Good Math, Bad Math

One of the most important things to recognize about Haskell's type system is that it's based on type inference. What that means is that in general, you don't need to provide type declarations. Based on how you use a value, the compiler can usually figure out what type it is. The net effect is that in many Haskell programs, you don't write any type declarations, but your program is still carefully type-checked. You can declare a type for any expression you want, but the only time you must is when something about your code is ambiguous enough that type inference can't properly select one type for an expression.

I'm not going to go into a lot of detail, but the basic idea behind Haskell's type system is that there's a basic propositional logic of types. Every type is represented as a statement in the type-logic. So as you'll see below, an integer is written as type Integer. A function from integers to integers has a type which is an implication statement: "Integer -> Integer", which you can read in the logic as "If the input type is Integer then the output type is Integer". The way that type inference works is that values with known types become propositions; the basic type inference process s just logical inference over the type logic.

The type logic is designed so that the inference process, when it analyzes an expression, it will generate the most general type; that is, when it's inferring a type for an expression, it will always pick the most general, inclusive type that can match the expression. And that leads to one complication for beginners: Haskell's type system is almost two different type systems - a basic type system, and a meta-type system. The meta-type system is based on something called type classes, which group related families of types together. The most general types that come up as a result of type inference are frequently based on type classes, rather than on specific concrete types.

A type-class in code is basically a predicate over a type variable. For example, the type of the parameters to the "+" operator must be a numeric type. But since "+" can be used on integers, floats, complex, rationals, and so on, the type of the "+" operator's parameters needs to be something that includes all of those. The way that Haskell says that is the type is "Num a => a" - that is, that the type a is a member of the type-class Num. The way to understand that is "Some type 'a' such that 'a' is a numeric type.".

The thing to remember is that essentially, a type-class is a type for types. A type can be thought of as a predicate which is only true for members of that type; a type-class is essentially a second-order predicate over types, which is only true for types that are members of the type-class. What type-classes do is allow you to define a general concept for grouping together a family of conceptually related types. Then you can write functions whose parameter or return types are formed using a type-class; the type class defines a predicate which describes the properties of types that could be used in the function.

So if we write a function whose parameters need to be numbers, but don't need to be a specific kind of number, we would write it to use a type-class that described the basic properties of numbers. Then you'd be able to use any type that satisfied the type-predicate defined by the type-class "Num".

If you write a function that uses numeric operations, but you don't write a type declaration, then Haskell's type-inference system will infer types for it based on the "Num" type-class: "Num" is the most general type-class of numbers; the more things we actually do with numbers, the more constrained the type becomes. For example, if we use the "/" operator, instead of inferring that the type of parameter must be an instance of the "Fractional" type-class.

With that little diversion out of the way, we can get back to talking about how we'll use types in Haskell. Types start at the bottom with a bundle of primitive atomic types which are built in to the language: Integer, Char, String, Boolean, and quite a few more. Using those types, we can construct more interesting types. For now, the most important constructed type is a function type. In Haskell, functions are just values like anything else, and so they need types. The basic form of a simple single-parameter function is "a -> b", where "a" is the type of the parameter, and "b" is the type of the value returned by the function.

So now, let's go back and look at our factorial function. What's the type of our basic "fact" function? According to Hugs, it's "Num a => a -> a".

Definitely not what you might expect. What's happening is that the system is looking at the expression, and picking the most general type. In fact, the only things that are done with the parameter are comparison, subtraction, and multiplication: so the system infers that the the parameter must be a number, but it can't infer anything more than that. So it says that the type of the function parameter is a numeric type; that is, a member of the type-class "Num"; and that the return type of the function is the same as the type of the parameter. So the statement "Num a => a -> a" basically says that "fact" is a function that takes a parameter of some type represented by a type variable "a" and returns a value of the same type; and it also says that the type variable "a" must be a member of the meta-type "Num", which is a type-class which includes all of the numeric types. So according to Haskell, as written the factorial function is a function which takes a parameter of any numeric type, and returns a value of the same numeric type as its parameter.

If we look at that type, and think about what the factorial function actually does, there's a problem. That type isn't correct, because factorial is only defined for integers, and if we pass it a non-integer value as a parameter, it will never terminate! But Haskell can't figure that out for itself - all it knows is that we do three things with the parameter to our function: we compare it to zero, we subtract from it, and we multiply by it. So Haskell's most general type for that is a general numeric type. So since we'd like to prevent anyone from mis-calling factorial by passing it a fraction (which will result in it never terminating), we should put in a type declaration to force it to take the more specific type "Integer -> Integer" - that is, a function from an integer to an integer. The way that we'd do that is to add an explicit type declaration before the function:

	
> fact :: Integer -> Integer
> fact 0 = 1 
> fact n = n*fact(n-1)

That does it; the compiler accepts the stronger constraint provided by our type declaration.

So what we've seen so far is that a function type for a single parameter function is created from two other types, joined by the "->" type-constructor. With type-classes mixed in, that can be prefixed by type-class constraints, which specify the type-classes of any type variables in the function type.

Before moving on to multiple parameter functions, it's useful to introduce a little bit of syntax. Suppose we have a function like the following:

poly x y = x*x + 2*x*y + y*y 

That definition is actually a shorthand. Haskell is a lambda calculus based language, so semantically, functions are really just lambda expressions: that definition is really just a binding from a name to a lambda expression.

In lambda calculus, we'd write a definition like that as:

  poly ≡ λ x y . x*x + 2*x*y + y*y

Haskell's syntax is very close to that. The definition in Haskell syntax using a lambda expression would be:

 poly = (\ x y -> x*x + 2*x*y + y*y)

The λ in the lambda expression is replaced by a backslash, which is the character on most keyboards that most resembles lambda; the "." becomes an arrow.

Now, with that out of the way, let's get back to multi-parameter functions. Suppose we take the poly function, and see what Hugs says about the type:

    poly x y = x*x + 2*x*y + y*y 

hugs>  Main> :type poly

hugs>  poly :: Num a => a -> a -> a

This answer is very surprising to most people: it's a two parameter function. So intuitively, there should by some grouping operator for the two parameters, to make the type say "a function that takes two a's, and returns an a"; something like "(a,a) -> a".

But functions in Haskell are automatically curried. Currying is a term from mathematical logic; it's based on the idea that if a function is a value, then you don't ever need to be able to take more than one parameter. Instead of a two parameter function, you can write a one-parameter function that returns another one-parameter function. Since that sounds really confusing, let's take a moment and look again at our "poly" example:

    poly x y = x*x + 2*x*y + y*y 

Now, suppose we knew that "x" was going to be three. Then we could write a special one-parameter function:

	poly3 y = 3*3 + 2*3*y + y*y

But we don't know "x". But what we can do is write a function that takes a parameter "x", and returns a function where all of the references to x are filled in, and given a y value, will return the value of the polynomial for x and y:

	polyn x = (\y -> x*x + 2*x*y + y*y)

If we call "polyn 3", the result is exactly what we wrote for "poly3". If we call "polyn a b", it's semantically exactly the same thing as "poly a b". (That doesn't mean that the compiler actually implements multi-parameter functions by generating single-parameter functions; it generates multi-parameters functions the way you'd expect. But everything behaves as if it did.) So what's the type of polyn? Well, it's a function that takes a parameter of type a; and returns a function of type "Num a => a -> a". So, the type of polyn is "Num a => a -> (a -> a)"; and since the precedence and associativity rules are set up to make currying convenient, the parents in that aren't necessary - that's the same as "Num a => a -> a -> a". Since "poly" and "polyn" are supposed to be semantically equivalent, that means that "poly"'s type must also be "Num a => a -> a -> a'".