New-to-Haskell, i.e. elementary, Haskell questions; extended discussions

Roman Cheplyaka | 9 Oct 2012 11:11

Type of scramblings

I am reading through Oleg's "Eliminating translucent existentials"[1].

[1]: http://okmij.org/ftp/Computation/Existentials.html#eliminating-translucent

He draws a distinction between

  forall a . [a] -> [a]

and

  forall a . [a]^n -> [a]

as types of "scramblings". This is something I'm struggling to understand.

First of all, I think here we're talking about total functions, otherwise
there's no point in introducing dependent types.

There are of course more total functions of type `[a]^n -> [a]` than of type
`[a] -> [a]`, in the sense that any term of the latter type can be assigned the
former type. But, on the other hand, any total function `f :: [a]^n -> [a]`
has an "equivalent" total function

  g :: [a] -> [a]
  g xs | length xs == n = f xs
       | otherwise = xs

(The condition `length xs == n` can be replaced by a similar condition that also
works for infinite lists.)

The functions `f` and `g` are equivalent in the sense that for any list `xs` of

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Marcelo Sousa | 11 Oct 2012 17:15

Re: Type of scramblings

Hi Roman,

On Tue, Oct 9, 2012 at 12:11 PM, Roman Cheplyaka <roma <at> ro-che.info> wrote:
> I am reading through Oleg's "Eliminating translucent existentials"[1].
>
> [1]: http://okmij.org/ftp/Computation/Existentials.html#eliminating-translucent
>
> He draws a distinction between
>
>   forall a . [a] -> [a]
>
> and
>
>   forall a . [a]^n -> [a]
>
> as types of "scramblings". This is something I'm struggling to understand.
>
> First of all, I think here we're talking about total functions, otherwise
> there's no point in introducing dependent types.
>
> There are of course more total functions of type `[a]^n -> [a]` than of type
> `[a] -> [a]`, in the sense that any term of the latter type can be assigned the
> former type. But, on the other hand, any total function `f :: [a]^n -> [a]`
> has an "equivalent" total function
>
>   g :: [a] -> [a]
>   g xs | length xs == n = f xs
>        | otherwise = xs
>
> (The condition `length xs == n` can be replaced by a similar condition that also

(Continue reading)

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oleg | 13 Oct 2012 00:20

Re: Type of scramblings


Sorry for a late reply.

> There are of course more total functions of type `[a]^n -> [a]` than of type
> `[a] -> [a]`, in the sense that any term of the latter type can be assigned the
> former type. But, on the other hand, any total function `f :: [a]^n -> [a]`
> has an "equivalent" total function
>
>   g :: [a] -> [a]
>   g xs | length xs == n = f xs
>        | otherwise = xs

That is correct. It is also correct that f has another "equivalent"
total function

   g2 :: [a] -> [a]
   g2 xs | length xs == n = f xs
         | otherwise = xs ++ xs

and another, and another... That is the problem. The point of the
original article on eliminating translucent existentials was to
characterize scramblings of a list of a given length -- to develop an
encoding of a scrambling which uses only simple types.  Since the
article talks about isomorphisms, the encoding should be tight.

Suppose we have an encoding of [a] -> [a] functions, which represents
each [a] -> [a] function as a first-order data type, say. The encoding
should be injective, mapping g and g2 above to different
encodings. But for the purposes of characterizing scramblings of a
list of size n, the two encodings should be regarded equivalent. So,

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