Okay, quiz is over.

Looks like there are no obvious red flags.

Right.

Let's get back to SML, recursion, all of those things.

Now that we've kind of looked a little bit and you had a little bit of experience of

playing with SML, I hope, and the quiz looks like you do, and you did, is I would like

to come to a quote that you hear very often.

That is, recursion, that's this thing that universities insist upon us understanding

and it doesn't exist out there in business.

That is partially true, mostly because many students and sometimes even instructors believe

this.

And yes, you can get by without recursion.

Just like you can get by as a doctor who is afraid of seeing blood.

You cannot become somebody who cuts into other people, which sometimes is necessary, but

you just can't do it.

So there are places where induction, recursion, all the stuff we're doing here are very important.

Maybe people don't die because you can't make that final incision or something like this,

but it's something where you actually, where you can actually differentiate yourself.

There are the web programmers.

A web programmer doesn't need recursion.

They can sometimes profit from it, but they don't really need it.

If you're doing high-performance computing for nuclear efficient equation solving or

so, for loops are just what you need.

Okay?

It's just basically, there are doctors who look at your skin, right?

Nice.

You don't have to cut anybody.

You don't need to, you know, don't need to deal with blood.

Different specialization.

What we're doing here is for dealing with a certain kind of structures that we will call

inductive sets in math that we're going to call user-supplied data types in functional

programming languages.

That will reappear as graphs and trees and terms and languages and all of those kind

of things that we need in symbolic AI.

Or many language-driven things.

So that's what I'm going to try to show you today is essentially that computation by recursion

is really the same as induction on steroids.

That's what I'm going to try to convince you of today.

And for some things, that is actually useful to know.

And it's useful to have that as a skill set.

And so we're going to look at what is called inductive sets.

And we've already seen two inductive sets.

One of them is unary natural numbers.

That's why I always do this little dance that you can actually do everything with the S

rule and the zero rule.

And lists are an inductive set.

A little bit more complicated inductive sets.

As well, trees, sets, graphs, all of those kind of things.

Languages, all of those are inductive sets.

And if you have the right tools for doing that, both the math to reason about it and