So now let's go about the whole thing in a smarter idea, namely by introducing our consistency.

So we've already realized here that once I assign green to, which one is this again,

to Queensland, that already was basically a stupid move.

And the one flaw that forward checking has is that it basically only uses those variables

that we have already assigned a value to, to rule out values in the other variables.

And our consistency is basically a way to go about this whole thing smarter by not just

using those values that I've already assigned, but to get rid of all values that I already

know no matter if I've already assigned a value to them or not, that those values ultimately

are not going to be viable.

And the idea behind this is to use our consistency.

So here's the mathematical definition.

We say this whole thing is a binary predicate basically.

So we say a variable u is R consistent relative to some other variable v.

If and only if either we don't have a constraint between the two variables in the first place

or for every value that is viable for the variable u, there is at least one value in

the domain of v such that those two values together satisfy our constraints.

One thing to note here is that this is not asymmetric relation.

So it's possible for u to be R consistent relative to v and also v not being R consistent

to u.

That's possible.

So this is again just a summary of the basic idea.

I try to make sure that given those two variables, I know that every value in my domain of the

first variable, there is at least one possible assignment to the other value that satisfies

the constraint between the two.

So going back to the previous example, which I'll do in a moment.

So on the top middle, which is here, is v3 R consistent relative to v2.

Let's check this.

So we have three nodes, v1, v2, v3.

And our constraints are that v1 has to be smaller than v2 and v2 has to be smaller than

v3.

Now remember R consistency means for every value we're looking at R consistency of v3

relative to v2 in this picture here.

So remember it means for every value in v3, there has to be at least one value in v2 that

satisfies the constraint between them.

So is v3 R consistent relative to v2?

Any volunteers?

No?

Okay.

Ah, yeah.

There.

Because for the one and the two in v3, you can't find a value that is smaller in v2.

Yes exactly.

Right?

So just going over it in detail, we need to make sure for every one of those values, there's

one over here that satisfies the constraint.

So let's check the first one.

One, is there a value over here that makes that as consistent with one?

And well our constraint says v2 has to be smaller than v3, so nothing over here is smaller

than one, so that doesn't work.

Same thing for two.