Good.

And that also gives us essentially the math of this.

So we're defining constraint network, which has a set of variables, finitely many.

You have a finite set of domains, and you have a finite set of constraints, possibly

between any two variables.

What we're not allowing is constraints on itself, so no circles in our graphs, and the

constraints are essentially relations on the two domains.

And they can be any domain and any relation, the only thing we want is that we want the

graph to be, we want it to be unordered.

You can always just turn the relation around.

So binary constraint networks can be, no, binary constraint satisfaction problems can

be expressed as these constraint networks, and that's what we're going to do.

The thing you want to realize is that these relations really tell us what to do.

Remember relations mathematically are just sets of pairs, and we're going to think of

them as allowed pairs.

And if you think of, I don't know, think of your two domains, du and dv, so that actually

means we have kind of the relation being a subset here.

That means this constraints means that this u and that v are actually allowed together.

I don't know whether that helps you.

The, if you have the not equal relation, is essentially this part and that part are all

allowed except the diagonal.

So you could write all the constraints.

You want, and of course this is sufficient because you can prove that you can kind of,

with the idea I had on the last slide, you can actually get every higher order constraint

down to two, and unary constraints you can do by making use of, by restricting the main.

Okay, so let's see.

So how do we, using this stuff, formalize Sudoku?

So we've already seen we have variables for every cell here.

We have one, right?

We're going to call them vij, and they all have the same domain.

So mathematically, what are the constraints?

Your turn.

There can be no two numbers that are the same in a box, in one of the box.

Right.

And we write it down as Cuvijwij' maybe equals a set of pairs such that.

That's what I want to know.

We want to do the math of it.

We need to formalize this so that we can actually tell a system.

Imagine if you will, you're using a constraint solver like Minion, and you wanted to solve

Sudoku because you failed on one and want to be sure that there's no solution or something

like this.

Why do you write down?

What are the constraints mathematically?

Yes please, you have the ball, the cube.

Cuvij is not equal to wij.

Anybody happy with that?

What's the constraint you're expressing here?

For which v's and j's?

I and I, a strich is the same or j and j' is the same.

And if they are not, and for the cubes you have to do some, a modeloo, and so you can