Is there a Haskell equivalent of OOP’s abstract classes, using algebraic data types or polymorphism?

Yes, you are correct, you are looking for algebraic data types. There is a great tutorial on them at Learn You a Haskell.

For the record, the concept of an abstract class from OOP actually has three different translations into Haskell, and ADTs are just one. Here is a quick overview of the techniques.

Algebraic Data Types

Algebraic data types encode the pattern of an abstract class whose subclasses are known, and where functions check which particular instance the object is a member of by down-casting.

abstract class IntBox { }

class Empty : IntBox { }

class Full : IntBox {
    int inside;
    Full(int inside) { this.inside = inside; }
}

int Get(IntBox a) {
    if (a is Empty) { return 0; }
    if (a is Full)  { return ((Full)a).inside; }
    error("IntBox not of expected type");
}

Translates into:

data IntBox = Empty | Full Int

get :: IntBox -> Int
get Empty = 0
get (Full x) = x

Record of functions

This style does not allow down-casting, so the Get function above would not be expressible in this style. So here is something completely different.

abstract class Animal { 
    abstract string CatchPhrase();
    virtual void Speak() { print(CatchPhrase()); }
}

class Cat : Animal {
    override string CatchPhrase() { return "Meow"; }
}

class Dog : Animal {
    override string CatchPhrase() { return "Woof"; }
    override void Speak() { print("Rowwrlrw"); }
}

Its translation in Haskell doesn’t map types into types. Animal is the only type, and Dog and Cat are squashed away into their constructor functions:

data Animal = Animal {
    catchPhrase :: String,
    speak       :: IO ()
}

protoAnimal :: Animal
protoAnimal = Animal {
    speak = putStrLn (catchPhrase protoAnimal)
}

cat :: Animal
cat = protoAnimal { catchPhrase = "Meow" }

dog :: Animal
dog = protoAnimal { catchPhrase = "Woof", speak = putStrLn "Rowwrlrw" }

There are a few different permutations of this basic concept. The invariant is that the abstract type is a record type where the methods are the fields of the record.

EDIT: There is a good discussion in the comments on some of the subtleties of this approach, including a bug in the above code.

Typeclasses

This is my least favorite encoding of OO ideas. It is comfortable to OO programmers because it uses familiar words and maps types to types. But the record of functions approach above tends to be easier to work with when things get complicated.

I’ll encode the Animal example again:

class Animal a where
    catchPhrase :: a -> String
    speak       :: a -> IO ()

    speak a = putStrLn (catchPhrase a)

data Cat = Cat 
instance Animal Cat where
    catchPhrase Cat = "Meow"

data Dog = Dog
instance Animal Dog where
    catchPhrase Dog = "Woof"
    speak Dog = putStrLn "Rowwrlrw"

This looks nice, doesn’t it? The difficulty comes when you realize that even though it looks like OO, it doesn’t really work like OO. You might want to have a list of Animals, but the best you can do right now is Animal a => [a], a list of homogeneous animals, eg. a list of only Cats or only Dogs. Then you need to make this wrapper type:

{-# LANGUAGE ExistentialQuantification #-}

data AnyAnimal = forall a. Animal a => AnyAnimal a
instance Animal AnyAnimal where
    catchPhrase (AnyAnimal a) = catchPhrase a
    speak (AnyAnimal a) = speak a

And then [AnyAnimal] is what you want for your list of animals. However, it turns out that AnyAnimal exposes exactly the same information about itself as the Animal record in the second example, we’ve just gone about it in a roundabout way. Thus why I don’t consider typeclasses to be a very good encoding of OO.

And thus concludes this week’s edition of Way Too Much Information!