Why are Haskell algebraic data types "closed"?

Correct me if I'm wrong, but it seems like algebraic data types in Haskell are useful in many of the cases where you would use classes and inheritance in OO languages. But there is a big difference: once an algebraic data type is declared, it can not be extended elsewhere. It is "closed". In OO, you can extend already defined classes. For example:

data Maybe a = Nothing | Just a

There is no way that I can somehow add another option to this type later on without modifying this declaration. So what are the benefits of this system? It seems like the OO way would be much more extensible.

The answer has to do with in what ways the code is easy to extend, a tension between classes and algebraic data types that Phil Wadler dubbed "the expression problem":

• With algebraic data types,

  • It is very cheap to add a new operation on things: you just define a new function. All the old functions on those things continue to work unchanged.

  • It is very expensive to add a new kind of thing: you have to add a new constructor an existing data type, and you have to edit and recompile every function which uses that type.

• With classes,

  • It is very cheap to add a new kind of thing: just add a new subclass, and as needed you define specialized methods, in that class, for all the existing operations. The superclass and all the other subclasses continue to work unchanged.

  • It is very expensive to add a new operation on things: you have to add a new method declaration to the superclass and potentially add a method definition to every existing subclass. In practice, the burden varies depending on the method.

So, algebraic data types are closed because a closed type supports certain kinds of program evolution well. For example, if your data types define a language, it is easy to add new compiler passes without invalidating old ones or changing the data.

It is possible to have "open" data types, but except under carefully controlled circumstances, the type checking becomes difficult. Todd Millstein did some very beautiful work on a language design that supports open algebraic types and extensible functions, all with a modular type checker. I found his paper a great pleasure to read.

The fact that ADT are closed makes it a lot easier to write total functions. That are functions that always produce a result, for all possible values of its type, eg.

maybeToList :: Maybe a -> [a]
maybeToList Nothing  = []
maybeToList (Just x) = [x]

If Maybe were open, someone could add a extra constructor and the maybeToList function would suddenly break.

In OO this isn't an issue, when you're using inheritance to extend a type, because when you call a function for which there is no specific overload, it can just use the implementation for a superclass. I.e., you can call printPerson(Person p) just fine with a Student object if Student is a subclass of Person.

In Haskell, you would usually use encapsulation and type classes when you need to extent your types. For example:

class Eq a where
   (==) :: a -> a -> Bool

instance Eq Bool where
  False == False = True
  False == True  = False
  True  == False = False
  True  == True  = True

instance Eq a => Eq [a] where
  []     == []     = True
  (x:xs) == (y:ys) = x == y && xs == ys
  _      == _      = False

Now, the == function is completely open, you can add your own types by making it an instance of the Eq class.

---

Note that there has been work on the idea of extensible datatypes, but that is definitely not part of Haskell yet.

If you write a function like

maybeToList Nothing = []
maybeToList (Just x) = [x]

then you know that it's never going to produce a runtime error because you've covered all of the cases. This ceases to be true as soon as the Maybe type is extensible. In those cases where you need an extensible type (and they are rarer than you think), the canonical Haskell solution is to use a type class.

Check "Open data types and open functions" http://lambda-the-ultimate.org/node/1453

> In object-oriented languages, it is > easy to extend data by defining new > classes, but it is difficult to add > new functions. In functional > languages, the situation is reversed: > adding new functions poses no > problems, but extending data (adding > new data constructors) requires > modifying existing code. The problem > of supporting both directions of > extensibility is known as the > expression problem. We present open > data types and open functions as a > lightweight solution to the expression > problem in the Haskell language. The > idea is that constructors of open data > types, and equations of open functions > can appear scattered throughout a > program. In particular, they may > reside in different modules. The > intended semantics is as follows: the > program should behave as if the data > types and functions were closed, > defined in one place. The order of > function equations is determined by > best-fit pattern matching, where a > specific pattern takes precedence over > an unspecific one. We show that our > solution is applicable to the > expression problem, generic > programming, and exceptions. We sketch > two implementations. A simple one, > derived from the semantics, and one > based on mutually recursive modules > that permits separate compilation.

Some excellent answers on this (admittedly old) question, but I feel I have to throw my few cents in.

> There is no way that I can somehow add another option to this type later on without modifying this declaration. So what are the benefits of this system? It seems like the OO way would be much more extensible.

The answer to this, I believe, is that the sort of extensibility that open sums give you is not always a plus, and that, correspondingly, the fact that OO forces this on you is a weakness.

The advantage of closed unions is their exhaustiveness: if you have fixed all the alternatives at compilation time, then you can be certain that there will be no unforeseen cases that your code cannot handle. This is a valuable property in many problem domains, for example, in abstract syntax trees for languages. If you're writing a compiler, the expressions of the language fall into a predefined, closed set of subcases—you do not want people to be able to add new subcases at runtime that your compiler doesn't understand!

In fact, compiler ASTs are one of the classic Gang of Four motivating examples for the Visitor Pattern, which is the OOP counterpart to closed sums and exhaustive pattern matching. It is instructive to reflect on the fact that OO programmers ended up inventing a pattern to recover closed sums.

Likewise, procedural and functional programmers have invented patterns to obtain the effect of sums. The simplest one is the "record of functions" encoding, which corresponds to OO interfaces. A record of functions is, effectively, a dispatch table. (Note that C programmers have been using this technique for ages!) The trick is that there is very often a large number of possible functions of a given type—often infinitely many. So if you have a record type whose fields are functions, then that can easily support an astronomically large or infinite set of alternatives. And what's more, since records are created at runtime and can be done flexibly based on runtime conditions, the alternatives are late bound.

The final comment I'd make is that, in my mind, OO has made too many people believe that extensibility is synonymous with late binding (e.g., the ability to add new subcases to a type at runtime), when this just isn't generally true. Late binding is one technique for extensibility. Another technique is composition—building complex objects out of a fixed vocabulary of building blocks and rules for assembling them together. The vocabulary and rules are ideally small, but designed so that they have rich interactions that allow you to build very complex things.

Functional programming—and the ML/Haskell statically typed flavors in particular—have long emphasized composition over late binding. But in reality, both kinds of techniques exist in both paradigms, and should be in the toolkit of a good programmer.

It's also worth noting that programming languages themselves are fundamentally examples of composition. A programming language has a finite, hopefully simple syntax that allows you to combine its elements to write any possible program. (This in fact goes back to the compilers/Visitor Pattern example above and motivates it.)

First, as counterpoint to Charlie's answer, this isn't intrinsic to functional programming. OCaml has the concept of open unions or polymorphic variants, which essentially do what you want.

As for why, I believe this choice was made for Haskell because

• this lets types be predictable - their are only a finite number of constructors for each
• it's easy to define your own types.
• many Haskell functions are polymorphic, and classes let you extend custom types to fit function parameters (think Java's Interfaces).

So if you'd rather have a data Color r b g = Red r | Blue b | Green g type, it's easy to make, and you can easily make it act like a monad or a functor or however other functions need.

Another (more or less) intuitive way of looking at data types and typeclasses versus object oriented classes is the following:

A class Foo in an OO language represents both a concrete type Foo as well as the class of all Foo-types: those which are directly or indirectly derived from Foo.

In OO languages, you just happen to implicitly program against the class of Foo-types which allows you to "extend" Foo.

Okay, by "open" here you're meaning "can be derived from" and not open in the sense of Ruby and Smalltalk, where you can extend a class with new methods at run-time, right?

In any case, notice two things: first, in most OO languages that are primarily inheritance-based, there is a way to declare a class to restrict it's ability to be inherited. Java has "final", and there are hacks for this in C++. So on that, it's just making as the default an option in other OO languages.

Second, you can still create a new type that uses the closed ADT and adds other methods or different implementations. So you aren't really restricted in that way. Again, they seem to formally have the same strength; what you can express in one can be expressed in the other.

The real thing is that functional programming really is a different paradigm ("pattern"). If you go into it with the expectation that it ought to be like an OO language, you'll be surprised regularly.