Here is a malformed question that occurs to me just now:
In a language like python i can form instances of a product type using tuples, but making instances of a sum (variant) type ... ?
How is it that the duality is broken when I no longer have variable declarations? Or is every object instance already a variant (of every possible subclass) ?
In object-oriented languages, I think the closest analogue to sum types are type hierarchies. It's more verbose and cumbersome, but you can get the same effect through dynamic polymorphism. Consider the example from the post:
type AST = Expr of ...
| Stmt of ...
| Comment of ...
let rec evaluate e =
match e with
| Expr # evaluate the ... of an Expr
| Stmt # ditto
| Comment # ditto
In the object-oriented world, I think this most naturally maps to (forgive me for not using Python syntax; I love Python, but I rarely do much OO in it):
class AST {
public:
virtual bool evaluate() = 0; // again, forgive the C++ syntax
};
class Expr: public AST {
public:
virtual bool evaluate()
{
// Expr specific evaluate code
}
};
class Stmt: public AST {
public:
virtual bool evaluate()
{
// Ditto
}
};
class Comment: public AST {
public:
virtual bool evaluate()
{
// Ditto
}
};
As seanmcdirmid points out, this is not the same, as the discrimination happens at runtime. But, I think it's the closest analogue. Also note that in the algebraic data-type world, we normalize on functions. That is, what to do in order to evaluate any given AST exists in the one evaluate function. But, in the OO world, we normalize on the types; how we evaluate a given AST is spread out over the many evaluate methods.
Yes. I was thinking there was an important distinction about knowing the types statically, and being able to make sure that all options have been handled. But, if in the interface you say "everyone must implement this method" (as I did), and then you don't, the compiler will issue an error in much the same way.
What I was getting at is that when we use the algebraic version of AST, we know exhaustively what forms it can take. The definition tells us. In the OO version, this is note true. If we restrict our use to the polymorphic functions, then this is fine. But if we try to inspect the OO version of AST, ask its true type, and then act on it, then we may get into trouble.
In a language like Python you can consider there to be one type called, say, ThePythonType, or which all types are a variant. Then all objects have type ThePythonType.
A more useful type system would have "union types". They differ from sum types in that you can construct a union type from existing types in an ad-hoc manner. For example, if you have a function that returns a string or nil you can construct a union type "Union of string or nil". To do this with a sum type you'd have to have a common super type of string and nil and no other types.
There are also intersection types, about which I know very little.
> To do this with a sum type you'd have to have a common super type of string and nil and no other types.
I'm not sure you understand sum types. Sum types subsume normal unions; sum types and unions are essentially the same thing and capturing this kind of return type in Haskell or ML is extremely easy. "Supertypes" are completely unrelated and subtyping is not actually needed in languages with sum types (for example Haskell has sum types but doesn't have any real form of subtyping).
For example in Haskell you'd use the `Maybe` type constructor. If your type returns a `Maybe String` that means it either returns a string or nothing.
Sort of. You can effectively do so in a language like Java, which allows you to inspect the types of objects at runtime. For example if you have a class `Foo` with two subclasses `SubA` and `SubB` you could do something like this:
public static void test(Foo f) {
if (f isinstanceof SubA) {
SubA a = (SubA)f;
/* ... */
} else if (f isinstanceof SubB) {
SubB b = (SubB)f;
}
}
This works okay but you have to implement all the machinery yourself, and for a statically typed language Java is notoriously type-unsafe so if you make a mistake with your casts you're basically not going to find out until you get an exception at runtime. And of course there might be other existing subclasses of `Foo` which means that you get no compile-time guarantee that you didn't miss a possible case.
Standard ML approaches subtyping in a very different way that renders this kind of approach impossible. It gives you less freedom in that it eliminates downcasting, but that ultimately results in a safer language. Of course Standard ML has proper sum types with all of their compile-time guarantees so there's little reason for it anyway.
The way I was envisioning it, there needn't be any down-casting at the language level. Particularly if you want to close the sum type (which you need anyway if you want exhaustiveness checks):
public static void test(MaybeFoo mf) {
switch(mf.type) {
case MaybeFoo.NOTHING:
/* ... */
break;
case MaybeFoo.JUST:
Foo f = mf.getFoo();
/* ... */
break;
}
}
Still has exhaustiveness checking and no downcasts. Still can throw if you use the wrong get* under one of those branches, so I'm certainly not saying it's ideal, but I think it legitimately expresses a sum type and can be implemented in most any language with small adjustments.
This is sort of a different way of saying the same thing: this is effectively no different from the standard tagged-union approach, with the exact same danger of calling the wrong `get` function and triggering a runtime error that you have in C et al. Don't get me wrong, it's certainly a valid representation of a sum type, but when you hear fans of Haskell and ML talking about how useful sum types are, it's generally because the compiler provides some support to ensure that they're safe and usable.
I would note that when I'm dealing with tagged unions in C, as soon as the operation gets at all complicated (more than maybe 3 lines?) I try to pass it off to another function operating on the internal value which restores safety outside the boilerplate. I agree that support for a true tagged union is ideal, though. (Actually, this might make a good attribute in GCC...)
I'd never thought about this before in the context of Java. IIRC, Java can not have type-parameterized enums (i.e. generic enums), so a Maybe<T> isn't possible. I hope I'm wrong — it's been a while since I did much Java.
Well, you certainly can by expanding the parameterization manually like I did above. I think you're otherwise correct, though - I don't think Java will presently do it for you.
Checking whether mf.type is NOTHING or JUST is your boolean. It's linked by providence to whether or not getFoo() is a runtime exception. With a true ADT getFoo would not be accessible unless mf.type == JUST, but in this example it's up to the programmer to enforce that providence.
So that's the blindness. The pre/post conditions around how mf.type relates to its surroundings cannot be expressed first-class in the code. You are required to do risky things like call potentially partial code.
So it's specifically the separation of the check and the action? I totally agree this is undesirable; I understood "boolean blindness" to be closer to "bare primitive types lack meaning", specialized to Bool.
> In a language like Python you can consider there to be one type called, say, ThePythonType, or which all types are a variant. Then all objects have type ThePythonType.
This is only if you reject the notion of a dynamic type (as most type theorists do, but programmers do not). Otherwise, Python has a rich type system that is checked at run-time. It lacks sum types, but we could easily imagine a dynamically checked realization.
A good way of getting at this is to call things at runtime "classes" or "tags" to make it apparent that their discrimination happens at runtime, not compile-time. The problem is that if you call both things types, you quickly run into a situation where you are essentially talking about different concepts entirely.
Note that most sum-types are implemented by having a runtime tag on which you can discriminate.
I don't think type theorists reject that at all. They just don't call them "types". Which I don't know why anyone would care in the context of a technical argument.
If an "average programmer" here and a type theorist spent 1 minute at the beginning of the conversation normalizing terms then nobody would have any argument here. I don't think anyone tries to claim that "types" and "static types" (or, in the type theorist's parlance, "tags" and "types") are the same... they just try to argue whether or not it's "right" to call one or the other by the name "type".
Most type theorists reject that dynamic typing is meaningful and not a misnomer, hence Noel Welsh's assertion that Python has only one type. It doesn't start the conversation off very well.
I don't think anyone argues that Python has more than one (static) type. It however has a dynamic type system. Significantly, static types and dynamic types do not behave the same way and their corresponding theories have somewhat minor overlap.
In particular, many things I can state about static types do not carry over to dynamic types and visa versa.
I believe many conversations get derailed early because among static type researchers, static types are known merely as "types" and dynamic types as "tags" or "classes". There are meaningful ways to compare these two kinds of things but somehow or another they ought to have different words since confusing the concepts will inhibit understanding quite severely.
All but the "if we just listened to each other, we'd understand each other." The type theorist's and programmer's different perspective s means basically that we'd fight over the meaning of type for our entire conversation.
It wasn't "if we listened", but "if we normalized terms". I agree that getting people to normalize terms with the aim of successfully communicating can be difficult...
"Arrow theory" here is better known as Category Theory. I highly, highly recommend learning more about it if you're interested. I cannot understate its beauty.
As a starter guide, I highly recommend reading Lawvere's Conceptual Mathematics.
Here is a malformed question that occurs to me just now:
In a language like python i can form instances of a product type using tuples, but making instances of a sum (variant) type ... ?
How is it that the duality is broken when I no longer have variable declarations? Or is every object instance already a variant (of every possible subclass) ?