As I said in the last post, Church came up with λ-calculus, which looks like it’s a great formal model of computation. But – there was a problem. Church struggled to find a model. What’s a model, and why would that matter? That’s the point of this post. To get a quick sense of what a model is, and why it matters?
A model is basically a mapping from the symbols of a logical system to some set off objects, such that all statements that you can prove in the logical system will be true about the corresponding objects. Note that when I say object here, I don’t necessarily mean real-world physical objects – they’re just something that we can work with, which is well-defined and consistent.
Why does it matter? Because the whole point of a system like λ-calculus is because we want to use it for reasoning. When you have a logical system like λ-calculus, you’ve built this system with its rules for a reason – because you want to use it as a tool for understanding something. The model provides you with a way of saying that the conclusions you derive using the system are meaningful. If the model isn’t correct, if it contains any kind of inconsistency, then your system is completely meaningless: it can be used to derive anything.
So the search for a model for λ-calculus is really important. If there’s a valid model for it, then it’s wonderful. If there isn’t, then we’re just wasting our time looking for one.
So, now, let’s take a quick look at a simple model, to see how a problem can creep in. I’m going to build a logic for talking about the natural numbers – that is, integers greater than or equal to zero. Then I’ll show you how invalid results can be inferred using it; and finally show you how it fails by using the model.
One quick thing, to make the notation easier to read: I’m going to use a simple notion of types. A type is a set of atoms for which some particular one-parameter predicate is true. For example, if is true, I’ll say that x is a member of type P. In a quantifier, I’ll say things like to mean . Used this way, we can say that P is a type predicate.
How do we define natural numbers using logic?
First, we need an infinite set of atoms, each of which represents one number. We pick one of them, and call it zero. To represent the fact that they’re natural numbers, we define a predicate , which is true if and only if x is one of the atoms that represents a natural number.
Now, we need to start using predicates to define the fundamental properties of numbers. The most important property of natural numbers is that they are a sequence. We define that idea using a predicate, , where is true if and only if x = y + 1. To use that to define the ordering of the naturals, we can say: .
Or in english: every natural number has a successor – you can always add one to a natural number and get another natural number.
We can also define predecessor similarly, with two statements:
So every number has a predecessor, and every number has a successor, and x is the predecessor of y if y is the successor of x.
To be able to define things like addition and subtraction, we can use successor. Let’s define addition using a predicate Sum(x,y,z) which means “z = x + y”.
Again, in english: for any two natural numbers, there is a natural number that it their sum; x + 0 always = x; and for any natural number, x + y = z is true if (x + 1) + (y – 1) = z.
Once we have addition, subtraction is easy:
That’s: x-y=z if and only if x=y+z.
We can also define greater than using addition: