- cross-posted to:
- [email protected]
- cross-posted to:
- [email protected]
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.
Yes, this pattern is covered in my post on algebraic data types (linked at the start of the object algebras post.) The problem you mention about adding new data variants is exactly what object algebras solves. With object algebras data and functions are only co-located at the smallest possible granularity, so the desired functions and the desired data types can be composed as needed.
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 expression2 + 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: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:
The ability to express multi-sorted algebras
The ability to extend algebras with new sorts
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.