Re: [Caml-list] Weird types

[Tyng-Ruey Chuang <trc@iis.sinica.edu.tw>]

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
-------------------
Bug reports: http://caml.inria.fr/bin/caml-bugs  FAQ: http://caml.inria.fr/FAQ/
To unsubscribe, mail caml-list-request@inria.fr  Archives: http://caml.inria.fr