Re: [isabelle] Type classes vs. locales?



Dear Holden,

In your situation of abstract algebra, locales are better than type classes, as you want to talk about carriers, sub-structures, and existence of structures. However, some of your comments need some refinement.

>     2. Anything involving types that depend on 2+ types cannot be expressed
>     using typeclasses (ex. (a)-(b), a product of 2 arbitrary rings cannot be
>     interpreted as a type of class ring, "arbitrary" meaning I haven't chosen a
>     specific instance).
This is not correct. You can define a type ('a, 'b) prod_ring (e.g., via typedef) and do an instance declaration for ring:

instantiation prod_ring (ring, ring) ring begin
(* definition of ring operations on product *)
instance (* proof *)
end

This declaration is valid for any two types with the ring structure without saying which ring they are.


>     4. However, there is no automated way to construct instances of
>     typeclasses (creating classes is allowed only at top-level, so there is no
>     dependence on parameters) or talk about their existence. Using a typeclass
>     representing finite fields, you can't say "for every prime p, there exists
>     a field F_p with p elements," you can only say "given a finite field with p
>     elements..."
You can phrase a statement like "for every prime p, there exists a field F_p with p elements" on the type level as follows. Define prime :: "nat => nat" as the distinct enumeration of all the primes, e.g., prime n is the n-th prime. Then, define a type 'a F_p as a type that contains prime CARD('a) elements. Then, you can do

instantiation F_p (finite) field begin

Since Isabelle has types with exactly n elements for any natural number n > 0, you get what you want. But I admit that this is a bit cumbersome.



For completeness, there are a few cases where type classes are superior to locales.

1. Code generation within locales with assumptions essentially does not work. You can only generate code for interpretations for which you have to manually redefine the constants introduced in the locale. With type classes, the code generator takes care of overloaded operations.

2. Overloading of polymorphic operations. If an overloaded operation (like addition in a ring) occurs with multiple different instances, the locale approach requires you to use a different constant for each instance. In that case, you normally have to introduce a locale that combines multiple instances, too. Here's an example with type classes:

lemma fixes f :: "'a :: ring => 'b :: ring => 'c :: ring"
  assumes "!!a b c d. f (a + b) (c + d) = f a c + f b d"
  shows "..."

Best,
Andreas

On 26/08/14 15:24, Holden Lee wrote:
In Isabelle, (A) what can be done using locales that can't be done using
typeclasses, and (B) what can be done using locales that would be
inconvenient with typeclasses?

I've read about both of them, but am still trying to find a clear-cut
answer, and hoping some discussion will clarify things. Here are some of my
thoughts, which might or might not be correct, so comment and add your own
thoughts.

I'm mostly interested in constructions from abstract algebra. Here are some
examples to draw on: (a) ring homomorphism, (b) the product of 2 arbitrary
rings, (c) the polynomial algebra R[X_1,...,X_n] over R, (d) the finite
fields F_p, (e) the localization S^{-1}R (a ring constructed from R and a
multiplicative subset S of R, or any subset, if you prefer), combinations
and reinterpretations of these (ex. the vector space F_p[X_1,...,X_n] over
F_p)


    1. In general, locales are more powerful, but lack the automatic
    type-checking that types give.
    2. Anything involving types that depend on 2+ types cannot be expressed
    using typeclasses (ex. (a)-(b), a product of 2 arbitrary rings cannot be
    interpreted as a type of class ring, "arbitrary" meaning I haven't chosen a
    specific instance). Anything depending on 1 type can be expressed using
    typeclasses, if the representation has to be done using typeclasses every
    step of the way. Are there more theoretical restrictions?
    3. You can't use a parameter to construct a type (ex. finite field F_p).
    (i) Using locales, this isn't a problem because you can certainly construct
    a ring_scheme, etc. from a parameter, using an ordinary function. (ii)
    Using typeclasses, you would have to "bake in" a parameter, for instance
    have a function card_Fp::'a=>nat discarding the input of type 'a and just
    giving the cardinality of the field (actually here you can sidestep this
    and just return the CARD of 'a, but you can't do this in general, like
    recovering S in (e)).
    4. However, there is no automated way to construct instances of
    typeclasses (creating classes is allowed only at top-level, so there is no
    dependence on parameters) or talk about their existence. Using a typeclass
    representing finite fields, you can't say "for every prime p, there exists
    a field F_p with p elements," you can only say "given a finite field with p
    elements..." This is mathematically a bit odd because you hypothesize
    everything you need about the type you want (ex. finite fields) as
    assumptions in the class (including everything necessary to recover the
    original input data), whereas in the locale case you can construct it
    first, and then prove that it satisfies the assumptions you want.
    5. A result proven with locales can always be converted to the
    equivalent result with typeclasses by interpretation. However, a result
    proved with typeclasses cannot be converted to locales, because
    instantiation of a typeclass is not allowed inside a proof.


-Holden





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