =
I of 'a * 'b * 'c
|  T of ('b, 'a, 'c) sigma
|  P of ('b, 'c, 'a) sigma

one need to define the 6 equivalent "length" functions gamma_xxx,
where xxx ranges from {abc, acb, bac, bca, cab, cba}, by

let rec gamma_abc s =
match s with
I _ -> 0
| T x -> 1 + gamma_bac x
| P x -> 1 + gamma_bca x
and gamma_acb s =
match s with
I _ -> 0
| T x -> 1 + gamma_cab x
| P x -> 1 + gamma_cba x
and gamma_bac s =
match s with
I _ -> 0
| T x -> 1 + gamma_abc x
| P x -> 1 + gamma_acb x
and gamma_bca s =
match s with
I _ -> 0
| T x -> 1 + gamma_cba x
| P x -> 1 + gamma_cab x
and gamma_cab s =
match s with
I _ -> 0
| T x -> 1 + gamma_acb x
| P x -> 1 + gamma_abc x
and gamma_cba s =
match s with
I _ -> 0
| T x -> 1 + gamma_bca x
| P x -> 1 + gamma_bac x

For the original definition of 7-ary sigma

type ('a,'b,'c,'d,'e,'f,'g) sigma =
I of 'a * 'b * 'c * 'd * 'e * 'f * 'g
| T of ('b,'a,'c,'d,'e,'f,'g) sigma
| P of ('b,'c,'d,'e,'f,'g,'a) sigma

one probably will need 7! = 5040 equivalent "length" functions
that are recursively defined among themselves!

I guess language-supported polymorphic recursions will help here.
However, I believe the general problem of typing polymorphic recursive
functions had been shown to be undecidable.

Tyng-Ruey Chuang
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```