Making this super easy and short to express makes a lot of designs clearer.
There's also an interesting tradeoff here between sum types and abstract interfaces:
- Abstract interfaces: "We have an unknown number of cases/subclasses, but they each support this fixed set of methods." The set of implementations is open, but the set of operations is harder to update.
- Sum types (aka Rust enums): "We know all the cases, but there may be many different methods." The set of cases is harder to extend, but the set of methods manipulating them is completely open.
A good example of the latter pattern is actually LLVM! LLVMs' intermediate representation is a essentially a small assembly language with a fixed set of operations (and an escape hatch). But because the set of basic instructions is small and known, it's possible to write dozens and dozens of optimization passes that each manipulate that small set of instructions.
And a surprising number of programs work well with an LLVM-like structure.
So in a language like Rust, which supports both abstract interfaces (via traits) and sum types (via enums), the tradeoff is whether you think you have many different types of values that you'll manipulate in a limited number of ways, or a small number of types of values that you'll manipulate in many different ways.
(Oh, and the metaphor of "algebraic types" goes deeper than you might think; there are some super interesting data structures based on taking the "derivatives" of simple algebraic types.)
We use something similar in Solvespace. Where one might have used polymorphism, there is one class with a type enum and a bunch of members that can be used for different things depending on the type. Definitely some disadvantages (what does valA hold in this type?) but also some advantages from the consistency.
However, if you consider programming to not only be about expressing what you can do to stateful objects but also expressing values and their operations, then algebraic data types and traits/modules/type classes become the natural basic vocabulary as opposed to classes and interfaces. When dealing with first-order algebraic data types, products (i.e. records) and coproducts (i.e. sum types) are unavoidable. A list is one of the simplest algebraic data types which use both products and coproducts:
data List a = Nil | Cons a (List a)
One trait of lists is that lists of totally ordered things are themselves totally ordered using lexicographic ordering, and this is in Haskell expressed as a type class instance
instance Ord a => Ord (List a) where ...
Crucially, the type class is removed from the definition of what a list is, which enables us to still talk about lists of e.g. functions which do not have an order on them. These lists do not have the Ord trait, but they are still lists.
Another important distinction between traits and interfaces is that some behavior of value types consist of simply identifying special values. For example, the Monoid type class in Haskell is
typeclass Monoid a where
mempty :: a
mappend :: a -> a -> a
Many languages now have proper support for both objects and values in that they have better primitives for defining both products and coproducts, but I still think that most mainstream languages are missing proper primitives for defining traits.
Just kidding but algebraic types are a great abstraction paradigm,
raxxorraxor•2d ago
My math prof did say the exact same while torturing students with question about proofs about their arcane set of arbitrary numbers and if they can be considered a field or ring or a group or everything at the same time.
Sure, just some + and *...
And sure, for a programmer it is mostly about which operations are defined on the type. But with just a few tweaks here and there you can transform a tool into a torture device...
Jokes aside, I think this is a good explanation about the concepts and parallels.
HelloNurse•1d ago
A toolbag of abstract theory and tools that can be straightforwardly applied to any "arcane set" is the value that upgrading from arithmetic to algebra provides.