On Tue, 2007-12-18 at 23:04 -0800, Don Stewart wrote:

jules:

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Brad Larsen wrote:

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Hi there list,

How would one go about creating a new type for a subset of the integers, for (contrived) example just the even integers? I was thinking of making a new type

newtype EvenInt = EvenInt Integer

but the problem with this is that it accepts any integer, even odd ones. So to prevent this, the module does not export the constructor for it---rather, the module exports a function `mkEvenInt' that creates an EvenInt if the given value is acceptable or raises an error otherwise.

What's the right way to do this? Thanks!

There are two ways:

(1) Have a representation which admits invalid values, and provide combinators which only perfect validity, and prove that consumers using your combinators can't produce invalid values.

(2) Have a cleverly designed representation such that every representation is valid.

An example here, for (2) would be to store n/2; there is a bijection between 'Integer' and 'EvenInt' given by n/2.

In real, more complex problems, (2) often isn't possible and we resort to (1). E.g. the representation of balanced trees (AVL? RedBlack?) admits invalid values (both unbalanced trees and out-of-order trees) and we rely on the reduced set of combinators never to generate one.

1) is always a challenge to particular types (heh) of people to do 2)

And indeed there are two distinct ways of enforcing balance in AVL trees using the type system. And of course, it would be "trivial" to do with dependent types. Usually the problem isn't so much that it is not possible, but that it is cumbersome and/or inefficient. http://www.haskell.org/pipermail/haskell/2003-April/011621.html http://www.haskell.org/pipermail/haskell/2003-April/011693.html (random) Also that month, lambdabot 1.0! http://www.haskell.org/pipermail/haskell/2003-April/011582.html

Brad Larsen wrote:

On Wed, 19 Dec 2007 02:00:53 -0500, Jules Bean wrote:

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Brad Larsen wrote:

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Hi there list, How would one go about creating a new type for a subset of the integers, for (contrived) example just the even integers? I was thinking of making a new type newtype EvenInt = EvenInt Integer but the problem with this is that it accepts any integer, even odd ones. So to prevent this, the module does not export the constructor for it---rather, the module exports a function `mkEvenInt' that creates an EvenInt if the given value is acceptable or raises an error otherwise. What's the right way to do this? Thanks!

There are two ways:

(1) Have a representation which admits invalid values, and provide combinators which only perfect validity, and prove that consumers using your combinators can't produce invalid values.

(2) Have a cleverly designed representation such that every representation is valid.

An example here, for (2) would be to store n/2; there is a bijection between 'Integer' and 'EvenInt' given by n/2.

To make sure I understand, an example of (1) would be to export only a ``smart constructor'' that somehow converts invalid values to valid ones (say, add 1 to arguments that are odd)?

I just meant one that throws a runtime error if they are not.

In your example of 2, how would you go about storing n/2? Store just n as in (newtype EvenInt = EvenInt Integer) and then write all functions that deal with EvenInts so that they account for the division by two?

You would store n/2. So "8" would be represented as EvenInt 4, "14" as EvenInt 7. Then you write the num instance to account for the division. Check out the num instance in Data.Fixed for an example.

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In real, more complex problems, (2) often isn't possible and we resort to (1). E.g. the representation of balanced trees (AVL? RedBlack?) admits invalid values (both unbalanced trees and out-of-order trees) and we rely on the reduced set of combinators never to generate one.

Jules

In my particular case, or what I actually want to do, is define a finite segment of the integers (0-42, say) as a new type and have that checked at compile time. Any way of doing this w/o defining Peano numbers or a whole bunch of nullary constructors (i.e. I'm hoping to be able to define a type whose constructor will accept only Integer arguments between 0 and 42)?

Not at compile-time, no. The best you can do is runtime error. There is no compile-time facility to check that an Int lies in a particular range. Consider, for example, that the int might not be known at compiletime: do id <- myThreadID return (MkSmallInt id) ^^ how can the compiler know? In general, of course, the question of calculating if an int is under 42 at compile time is the question of partial evaluation. Partial evaluation at compile time is flat-out impossible for IO actions, and although it's possible for pure functions, GHC doesn't do it to any significant extent. (It would make the compiler potentially non-terminating, of course!) Jules