this post was submitted on 11 Aug 2024
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[self post] Extensible TypeScript with Object Algebras (jnkr.tech)
submitted 3 months ago by jnkrtech@programming.dev to c/programming@programming.dev
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I think object algebras have huge potential to improve the way complex software is written but I've never seen them used in practice. I think one reason why is that the research paper which introduced them is pretty hard to read. This post is my attempt to change that.

I've been working on this post off and on for like two years so I'm really excited to share it with people. It is very long. There's a lot of ground to cover.

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[โ€“] expr@programming.dev 1 points 3 months ago (1 children)

Your post only showed adding functionality over the algebra, not new types on which the algebra operates (or "sorts", as they are otherwise known). In other words, you can't easily extend Expr to support Boolean logic in addition to addition itself. For a concrete example, how could you represent ternary operators like in the expression 2 + 2 == 4 ? 1 : 2, such that it's well typed and will never result in an exception? With GADTs, this is very simple to do:

data Expr a where
  Lit :: Int -> Expr Int
  Add :: Expr Int -> Expr Int -> Expr Int
  Eq :: Expr Int -> Expr Int -> Expr Bool
  If :: Expr Bool -> Expr Int -> Expr Int ->  Expr Int

eval :: Expr a -> a
eval expr = case expr of
  Lit n -> n
  Add a b -> eval a + eval b
  Eq a b -> eval a == eval b
  If p a b -> if eval p then eval a else eval b

-- >> eval example == 1 => true
example :: Expr Int
example =
  If ((Lit 2 `Add` Lit 2)  `Eq` Lit 4) (Lit 1) (Lit 2)
[โ€“] jnkrtech@programming.dev 1 points 3 months ago

lemmy seems to have lost my response to this, so I'll type it again and hope that it doesn't show up twice

There's three separate issues here:

  1. The ability to express multi-sorted algebras

  2. The ability to extend algebras with new sorts

  3. The ability to extend a single sort of an algebra with new variants

For point 1, object algebras do support this even though I didn't show it in the post:

interface ExprAlg<Num, Bool> {
    lit: (value: number) => Num;
    add: (left: Num, right: Num) => Num;
    eq: (left: Num, right: Num) => Bool;
    iff: (interrogee: Bool, then: Num, els: Num) => Num;
}

const evaluate: ExprAlg<number, boolean> = {
    lit: (value) => value,
    add: (left, right) => left + right,
    eq: (left, right) => left === right,
    iff: (interrogee, then, els) => interrogee ? then : els
}

function makeExample<Num, Bool>(alg: ExprAlg<Num, Bool>): Num {
    return alg.iff(
        alg.eq(
            alg.add(alg.lit(2), alg.lit(2)),
            alg.lit(4)),
        alg.lit(1),
        alg.lit(2))
}

console.log(makeExample(evaluate)); // prints 1

For point 2, you are correct that the original Java formulation of object algebras does not support data sort extensibility. I haven't tried to see if TS's more powerful generics change this or not.

For point 3, consider what happens if you want to add a new variant to the data type. In this case, add a Mult variant for multiplication. With GADTs this is not possible without modifying or duplicating the existing evaluation code, but I can do it with object algebras:

type ExtendedExprAlg<Num, Bool> = ExprAlg<Num, Bool> & {
    mult: (left: Num, right: Num) => Num;
}

const extendedEvaluate: ExtendedExprAlg<number, boolean> = Object.assign({}, evaluate, {
    mult: (left: number, right: number) => left * right
})

function makeExtendedExample<Num, Bool>( alg: ExtendedExprAlg<Num, Bool>): Num {
    const one = alg.mult(alg.lit(1), alg.lit(1));
    return alg.iff(
        alg.eq(one, makeExample(alg)),
        alg.lit(3),
        alg.mult(alg.lit(2), alg.lit(2))
    )
}

console.log(makeExtendedExample(extendedEvaluate)); // prints 3

This is the point of object algebras. ADTs and GADTs can't do this out of the box; this is the essence of the expression problem and is why more advanced techniques like final tagless encodings and datatypes a la carte exist.