So, last post, I finally defined closed cartesian categories. And I alluded to the fact that the CCCs are, essentially, equivalent to the simply typed λ calculus. But I didn’t really talk about what that meant.
Before I can get to that, you need to know what λ calculus is. Many readers are probably familiar, but others aren’t. And as it happens, I absolutely love λ calculus.
In computer science, especially in the field of programming languages, we tend to use λ calculus a whole lot. It’s also extensively used by logicians studying the nature of computation and the structure of discrete mathematics. λ calculus is great for a lot of reasons, among them:
- It’s very simple.
- It’s Turing complete: if a function can be computed by any possible computing device, then it can be written in λ-calculus.
- It’s easy to read and write.
- Its semantics are strong enough that we can do reasoning from it.
- It’s got a good solid model.
- It’s easy to create variants to explore the properties of various alternative ways of structuring computations or semantics.
The ease of reading and writing λ calculus is a big deal. It’s led to the development of a lot of extremely good programming languages based, to one degree or another, on the λ calculus: Lisp, ML, Haskell, and my current favorite, Scala, are very strongly λ calculus based.
The λ calculus is based on the concept of functions. In the pure λ calculus, everything is a function; there are no values except for functions. In fact, we can pretty much build up all of mathematics using λ-calculus.
With the lead-in out of the way, let’s dive in a look at λ-calculus. To define a calculus, you need to define two things: the syntax, which describes how valid expressions can be written in the calculus; and a set of rules that allow you to symbolically manipulate the expressions.
Lambda Calculus Syntax
The λ calculus has exactly three kinds of expressions:
- Function definition: a function in λ calculus is an expression, written: λ param . body, which defines a function with one parameter.
- Identifier reference: an identifier reference is a name which matches the name of a parameter defined in a function expression enclosing the reference.
- Function application: applying a function is written by putting the function value in front of its parameter, as in x y to apply the function x to the value y.
There’s a trick that we play in λ calculus: if you look at the definition above, you’ll notice that a function (lambda expression) only takes one parameter. That seems like a very big constraint – how can you even implement addition with only one parameter?
It turns out to be no problem, because of the fact that functions are, themselves, values. Instead of writing a two parameter function, you can write a one parameter function that returns a one parameter function, which can then operate on the second parameter. In the end, it’s effectively the same thing as a two parameter function. Taking a two-parameter function, and representing it by two one-parameter functions is called currying, after the great logician Haskell Curry.
For example, suppose we wanted to write a function to add x and y. We’d like to write something like: λ x y . x + y. The way we do that with one-parameter functions is: we first write a function with one parameter, which returns another function with one parameter.
Adding x plus y becomes writing a one-parameter function with parameter x, which returns another one parameter function which adds x to its parameter: λ x . (λ y . x + y).
Now that we know that adding multiple parameter functions doesn’t really add anything but a bit of simplified syntax, we’ll go ahead and use them when it’s convenient.
One important syntactic issue that I haven’t mentioned yet is closure or complete binding. For a λ calculus expression to be evaluated, it cannot reference any identifiers that are not bound. An identifier is bound if it a parameter in an enclosing λ expression; if an identifier is not bound in any enclosing context, then it is called a free variable. Let’s look quickly at a few examples:
- λ x . p x y: in this expression, y and p are free, because they’re not the parameter of any enclosing λ expression; x is bound because it’s a parameter of the function definition enclosing the expression p x y where it’s referenced.
- λ x y.y x: in this expression both x and y are bound, because they are parameters of the function definition, and there are no free variables.
- λ y . (λ x . p x y). This one is a tad more complicated, because we’ve got the inner λ. So let’s start there. In the inner λ, λ x . p x y, y and p are free and x is bound. In the full expression, both x and y are bound: x is bound by the inner λ, and y is bound by the other λ. “p” is still free.
We’ll often use “free(x)” to mean the set of identifiers that are free in the expression “x”.
A λ calculus expression is valid (and thus evaluatable) only when all of its variables are bound. But when we look at smaller subexpressions of a complex expression, taken out of context, they can have free variables – and making sure that the variables that are free in subexpressions are treated right is very important.
Lambda Calculus Evaluation Rules
There are only two real rules for evaluating expressions in λ calculus; they’re called α and β. α is also called “conversion”, and β is also called “reduction”.
α is a renaming operation; basically it says that the names of variables are unimportant: given any expression in λ calculus, we can change the name of the parameter to a function as long as we change all free references to it inside the body.
So – for instance, if we had an expression like:
λ x . if (= x 0) then 1 else x^2
We can do an α to replace X with Y (written “α[x/y]” and get):
λ y . if (= y 0) then 1 else y^2
Doing α does not change the meaning of the expression in any way. But as we’ll see later, it’s important because without it, we’d often wind up with situations where a single variable symbol is bound by two different enclosing λs. This will be particularly important when we get to recursion.
β reduction is where things get interesting: this single rule is all that’s needed to make the λ calculus capable of performing any computation that can be done by a machine.
β basically says that if you have a function application, you can replace it with a copy of the body of the function with references to the parameter identifiers replaced by references to the parameter value in the application. That sounds confusing, but it’s actually pretty easy when you see it in action.
Suppose we have the application expression: (λ x . x + 1) 3. By performing a beta reduction, we can replace the application by taking the body x + 1 of the function, and substituting (or αing) the value of the parameter (3) for the parameter variable symbol (x). So we replace all references to x with 3. So the result of doing a beta reduction xs 3 + 1.
A slightly more complicated example is the expression:
λ y . (λ x . x + y)) q
It’s an interesting expression, because it’s a λ expression that when applied, results in another λ expression: that is, it’s a function that creates functions. When we do beta reduction in this, we’re replacing all references to the parameter y with the identifier q; so, the result is λ x . x + q.
One more example, just for the sake of being annoying. Suppose we have: (λ x y. x y) (λ z . z * z) 3
That’s a function that takes two parameters, and applies the first one to the second one. When we evaluate that, we replace the parameter x in the body of the first function with λ z . z * z; and we replace the parameter y with 3, getting: (λ z . z * z) 3. And we can perform beta on that, getting 3 * 3.
Written formally, beta says: λ x . B e = B[x := e] if free(e) ⊂ free(B[x := e])
That condition on the end, “if free(e) ⊂ free(B[x := e]” is why we need α: we can only do beta reduction if doing it doesn’t create any collisions between bound identifiers and free identifiers: if the identifier “z” is free in “e”, then we need to be sure that the beta-reduction doesn’t make “z” become bound. If there is a name collision between a variable that is bound in “B” and a variable that is free in “e”, then we need to use α to change the identifier names so that they’re different.
As usual, an example will make that clearer: Suppose we have a expression defining a function, λ z . (λ x . x+z). Now, suppose we want to apply it: (λ z . (λ x . x + z)) (x + 2). In the parameter (x + 2), x is free. Now, suppose we break the rule and go ahead and do beta. We’d get “λ x . x + x + 2“. The variable that was free in x + 2 is now bound! We’ve changed the meaning of the function, which we shouldn’t be able to do. If we were to apply that function after the incorrect β, we’d get (λ x . x + x + 2) 3. By beta, we’d get 3 + 3 + 2, or 8.
What if we did α the way we were supposed to?
First, we’d do an α to prevent the name overlap. By α[x/y], we would get λ z . (λ y . y + z) (x+2).
Then by β, we’d get “λ y . y + x + 2“. If we apply this function the way we did above, then by β, we’d get 3+x+2.
3+x+2 and 3+3+2 are very different results!
And that’s pretty much it. There’s another optional rule you can add called η-conversion. η is a rule that adds extensionality, which provides a way of expressing equality between functions.
η says that in any λ expression, I can replace the value f with the value g if/f for all possible parameter values x, f x = g x.
What I’ve described here is Turing complete – a full effective computation system. To make it useful, and see how this can be used to do real stuff, we need to define a bunch of basic functions that allow us to do math, condition tests, recursion, etc. I’ll talk about those in my next post.
It’l also important to point out that while I’ve gone through a basic definition of λ calculus, and described its mechanics, I haven’t yet defined a model for λ-calculus. That’s quite an important omission! λ-calculus was played with by logicians for several years before they were able to come up with a complete model for it, and it was a matter of great concern that although it looked correct, the early attempts to define a model for it were failures! And without a valid model, the results of the system are meaningless. An invalid model in a logical system like calculus is like a contradiction in axioms: it means that nothing that it produces is valid.