So we're going to see, and what we're going to see is mostly examples of using kind of

prologue-like techniques in learning because horn clauses are nice because they are kind

of nicely directed and logic programming-like techniques are nice because they're quite

directed, that cuts down on search spaces. And the knowledge will really be mostly be used to

reduce the sizes of the hypothesis spaces. It's clear that if we're in logic we're essentially

learning things that relate to Boolean functions. And we know what the problem there is and we've

looked at the sizes in the discussion of PAC learning and we're going to see those kind of

things here as well. And here we're going to use knowledge to reduce hypothesis spaces and get

around this kind of PAC learning conundrum which says, well, if you want to learn something you

have to look at all possible examples, which doesn't help you because then you can just,

then you have a tabulation of everything that can happen. So let's make an example. Let's look

at trying to extract general rules from individual examples. Essentially like the caveman example,

they're seeing exactly once somebody using a good idea and then you say, ah, I can do this too.

Okay, and then you usually come up with a generalized rule. It doesn't help you to learn

that Zog can grill a fish with a stick. You usually learn something, ah, it's a good idea

to grill my food, that's a generalization from fish to food, on a stick. We also transform,

transferring essentially Zog can burn, no, grill his food without burning his fingers to, I can

grill my food, whatever it is, without burning my fingers, which is really what you care about.

You might care less for Zog's fingers than your own fingers. And these two things, these two

kind of generalizations and transfers are something that in learning you want to do. And in effect,

this transfer from Zog can grill without burning his fingers to I can grill without burning my

fingers is also related to generalization because the rule you really want to have is anybody can

grill any food without burning the respective fingers. And so we have observations, we want to

learn from them. And the first thing I would like to do is learn from single examples. And

since caveman life is relatively complicated, we're going to do something very simple, namely

derivatives. Okay, so the example is if you want to differentiate and simplify algebraic expressions,

then you have something like this. You have x, the polynomial x squared, and you want to differentiate

it with respect to x to get to x. Okay, that sounds like something you can write down in logic. And you

might actually say if you had a reasoning system, which you can query, then you could ask, then what

is the derivative of x squared with respect to x. And you give like in a prologue like system,

you give it you give a variable D, which then hopefully is bound to two times x. There are a

couple of problems in this and I've, I've, the answer, the example is made that way, is that

actually when you're doing logic, you're not actually getting 2x. But what you're getting is

x to the 2 minus 1 times 2 times 1, if you actually follow all the rules. So what you want is not only

a general rule, you need not only the general rule, but simplification rules that from this guy here

actually gets the simplified version 2x. And indeed you can do that. You can write down all the rules,

the rules you actually know, and we're going to see some of them. And then you can prove, you remember

there's always a proof behind an answer in logic, which is a nice thing, which in this case happens

to be relatively big, right? If you have kind of a write down, write a standard proof check and proof,

their improver, then you'll get things like 136 proof steps and in a huge tray which has 99 dead

end branches. Okay, and you can see that if you're only being told the rules, then you have to think

quite a lot. Okay? And as you get more familiar with derivatives, things become easier, right?

What you're doing is you're usually memorizing, or in CS we would say memorizing, stashing away little

nice things that make your life easier. So what you do in learning and in exploring a theory like

differentiability in, was it probably Math 1 or something like that, or high school, where you're

creating a little kind of a database of input-output rules, things that you remember and can actually

use without thinking much about them. And one of them eventually is something like that, differentiate

x squared to 2x. I'm pretty sure that everybody of you has this kind of a rule stashed away somewhere.

Okay, so that's really what we do. We have these general rules and we would like to have them cover

an entire class of rules. And one of, if we really write it down, a rule might actually look like