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[SQUEAKING]
[RUSTLING] [CLICKING]

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PROFESSOR: OK, why
don't I get started?

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So OK, what have we been doing?

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So last time, we
considered a bunch

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of procedures for testing
properties of various automata

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and grammars, the acceptance
problem for DFAs, for NFAs,

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the acceptance problem,
which is really

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degeneration problem for
context-free grammars,

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and emptiness problems for
DFAs and context-free grammars.

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And also, we showed that
A,TM is Turing-recognizable.

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There's a question here in
the chat already about that.

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Yes, we did show that.

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That was the universal
Turing machine

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that we presented at the end.

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That's what shows that A,TM
is Turing-recognizable.

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We mentioned that
A,TM is not decidable,

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which we promised
we will prove today.

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And we will do so.

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OK, so that is the plan--

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proving A,TM is undecidable.

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And we'll introduce the
method for doing that called

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the diagonalization method.

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I will also show the
complement of A,TM is

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Turing-unrecognizable.

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Even though A,TM
itself is recognizable,

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the complement is not.

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And then we will introduce
another method called

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the reducibility method for
showing other problems are not

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decidable and give one example
of another problem, which

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is not decidable, assuming
we have time to get to it.

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If not, we will delay that part
until next Tuesday's lecture.

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All right, acceptance
problem for Turing Machines.

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So just as I mentioned,
we showed that A,TM,

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which is the language of Turing
machines and inputs where

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the machine accepts the input.

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We showed that was recognizable.

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We claimed it was decidable.

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Today, we're going to
prove it's not decidable.

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All right, now the method
that we're going to use,

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which is really the
only method out there

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for proving a problem
as undecidable

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is called the
diagonalization method.

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I mean, ultimately, we're going
to show the reducibility method

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as well.

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But it really depends
on having already

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shown some other
problem undecidable

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via the diagonalization method.

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So ultimately, everything hinges
on the diagonalization method,

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which is really what we have.

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And so to introduce the
diagonalization method,

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I'm going to make a
bit of a digression

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into a branch of mathematics
called set theory

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or it's a part of
mathematical logic, where

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the method of
diagonalization was first

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conceived of back in the late
19th century by a mathematician

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called Georg Cantor.

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And Cantor was considering
the problem of,

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how do you compare the relative
sizes of infinite sets?

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For finite sets, the problem
of comparing-- somebody

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said that Cantor went crazy.

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That is true.

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And maybe I don't know
why he went crazy.

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But he did go--

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he had some mental
problems, unfortunately.

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And so how do we compare the
sizes of sets in general?

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If they're finite sets,
we can just count them up.

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We can say, whoa, this
set has 11 elements,

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and the other set
has 10 elements.

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So the one with 11 is bigger.

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Or if they both have 11,
they're the same size.

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Well, that's not going
to work for infinite sets

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because you can't count them up.

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And so he had--

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Cantor had the following
idea for comparing

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the sizes of infinite sets.

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And that was, basically,
to see whether you

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can have a function
that would map

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from one set to the other
set with certain properties.

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And those properties are
called, traditionally, well,

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I mean, in the past, have
been called the one-to-one

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and onto properties
for the function.

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I'll tell you what that means.

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But the concept is very simple.

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So a one-to-one
function is a function

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that's mapping from A to B.
Those are the two sets whose

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sizes we're trying to compare.

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And the function
being one-to-one just

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means that there
are no collisions.

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If you have two
different elements of A,

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they're never going to map
onto the same element of B.

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So two different
elements of A always

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map onto two different
elements of B.

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So that's the
one-to-one property.

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It's also called injective.

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And the other property is
called onto or surjective,

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which is that the range
of f has to be all of B.

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So you're not allowed
to miss any elements,

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but you have to hit everything.

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And when you have both
of those properties,

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the function is called a
one-to-one correspondence

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or a bijection also, OK?

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Now another way
of looking at it--

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I don't want to make this
more complicated than it

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needs to be.

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It just simply
means that two sets

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are considered to be the
same size if we can match up

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the elements with one set with
elements of the other set.

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You just pair them up.

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For example, well, if you
have finite sets, that idea,

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that informal idea just works
exactly as you would expect.

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For example, if we have two
sets-- here's a set of puppies.

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Here's a set of kittens.

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Now we want to show that those
two sets have the same size.

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We could count them up, as I
mentioned, and see that there

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are six elements in both.

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But counting up does not
work for infinite sets.

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So we can just match up
the elements of the puppies

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with the kittens,
and then we know

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we have the same number
of puppies as kittens.

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OK, now that has the
advantage of it making sense

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when you have infinite sets.

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So we're just going
to extend that idea

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and apply it to
infinite sets too.

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And then we'll have
a notion of what

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it means for two infinite
sets to have the same size.

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And you might wonder,
what do you get?

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Are all infinite
sets of the same size

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when you use this notion or not?

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What happens?

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Well, some strange
things do happen.

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But there actually are quite
some interesting structure

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there that emerges.

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So anyway, I don't
want to rush on.

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Questions on any of this?

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If you want to--

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quick question.

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This, hopefully, was not too
hard, but I want to make sure

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everybody's together with
me on it, so we can pop in--

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I'll give a few seconds for a
chat if you have any questions.

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The range of the set
is all of the elements

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that you hit as you look at
the different possible elements

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of A. So all of the things that
f hits, the standard notion

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of a range of a function.

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So the range of f
has to be equal to B.

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You have to hit everything.

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All right, can you think of
a one-to-one correspondence

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as a relabeling?

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Yeah, I'm not sure
that's helpful.

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But, yes, you can think of as
a relabeling of the elements

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in a sense, but yeah.

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I just think of it
as a matching up.

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All right, so let's move on.

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So coming out of
this notion of sets

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being of the same
size with this notion

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of countable sets, as
we'll see in a minute,

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so let's do an example.

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Let's take the set
of natural numbers--

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1, 2, 3, 4, dot, dot,
dot, dot, and the set

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of integers, which includes
the natural numbers

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but also the negative
numbers and zero.

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So the natural
numbers are typically

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referred to as N. The integers
are typically referred to as Z.

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And what do we think of the
relative sizes of N and Z?

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Well, N is a subset of Z.
It's a proper subset of Z.

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So you might, at first glance,
think that Z is larger,

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and they're not really going
to be of the same size.

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But actually, it turns out that
there's a very simple way--

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the arithmetic and the
properties of infinite sets

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can be a little bit surprising.

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And there is a way of matching
up all of the elements of Z

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with their own elements
of N. And so you

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can show that following that
definition, that these two sets

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are, in fact, of the same size.

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So let's just quickly go
make sure you see that.

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So here is N. Here
is Z I'm going

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to write down a table which
shows how they match up.

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Here is the function f of n
that I'm going to be describing.

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That's the one-to-one
correspondence.

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And so here are the
natural numbers--

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dot, dot, dot, 1 through
7, dot, dot, dot.

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And here are the
elements of Z that I'm

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going to be matching up.

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This is how the
function is working.

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So one, I'm going to--

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f of 1 is going to be 0.

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F of 2 is going to be minus 1.

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F of 3 is going to be 1.

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And I'm just giving
you a way to match up

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each of the natural
numbers with integers

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so that every single integer has
its own natural number and vice

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versa.

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So 4 goes to minus 2.

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5 goes to 2.

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6 goes to minus 3.

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7 goes to 3, and then minus
4, and then 4, and minus 5,

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and then 5, and so on.

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You're clearly
going to cover all

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of the integers in this way.

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And each of the
natural numbers is

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going to have its own integer.

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And there's never going
to be any collisions.

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There's never going to be
two natural numbers assigned

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to the same integer.

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So this meets the conditions
of a one-to-one correspondence

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and shows that the natural
numbers and the integers

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have the same size
following this definition.

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OK, let's do one that's slightly
more complicated and, perhaps,

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00:11:02,030 --> 00:11:06,230
slightly more surprising,
which is that if you consider

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00:11:06,230 --> 00:11:11,715
all of the rational numbers,
and then they have the same--

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that is a collection.

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00:11:12,590 --> 00:11:14,990
Even though the
rational numbers seem

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to be much richer and more
numerous than the integers,

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00:11:19,190 --> 00:11:21,500
from this perspective,
they have the same size.

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00:11:21,500 --> 00:11:25,110
And for the simplicity
of my presentation,

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00:11:25,110 --> 00:11:27,720
I'm going to consider only
the positive rational numbers,

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00:11:27,720 --> 00:11:30,080
which I'm going to write as Q+.

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00:11:30,080 --> 00:11:32,930
So those are all of the
positive rational numbers

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00:11:32,930 --> 00:11:36,530
that can be expressed as a
ratio of two natural numbers.

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00:11:36,530 --> 00:11:41,000
And I'm going to show that there
is a one-to-one correspondence

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00:11:41,000 --> 00:11:44,180
between these positive
rational numbers

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00:11:44,180 --> 00:11:46,820
and the natural numbers.

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00:11:46,820 --> 00:11:50,030
OK, so here, I'm
going to, again,

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00:11:50,030 --> 00:11:53,510
make a table, just
as I did up above--

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so comparing the
natural numbers and

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00:11:55,520 --> 00:11:57,530
the positive rational numbers.

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00:11:57,530 --> 00:12:01,010
And to do that, I'm going
to write down over here

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00:12:01,010 --> 00:12:02,990
on the side, just
to help you see

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00:12:02,990 --> 00:12:08,180
how I'm getting this table,
a separate table that

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has all of the positive
rational numbers appearing

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in a nicely organized way.

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00:12:15,170 --> 00:12:17,450
Here are all of the
rational numbers

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00:12:17,450 --> 00:12:20,960
here that have 1 as a numerator,
that have 2 as a numerator,

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00:12:20,960 --> 00:12:24,140
and so on, and going
through across the columns,

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00:12:24,140 --> 00:12:26,310
the different denominators.

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00:12:26,310 --> 00:12:29,367
So the numbers
inside here represent

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00:12:29,367 --> 00:12:30,950
all of the different
rational numbers.

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00:12:30,950 --> 00:12:32,900
And so whatever rational
number you have,

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00:12:32,900 --> 00:12:36,770
m over n, that's
going to appear down

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00:12:36,770 --> 00:12:39,018
the nth row in the nth column.

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00:12:39,018 --> 00:12:40,310
That number is going to appear.

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00:12:40,310 --> 00:12:42,200
So they're all going to show up.

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00:12:42,200 --> 00:12:49,070
And I'm going to use this table
here to fill out this function.

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00:12:49,070 --> 00:12:52,360
So here are all the
natural numbers.

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00:12:52,360 --> 00:12:56,950
And the way I'm going to assign
the rational numbers to appear

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00:12:56,950 --> 00:13:02,680
over here is I'm just going to
work my way in from the corner.

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00:13:02,680 --> 00:13:06,200
And I'm going to do
that by doing layers.

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00:13:06,200 --> 00:13:09,370
So first, I'll take
the 1 in this number

248
00:13:09,370 --> 00:13:10,330
here in this corner.

249
00:13:10,330 --> 00:13:13,150
Then, I'll do these
three that surrounded it,

250
00:13:13,150 --> 00:13:15,550
and then these guys
that surround that,

251
00:13:15,550 --> 00:13:18,910
and these guys sort of in shells
going around the previous ones

252
00:13:18,910 --> 00:13:20,830
that I've already covered.

253
00:13:20,830 --> 00:13:23,690
OK, so let me illustrate that.

254
00:13:23,690 --> 00:13:26,530
So here's 1 over 1, my
first rational number

255
00:13:26,530 --> 00:13:28,570
that I'm going to
enter into the table,

256
00:13:28,570 --> 00:13:29,920
appearing right over there.

257
00:13:29,920 --> 00:13:32,560
So next, I'm going
to go 2 over 1.

258
00:13:32,560 --> 00:13:35,270
That's going to appear
in the table over there.

259
00:13:35,270 --> 00:13:36,680
Now, we have 2 over 2.

260
00:13:36,680 --> 00:13:41,210
Now that's actually a little
bit problematic because 2 over 2

261
00:13:41,210 --> 00:13:43,530
has already been done.

262
00:13:43,530 --> 00:13:45,410
And if we put that in--

263
00:13:45,410 --> 00:13:47,060
because 2 over 2
equals 1 over 1.

264
00:13:47,060 --> 00:13:50,300
They're both the equal
to the rational number 1.

265
00:13:50,300 --> 00:13:52,910
So if we put 2 over 2
in this table over here,

266
00:13:52,910 --> 00:13:55,580
then we would no longer
have the one-to-one property

267
00:13:55,580 --> 00:13:59,850
because both 1 and 3 would
be mapping to the same point.

268
00:13:59,850 --> 00:14:04,850
So we're just going to
simply skip over 2 over 2.

269
00:14:04,850 --> 00:14:06,930
I'll show that as kind
of graying it out.

270
00:14:06,930 --> 00:14:10,010
So we're not going to add
that one on to the table

271
00:14:10,010 --> 00:14:12,650
onto my function.

272
00:14:12,650 --> 00:14:14,270
But so we'll skip over that.

273
00:14:14,270 --> 00:14:18,532
We'll go to 1 over 2, which
has not been seen before.

274
00:14:18,532 --> 00:14:20,990
And then we're just going to
continue doing the same thing,

275
00:14:20,990 --> 00:14:23,600
now going to this
next shell out here.

276
00:14:23,600 --> 00:14:28,760
3 over 1, 3 over 2, 3 over 3--

277
00:14:28,760 --> 00:14:31,840
same thing, we're going
to skip over that one--

278
00:14:31,840 --> 00:14:34,600
2 over 3, 1 over 3.

279
00:14:34,600 --> 00:14:36,680
I hope you're getting the idea.

280
00:14:36,680 --> 00:14:38,180
So I'm not going
to fill this table.

281
00:14:38,180 --> 00:14:41,410
I ran out of room to fill
out this table some more.

282
00:14:41,410 --> 00:14:44,140
But just to look at how the
process is going to continue

283
00:14:44,140 --> 00:14:45,580
here, we get 4 over 1.

284
00:14:45,580 --> 00:14:49,630
Now 4 over 2, again, is a
number we previously seen

285
00:14:49,630 --> 00:14:52,300
because that's the
rational number 2.

286
00:14:52,300 --> 00:14:54,400
We saw that up here
when we had 2 over 1,

287
00:14:54,400 --> 00:14:56,170
so we're going to
have to skip that one.

288
00:14:56,170 --> 00:14:58,480
4 over 3, 4 over 4--

289
00:14:58,480 --> 00:15:00,220
that was going to skip.

290
00:15:00,220 --> 00:15:02,760
3 over 4, and so on.

291
00:15:02,760 --> 00:15:03,260
OK?

292
00:15:03,260 --> 00:15:07,040
So by following
this procedure, I'm

293
00:15:07,040 --> 00:15:08,978
going to be able to
define this function.

294
00:15:08,978 --> 00:15:10,520
Now this function
is not particularly

295
00:15:10,520 --> 00:15:14,120
nice in terms of having
an elegant, closed form.

296
00:15:14,120 --> 00:15:16,820
But it is a totally
legitimate function

297
00:15:16,820 --> 00:15:20,000
in the sense of being a
mapping from natural numbers

298
00:15:20,000 --> 00:15:23,630
to the positive
rational numbers.

299
00:15:23,630 --> 00:15:27,530
And it has the one-to-one
correspondence property.

300
00:15:27,530 --> 00:15:31,340
So it pairs up each
of the natural numbers

301
00:15:31,340 --> 00:15:36,740
with each of the positive
rational numbers.

302
00:15:36,740 --> 00:15:40,217
They each get their
own mate, in a sense.

303
00:15:40,217 --> 00:15:42,050
And so that shows that
the rational numbers,

304
00:15:42,050 --> 00:15:44,150
despite the fact
that they're dense,

305
00:15:44,150 --> 00:15:48,020
and they have all sorts of
very more much bigger seeming,

306
00:15:48,020 --> 00:15:51,110
they really, from this
perspective of the sizes

307
00:15:51,110 --> 00:15:56,210
of the sets, they have the same
size as the natural numbers do.

308
00:15:56,210 --> 00:16:00,020
And so with that, it suggests
that the following definition

309
00:16:00,020 --> 00:16:02,600
that a set is
countable if it has

310
00:16:02,600 --> 00:16:06,583
the same size as the natural
numbers or it's finite.

311
00:16:06,583 --> 00:16:08,000
From this perspective,
we're going

312
00:16:08,000 --> 00:16:10,220
to be focusing on infinite sets.

313
00:16:10,220 --> 00:16:13,940
But yeah, countable or countably
infinite, sometimes people say,

314
00:16:13,940 --> 00:16:15,948
if it has the same size
as the natural numbers.

315
00:16:15,948 --> 00:16:17,990
Or otherwise, you have to
include the finite sets

316
00:16:17,990 --> 00:16:18,650
as well.

317
00:16:18,650 --> 00:16:23,450
And countable means you can
just go through the elements

318
00:16:23,450 --> 00:16:27,380
of the set as a list.

319
00:16:27,380 --> 00:16:29,630
So you can count them-- the
first one, the second one,

320
00:16:29,630 --> 00:16:32,206
the third one, the fourth
one, dot, dot, dot.

321
00:16:32,206 --> 00:16:34,990
And once you can do
that, and that counting

322
00:16:34,990 --> 00:16:38,600
is going to hit everything,
then you know can match them--

323
00:16:38,600 --> 00:16:40,570
you can pair them up
with the natural numbers.

324
00:16:40,570 --> 00:16:42,760
And so therefore, you
have a countable set.

325
00:16:42,760 --> 00:16:45,190
OK.

326
00:16:45,190 --> 00:16:49,750
So as we've shown,
both Z, the integers,

327
00:16:49,750 --> 00:16:53,690
and the positive rational
numbers are countable sets.

328
00:16:53,690 --> 00:16:54,190
OK?

329
00:16:54,190 --> 00:16:57,840
Now you might think
that, well, since we're

330
00:16:57,840 --> 00:17:00,720
allowing ourselves
to do something

331
00:17:00,720 --> 00:17:06,540
as arbitrary, in a way,
as this kind of a function

332
00:17:06,540 --> 00:17:10,740
to match up the natural numbers
with whatever set you're trying

333
00:17:10,740 --> 00:17:15,572
to measure, that every set
is going to be countable,

334
00:17:15,572 --> 00:17:18,030
if you think about it that way
because it seems like you're

335
00:17:18,030 --> 00:17:20,452
allowing too many possibilities.

336
00:17:20,452 --> 00:17:21,869
And then all the
infinite sets are

337
00:17:21,869 --> 00:17:24,599
going to end up being the same
size as the natural numbers.

338
00:17:24,599 --> 00:17:28,000
Well, that is, in
fact, is not true.

339
00:17:28,000 --> 00:17:29,970
And I don't know.

340
00:17:29,970 --> 00:17:32,460
Cantor, is the one
who discovered that.

341
00:17:32,460 --> 00:17:35,500
I don't know if that
was surprising or not.

342
00:17:35,500 --> 00:17:39,450
But it is interesting,
I think, that there are

343
00:17:39,450 --> 00:17:41,940
different sizes of infinities.

344
00:17:41,940 --> 00:17:46,870
And so we're going to now
take a look and see that.

345
00:17:46,870 --> 00:17:51,660
But I want to, again,
I want to stop and make

346
00:17:51,660 --> 00:17:52,980
sure we're all together.

347
00:17:52,980 --> 00:17:54,690
Can we always find
a closed formula

348
00:17:54,690 --> 00:17:56,850
for f, somebody's asking me.

349
00:17:56,850 --> 00:17:58,200
I don't know.

350
00:17:58,200 --> 00:18:03,460
For this particular
f, you probably could,

351
00:18:03,460 --> 00:18:07,600
but it would probably
be very messy.

352
00:18:07,600 --> 00:18:09,940
But in general, that's
not a requirement,

353
00:18:09,940 --> 00:18:13,810
having a closed form for
the mapping function.

354
00:18:13,810 --> 00:18:17,320
Any function is allowed as long
as a one-to-one correspondence.

355
00:18:20,530 --> 00:18:22,120
Are we all together on this?

356
00:18:22,120 --> 00:18:23,920
Are we comfortable
with the notion

357
00:18:23,920 --> 00:18:27,927
of some infinite sets
having the same size

358
00:18:27,927 --> 00:18:30,010
as the natural numbers,
and therefore, we're going

359
00:18:30,010 --> 00:18:31,420
to call them countable sets?

360
00:18:31,420 --> 00:18:33,760
And we're going to show
some other sets are

361
00:18:33,760 --> 00:18:36,520
going to have more elements.

362
00:18:36,520 --> 00:18:38,200
They're going to be uncountable.

363
00:18:38,200 --> 00:18:39,550
They're going to be beyond--

364
00:18:39,550 --> 00:18:43,300
strictly speaking, strictly
larger sets in the sense

365
00:18:43,300 --> 00:18:44,740
that we won't be
able to put them

366
00:18:44,740 --> 00:18:48,230
into a one-to-one correspondence
with the integers.

367
00:18:48,230 --> 00:18:49,540
So is N the smallest infinity?

368
00:18:49,540 --> 00:18:52,810
Yes, N is the smallest one.

369
00:18:52,810 --> 00:18:56,920
So any infinity
is going to have--

370
00:18:56,920 --> 00:18:58,107
you're going to always--

371
00:18:58,107 --> 00:18:59,440
I don't think you even need it--

372
00:18:59,440 --> 00:19:02,620
you need a special--

373
00:19:02,620 --> 00:19:04,600
whenever you have an
infinite collection,

374
00:19:04,600 --> 00:19:06,880
you can always
find a subset which

375
00:19:06,880 --> 00:19:12,447
is going to be a
countable subset

376
00:19:12,447 --> 00:19:14,530
and is going to be the
smallest of the infinities,

377
00:19:14,530 --> 00:19:16,647
but there are bigger ones.

378
00:19:16,647 --> 00:19:17,980
All right, why don't we move on?

379
00:19:20,680 --> 00:19:27,280
OK, so the example of a set that
we will show is not countable--

380
00:19:27,280 --> 00:19:31,300
an uncountable set,
as we call it--

381
00:19:31,300 --> 00:19:37,910
is a set of real numbers, which
we all know what these are,

382
00:19:37,910 --> 00:19:38,680
I hope.

383
00:19:38,680 --> 00:19:40,990
These are all of
the numbers that you

384
00:19:40,990 --> 00:19:44,620
can express by possibly
infinite decimal representations

385
00:19:44,620 --> 00:19:50,890
like pi, or E squared of 2, or
any of the other familiar ones.

386
00:19:50,890 --> 00:19:56,590
Rational numbers are also
members of or also count

387
00:19:56,590 --> 00:19:59,410
as real numbers
and integers too.

388
00:19:59,410 --> 00:20:04,390
And so but there are
these additional numbers

389
00:20:04,390 --> 00:20:07,240
that you can get by the
decimal expansions which

390
00:20:07,240 --> 00:20:09,520
may not be rational.

391
00:20:09,520 --> 00:20:12,100
And that collection, even
though, in some ways,

392
00:20:12,100 --> 00:20:16,060
it looks somewhat similar
to the rational numbers,

393
00:20:16,060 --> 00:20:17,328
the real numbers--

394
00:20:17,328 --> 00:20:18,370
I hope I said that right.

395
00:20:18,370 --> 00:20:22,900
The real numbers,
which is the set

396
00:20:22,900 --> 00:20:25,630
I'm considering here, the
ones with the infinite decimal

397
00:20:25,630 --> 00:20:30,200
expansions, they
actually are much larger.

398
00:20:30,200 --> 00:20:36,650
So the theorem is-- and this
was discovered by Cantor--

399
00:20:36,650 --> 00:20:39,720
that R is an uncountable set.

400
00:20:39,720 --> 00:20:42,110
And the reason why
I'm introducing

401
00:20:42,110 --> 00:20:47,780
this is, besides that I think
it's interesting and has

402
00:20:47,780 --> 00:20:51,360
some relation to the kinds
of things we're doing,

403
00:20:51,360 --> 00:20:53,630
but it's really for
the purposes right now,

404
00:20:53,630 --> 00:20:56,050
is to introduce this method
called diagonalization.

405
00:20:59,130 --> 00:21:02,100
OK, that's what we're going
to use later on again.

406
00:21:02,100 --> 00:21:04,530
But this is the first
time it appeared.

407
00:21:04,530 --> 00:21:08,190
So we're going to use a proof
by contradiction in order

408
00:21:08,190 --> 00:21:12,060
to show that R, the
collection of real numbers,

409
00:21:12,060 --> 00:21:13,740
is uncountable.

410
00:21:13,740 --> 00:21:15,840
And we'll assume,
for contradiction,

411
00:21:15,840 --> 00:21:17,910
that R countable.

412
00:21:17,910 --> 00:21:20,250
OK?

413
00:21:20,250 --> 00:21:23,340
So if we assume
that R is countable,

414
00:21:23,340 --> 00:21:26,640
it means we have, by definition,
a one-to-one correspondence

415
00:21:26,640 --> 00:21:29,530
from the natural numbers
to the real numbers.

416
00:21:29,530 --> 00:21:32,560
OK, so we can match up
all of the natural numbers

417
00:21:32,560 --> 00:21:37,520
with the real numbers in a
one-to-one and onto fashion.

418
00:21:37,520 --> 00:21:40,640
So we can pair up the natural
numbers with the real numbers.

419
00:21:40,640 --> 00:21:44,240
And we will cover all of the
real numbers by doing that.

420
00:21:44,240 --> 00:21:47,840
And we can make a table,
just like I did before.

421
00:21:47,840 --> 00:21:50,770
Here it is.

422
00:21:50,770 --> 00:21:54,700
And you can fill this out
with all of the real numbers.

423
00:21:54,700 --> 00:21:57,940
And that was what
the assumption means.

424
00:21:57,940 --> 00:22:00,130
And I'm going to show you
that that's impossible.

425
00:22:00,130 --> 00:22:04,700
Something is going to go
terribly wrong if you do that.

426
00:22:04,700 --> 00:22:07,410
Now you might disagree.

427
00:22:07,410 --> 00:22:08,580
You might be surprised.

428
00:22:08,580 --> 00:22:13,200
You might think,
well, Professor Sipser

429
00:22:13,200 --> 00:22:17,640
is wrong, that I'm going
to work on this overnight

430
00:22:17,640 --> 00:22:19,080
and forget the
rest of my classes,

431
00:22:19,080 --> 00:22:24,040
and I'm going to come up with
a list of all the real numbers.

432
00:22:24,040 --> 00:22:26,280
I'm going to fill
them out here and show

433
00:22:26,280 --> 00:22:28,020
that that's impossible.

434
00:22:28,020 --> 00:22:31,740
So let's say, hypothetically,
just for my example's sake,

435
00:22:31,740 --> 00:22:35,280
here is the list that
you came up with.

436
00:22:35,280 --> 00:22:37,200
So you have to match up
something with the 1.

437
00:22:37,200 --> 00:22:39,480
Let's say, E. That
was a special case,

438
00:22:39,480 --> 00:22:41,550
so you're going to
stick that with 1.

439
00:22:41,550 --> 00:22:43,320
And then pi came
out to be the number

440
00:22:43,320 --> 00:22:45,982
that you connected up with 2.

441
00:22:45,982 --> 00:22:47,440
You understand what
I'm doing here?

442
00:22:47,440 --> 00:22:48,330
I'm making a list.

443
00:22:48,330 --> 00:22:51,660
I'm trying to make my
correspondence-- my matching up

444
00:22:51,660 --> 00:22:54,870
between the natural
numbers and my real numbers

445
00:22:54,870 --> 00:22:59,440
that I'm hypothesizing to
exist for a contradiction.

446
00:22:59,440 --> 00:23:03,000
So 3, I don't know, 3
is gets connected to 0.

447
00:23:03,000 --> 00:23:05,820
I'm making this up, obviously--

448
00:23:05,820 --> 00:23:06,660
not on the spot.

449
00:23:06,660 --> 00:23:09,210
I wrote the slide,
you know, yesterday.

450
00:23:09,210 --> 00:23:13,320
But here is the square root of 2
showing up for whatever reason.

451
00:23:13,320 --> 00:23:15,960
Here is 1/7.

452
00:23:15,960 --> 00:23:18,660
Here is some other number,
which, I'll be interested

453
00:23:18,660 --> 00:23:20,407
if you recognize that one.

454
00:23:20,407 --> 00:23:21,240
That's a subtle one.

455
00:23:23,990 --> 00:23:29,930
This is 1.234-- yeah, some
familiar looking sequence here.

456
00:23:29,930 --> 00:23:31,610
And whatever it is--

457
00:23:31,610 --> 00:23:33,770
whatever it is, this is
what you came up with.

458
00:23:33,770 --> 00:23:37,160
You claim that this
is going to work as--

459
00:23:37,160 --> 00:23:37,940
very good.

460
00:23:37,940 --> 00:23:39,110
Somebody got it.

461
00:23:39,110 --> 00:23:40,640
It's i to the ith power.

462
00:23:40,640 --> 00:23:44,640
But let's not get hung
up on that, please.

463
00:23:44,640 --> 00:23:46,100
I to the ith power,
oddly enough,

464
00:23:46,100 --> 00:23:48,230
can be seen as a
real number under--

465
00:23:48,230 --> 00:23:55,560
if you define the imaginary
exponentiation, which is weird.

466
00:23:55,560 --> 00:23:59,900
But anyway, let's not
get too distracted.

467
00:23:59,900 --> 00:24:03,230
So here's my list of
numbers, my real numbers

468
00:24:03,230 --> 00:24:07,950
that I've matched up in my table
here with the natural numbers.

469
00:24:07,950 --> 00:24:09,710
Now I claim
something goes wrong.

470
00:24:09,710 --> 00:24:10,610
What goes wrong?

471
00:24:10,610 --> 00:24:15,830
I'm going to show you
that you actually did not

472
00:24:15,830 --> 00:24:19,400
do as you claimed,
that there's actually

473
00:24:19,400 --> 00:24:23,022
at least one number that's
missing from this list.

474
00:24:23,022 --> 00:24:24,230
And I'll exhibit that number.

475
00:24:24,230 --> 00:24:25,980
I'm going to show you
what that number is.

476
00:24:25,980 --> 00:24:27,940
I'm going to explicitly
come up with a number

477
00:24:27,940 --> 00:24:29,690
and show you that
there's that number it's

478
00:24:29,690 --> 00:24:31,500
missing from the list.

479
00:24:31,500 --> 00:24:34,910
So here, it's going
to be a number x.

480
00:24:34,910 --> 00:24:36,840
X is the one that's missing.

481
00:24:36,840 --> 00:24:38,420
So how am I going to get x?

482
00:24:38,420 --> 00:24:41,300
So x, well, I'm going to
start it off with 0 point,

483
00:24:41,300 --> 00:24:43,550
and then I'm going to fill
out the rest of the places.

484
00:24:43,550 --> 00:24:44,990
And how am I going
to get those places?

485
00:24:44,990 --> 00:24:46,490
I'm going to look
at the table here.

486
00:24:48,940 --> 00:24:50,160
So I want to look at--

487
00:24:50,160 --> 00:24:53,610
to get my first digit after
the decimal point of x,

488
00:24:53,610 --> 00:24:55,110
I'm going to look
at the first digit

489
00:24:55,110 --> 00:24:59,390
after the decimal point
of the very first number.

490
00:24:59,390 --> 00:25:01,540
So I take the first number,
look at the first digit

491
00:25:01,540 --> 00:25:05,410
after the decimal point,
which happens to be a 7.

492
00:25:05,410 --> 00:25:07,510
And the number I'm
going to put in for x

493
00:25:07,510 --> 00:25:08,950
It's not going to be 7.

494
00:25:08,950 --> 00:25:11,890
It's going to be
anything except 7.

495
00:25:11,890 --> 00:25:12,460
Let say, 8.

496
00:25:15,410 --> 00:25:18,720
For my next digit of x,
I'm going to look at--

497
00:25:18,720 --> 00:25:20,770
so my second digit
of x, I'm going to--

498
00:25:20,770 --> 00:25:22,770
everything's going to be
up to the decimal point

499
00:25:22,770 --> 00:25:24,860
now, so I'm not going
to keep saying that.

500
00:25:24,860 --> 00:25:26,510
I'm going to look
at the second digit

501
00:25:26,510 --> 00:25:31,880
of the second number, which is
a 4 after the decimal point.

502
00:25:34,430 --> 00:25:36,260
There it is.

503
00:25:36,260 --> 00:25:38,300
And for the second
digit of x, I'm

504
00:25:38,300 --> 00:25:42,290
going to use something which
is anything besides 4--

505
00:25:42,290 --> 00:25:44,150
5.

506
00:25:44,150 --> 00:25:46,492
I have some flexibility here.

507
00:25:46,492 --> 00:25:47,450
So I could have used 6.

508
00:25:47,450 --> 00:25:49,820
I could have used 7.

509
00:25:49,820 --> 00:25:51,860
For those of you who
have seen this before,

510
00:25:51,860 --> 00:25:54,440
who are going to
nitpick on me, I'm

511
00:25:54,440 --> 00:25:58,040
going to stay away from 9
and 0 just because there

512
00:25:58,040 --> 00:26:01,167
is some little edge cases that
arise there, which I don't want

513
00:26:01,167 --> 00:26:02,750
to get into because
I don't think it's

514
00:26:02,750 --> 00:26:04,910
interesting for this argument.

515
00:26:04,910 --> 00:26:06,560
But since I have
some flexibility,

516
00:26:06,560 --> 00:26:08,270
I'm going to avoid
zeros and nines--

517
00:26:08,270 --> 00:26:10,140
probably just nines
is all I really need.

518
00:26:10,140 --> 00:26:15,800
And then the argument is
just going to work fine, OK?

519
00:26:15,800 --> 00:26:20,290
So the next digit is a 0.

520
00:26:20,290 --> 00:26:22,900
The third digit, or the
third number is a 0.

521
00:26:22,900 --> 00:26:25,585
The third digit of x is going
to be anything except for 0.

522
00:26:25,585 --> 00:26:27,370
It could be a 1.

523
00:26:27,370 --> 00:26:30,760
Then I have a 2 here.

524
00:26:30,760 --> 00:26:36,110
Anything except a 2, 6.

525
00:26:36,110 --> 00:26:39,110
Then a 5-- anything
except a 5, a 1.

526
00:26:39,110 --> 00:26:43,670
A 9-- anything except
a 9, an 8, and so on.

527
00:26:43,670 --> 00:26:45,470
There's an 8.

528
00:26:45,470 --> 00:26:47,270
There's a 2, and dot, dot, dot.

529
00:26:47,270 --> 00:26:49,270
I'm just going to
keep doing that.

530
00:26:49,270 --> 00:26:54,330
And I'm going to say you missed
that number, whatever it is.

531
00:26:54,330 --> 00:26:55,970
But that number,
that's a number.

532
00:26:55,970 --> 00:26:57,770
After I'm all done,
it's going to be

533
00:26:57,770 --> 00:27:00,200
the decimal representation
of something.

534
00:27:00,200 --> 00:27:06,480
And that number, I claim,
is absent from the table.

535
00:27:06,480 --> 00:27:12,570
And you might say,
well, it's really there.

536
00:27:12,570 --> 00:27:17,250
It's just on the 23rd row.

537
00:27:17,250 --> 00:27:20,040
You just didn't get
to it and your slide.

538
00:27:20,040 --> 00:27:21,180
But it's there.

539
00:27:21,180 --> 00:27:24,360
Well, I'm going to say I
know it's not in the 23rd row

540
00:27:24,360 --> 00:27:27,270
because it differs from
the number in the 23rd row

541
00:27:27,270 --> 00:27:31,920
at the 23rd digit because
I constructed it that way.

542
00:27:31,920 --> 00:27:34,740
This number is
designed explicitly

543
00:27:34,740 --> 00:27:37,680
to be different from each of
the numbers that's on the list.

544
00:27:41,255 --> 00:27:42,880
So it can't be on
the list because it's

545
00:27:42,880 --> 00:27:44,680
been constructed to be
different from each of them

546
00:27:44,680 --> 00:27:45,597
in at least one place.

547
00:27:48,290 --> 00:27:50,410
So I think this is a
beautiful argument,

548
00:27:50,410 --> 00:27:53,500
and it shows that
no matter how you

549
00:27:53,500 --> 00:27:57,580
try to come up with a
one-to-one correspondence,

550
00:27:57,580 --> 00:27:58,990
you're going to fail.

551
00:27:58,990 --> 00:28:03,780
You might say, oh,
like, just one number?

552
00:28:03,780 --> 00:28:04,783
I did so much hard work.

553
00:28:04,783 --> 00:28:05,950
I'm missing just one number.

554
00:28:05,950 --> 00:28:09,645
Can't I get partial credit
and put this one on the list

555
00:28:09,645 --> 00:28:10,270
now I fixed it?

556
00:28:10,270 --> 00:28:10,630
No.

557
00:28:10,630 --> 00:28:12,730
Obviously, there are many,
many numbers that are missing.

558
00:28:12,730 --> 00:28:14,155
If you put this
one on the list, I

559
00:28:14,155 --> 00:28:16,570
was going to construct another
number that was missing.

560
00:28:16,570 --> 00:28:24,310
So there's just not no
possibility of fixing this.

561
00:28:24,310 --> 00:28:29,323
And in fact, the real
numbers are uncountable--

562
00:28:29,323 --> 00:28:31,240
cannot be matched up
with the natural numbers.

563
00:28:31,240 --> 00:28:33,010
And there's nothing.

564
00:28:33,010 --> 00:28:34,135
It doesn't even come close.

565
00:28:36,650 --> 00:28:41,050
So here, a summary
of what I just said.

566
00:28:41,050 --> 00:28:42,632
F is not a one-to-one
correspondence

567
00:28:42,632 --> 00:28:43,840
no matter what you try to do.

568
00:28:43,840 --> 00:28:45,965
You can't come up with a
one-to-one correspondence.

569
00:28:45,965 --> 00:28:48,900
That's the contradiction.

570
00:28:48,900 --> 00:28:53,310
So that proves that
R is uncountable.

571
00:28:53,310 --> 00:28:56,760
And that's why, by the way,
we call this a diagonalization

572
00:28:56,760 --> 00:28:58,950
because we're going
down this diagonal here

573
00:28:58,950 --> 00:29:02,990
that's where the term-- that's
where the name is come from.

574
00:29:02,990 --> 00:29:08,720
OK, so I'm happy to--

575
00:29:08,720 --> 00:29:11,240
[CHUCKLES] OK, that's a--

576
00:29:14,260 --> 00:29:17,260
somebody's asking me--
[CHUCKLES] I'm actually--

577
00:29:17,260 --> 00:29:20,570
I have a-- is it here?

578
00:29:20,570 --> 00:29:21,820
I can't remember if it's here.

579
00:29:21,820 --> 00:29:25,090
I have a check-in
coming about this.

580
00:29:25,090 --> 00:29:27,640
And somebody is
anticipating that

581
00:29:27,640 --> 00:29:31,450
by asking the actually rather
famous question about there

582
00:29:31,450 --> 00:29:36,550
being a possible infinity in
between the natural numbers

583
00:29:36,550 --> 00:29:37,985
and the real numbers.

584
00:29:37,985 --> 00:29:39,610
We know the real
numbers now are bigger

585
00:29:39,610 --> 00:29:40,657
than the natural numbers.

586
00:29:40,657 --> 00:29:42,490
But is there something
that's an in between?

587
00:29:42,490 --> 00:29:46,000
I mean, this is a very,
very famous problem,

588
00:29:46,000 --> 00:29:49,090
which I'll talk about when
we get to our check-in, which

589
00:29:49,090 --> 00:29:49,810
is coming up.

590
00:29:55,020 --> 00:29:56,940
Somebody is objecting
just the way

591
00:29:56,940 --> 00:29:59,700
I've defined x using
this procedure that

592
00:29:59,700 --> 00:30:03,700
really can't, in a sense, state,
you know-- but x is a number.

593
00:30:03,700 --> 00:30:07,980
X is the result of
what this procedure is.

594
00:30:07,980 --> 00:30:09,730
Following this procedure,
we're converging

595
00:30:09,730 --> 00:30:11,260
on some particular value.

596
00:30:11,260 --> 00:30:13,690
And so that is a value.

597
00:30:13,690 --> 00:30:18,372
If you want, we can make a more
precise way of determining.

598
00:30:18,372 --> 00:30:20,080
We don't have the
flexibility of choosing

599
00:30:20,080 --> 00:30:21,620
the way I did in my example.

600
00:30:21,620 --> 00:30:24,850
We can make a precise
procedure for coming up

601
00:30:24,850 --> 00:30:25,820
with these digits.

602
00:30:25,820 --> 00:30:29,075
But I don't think there's
anybody thinks there's anything

603
00:30:29,075 --> 00:30:29,575
that's--

604
00:30:32,460 --> 00:30:35,700
there's any shortcoming
in this argument in terms

605
00:30:35,700 --> 00:30:37,470
of the way we're defining x.

606
00:30:37,470 --> 00:30:40,860
I think it's worth understanding
this because it's really going

607
00:30:40,860 --> 00:30:47,430
to set the groundwork for our
proof that A,TM is undecidable,

608
00:30:47,430 --> 00:30:49,020
which is a little,
I think, perhaps,

609
00:30:49,020 --> 00:30:52,350
slightly more abstract in a
way in the way it sort of comes

610
00:30:52,350 --> 00:30:53,220
across.

611
00:30:53,220 --> 00:30:56,250
I'm going to try
to connect the two.

612
00:30:56,250 --> 00:30:58,320
But I think it's
helpful to understand

613
00:30:58,320 --> 00:31:00,540
at least this argument
because this argument is

614
00:31:00,540 --> 00:31:04,470
diagonalization in
its most raw form.

615
00:31:09,600 --> 00:31:11,290
All right, I think we're good.

616
00:31:11,290 --> 00:31:14,220
So why don't we continue, then?

617
00:31:14,220 --> 00:31:16,560
So there are a
number of corollaries

618
00:31:16,560 --> 00:31:19,680
that follow from the statement
that the real numbers is

619
00:31:19,680 --> 00:31:20,490
uncountable.

620
00:31:23,910 --> 00:31:27,180
First of all, if
you let script L

621
00:31:27,180 --> 00:31:29,485
be the collection
of all languages,

622
00:31:29,485 --> 00:31:31,860
if you want to consider it
over some particular alphabet,

623
00:31:31,860 --> 00:31:32,370
that's fine.

624
00:31:32,370 --> 00:31:39,010
But that's not going to be
really important for this point

625
00:31:39,010 --> 00:31:40,600
that I'm going to make.

626
00:31:40,600 --> 00:31:46,540
So script L is the
collection of all languages.

627
00:31:46,540 --> 00:31:50,280
So every subset of sigma star--

628
00:31:50,280 --> 00:31:53,250
every subset of sigma
star is a language.

629
00:31:53,250 --> 00:31:56,760
So look at all those
possible subsets.

630
00:31:56,760 --> 00:31:59,040
So that includes 0
to the k, 1 to the k,

631
00:31:59,040 --> 00:32:01,860
plus every other language you
can ever think of and more--

632
00:32:01,860 --> 00:32:04,730
all possible languages.

633
00:32:04,730 --> 00:32:09,110
So the collection of all
languages is uncountable.

634
00:32:09,110 --> 00:32:12,110
There's uncountably many
different languages out there.

635
00:32:12,110 --> 00:32:13,942
I don't want to
belabor this point.

636
00:32:13,942 --> 00:32:15,650
You can just take this
if you don't quite

637
00:32:15,650 --> 00:32:18,530
follow the quick argument
I'm going to make here.

638
00:32:18,530 --> 00:32:23,420
But you can make a
one-to-one correspondence

639
00:32:23,420 --> 00:32:27,650
from all languages to the
reals so that each language

640
00:32:27,650 --> 00:32:30,328
gets its own real number.

641
00:32:30,328 --> 00:32:32,370
And the way I'm going to
think about that-- let's

642
00:32:32,370 --> 00:32:35,220
put the real numbers
into binary form.

643
00:32:35,220 --> 00:32:37,590
And if you imagine
here being sigma star,

644
00:32:37,590 --> 00:32:39,270
all of the possible
strings of sigma

645
00:32:39,270 --> 00:32:43,560
star written out in
their standard order.

646
00:32:43,560 --> 00:32:45,500
And now if you have
a language here, A,

647
00:32:45,500 --> 00:32:47,850
it's just some
arbitrary language.

648
00:32:47,850 --> 00:32:52,210
So that's going to have some
of the strings of sigma star

649
00:32:52,210 --> 00:32:52,710
appearing.

650
00:32:52,710 --> 00:32:55,350
Like 0 is appearing,
but 1 is not appearing.

651
00:32:55,350 --> 00:32:58,590
00 is appearing and 01,
but not these three.

652
00:33:01,270 --> 00:33:04,600
And I'm going to associate
to A, my language,

653
00:33:04,600 --> 00:33:11,440
some real number in binary by
putting in a 0 in the decimal

654
00:33:11,440 --> 00:33:13,900
representation-- well,
the binary representation,

655
00:33:13,900 --> 00:33:15,610
I should say--

656
00:33:15,610 --> 00:33:20,770
for that string if it's not
there and a 1 if it is there.

657
00:33:24,450 --> 00:33:27,860
And so a real number--
because there's infinitely

658
00:33:27,860 --> 00:33:31,880
many yes/no choices in
the binary representation

659
00:33:31,880 --> 00:33:35,090
can represent a
language because of each

660
00:33:35,090 --> 00:33:38,240
of the yes/no choices of a
string being present or not.

661
00:33:38,240 --> 00:33:40,450
I'm going to put a 1 for
when the string is present,

662
00:33:40,450 --> 00:33:41,750
a 0 when it's not present.

663
00:33:41,750 --> 00:33:44,822
So each language has
its own real number,

664
00:33:44,822 --> 00:33:46,280
and each real number
is going to be

665
00:33:46,280 --> 00:33:48,500
associated to its own language.

666
00:33:48,500 --> 00:33:52,420
Here, I'm restricting myself to
real numbers between 0 and 1.

667
00:33:52,420 --> 00:33:54,170
That's not going to
be an essential point.

668
00:33:54,170 --> 00:33:55,880
So let's not get
hung up on that.

669
00:34:00,050 --> 00:34:10,679
So the fact that the
languages can be putting it

670
00:34:10,679 --> 00:34:14,429
into a one-to-one correspondence
with the real numbers

671
00:34:14,429 --> 00:34:19,230
shows that the collection of all
languages is also uncountable.

672
00:34:19,230 --> 00:34:21,900
Now another observation here--

673
00:34:21,900 --> 00:34:25,050
another point worth noting is
that if you look at sigma star

674
00:34:25,050 --> 00:34:30,460
itself, the strings,
all possible strings,

675
00:34:30,460 --> 00:34:32,880
that's a countable set.

676
00:34:32,880 --> 00:34:35,340
The collection of all possible
languages is uncountable.

677
00:34:35,340 --> 00:34:38,340
But the collection of all
possible finite strings

678
00:34:38,340 --> 00:34:41,270
is countable because
I can just list them.

679
00:34:41,270 --> 00:34:43,370
Here's my list of all
possible strings, which

680
00:34:43,370 --> 00:34:44,870
you can put into a
table if you like

681
00:34:44,870 --> 00:34:46,995
to think of it matching up
with the natural numbers

682
00:34:46,995 --> 00:34:49,600
in that way.

683
00:34:49,600 --> 00:34:52,010
Now I'm trying to
make a point here,

684
00:34:52,010 --> 00:34:55,630
which is that if you take M,
which is all possible Turing

685
00:34:55,630 --> 00:34:58,030
machines-- script M
is all possible Turing

686
00:34:58,030 --> 00:35:00,850
machines-- the collection of
all possible Turing machines

687
00:35:00,850 --> 00:35:02,370
is countable.

688
00:35:02,370 --> 00:35:06,510
There are only countably many
different Turing machines.

689
00:35:06,510 --> 00:35:09,217
And you can argue that in all
sorts of messy different ways.

690
00:35:09,217 --> 00:35:10,800
But I think the most
simple way to see

691
00:35:10,800 --> 00:35:14,100
that is to think about
each Turing machine

692
00:35:14,100 --> 00:35:19,230
as having a description,
which is a string.

693
00:35:19,230 --> 00:35:22,530
So the collection of all
descriptions of Turing machines

694
00:35:22,530 --> 00:35:25,440
is just a subset of sigma
star, which we already

695
00:35:25,440 --> 00:35:26,730
know is countable.

696
00:35:26,730 --> 00:35:32,210
And the subset of any countable
set is going to be countable.

697
00:35:32,210 --> 00:35:41,250
So anyway, I think
it's worth remembering

698
00:35:41,250 --> 00:35:46,890
that the collection of old
Turing machines is countable.

699
00:35:46,890 --> 00:35:49,290
Whereas the collection of
all languages is uncountable.

700
00:35:49,290 --> 00:35:52,890
And that tells you right there
that some language is not

701
00:35:52,890 --> 00:35:59,830
decided because there are more
languages than Turing machines.

702
00:35:59,830 --> 00:36:02,100
We've unaccountably
many language, only

703
00:36:02,100 --> 00:36:05,470
countably many Turing machines,
so that's fewer Turing machines

704
00:36:05,470 --> 00:36:06,490
than languages.

705
00:36:06,490 --> 00:36:09,398
There's no way to map all the
languages onto Turing machines.

706
00:36:09,398 --> 00:36:11,440
So there's going to always
be some languages that

707
00:36:11,440 --> 00:36:13,330
got unmapped.

708
00:36:13,330 --> 00:36:15,318
And so, in particular,
there are going

709
00:36:15,318 --> 00:36:17,110
to be some languages
which are undecidable.

710
00:36:17,110 --> 00:36:18,860
There are going to be
some languages which

711
00:36:18,860 --> 00:36:20,080
are not Turing-recognizable.

712
00:36:20,080 --> 00:36:25,450
And anything based on some
automata kind of a definition

713
00:36:25,450 --> 00:36:28,727
process is going to be some
languages that they're not

714
00:36:28,727 --> 00:36:29,560
going to be defined.

715
00:36:32,160 --> 00:36:34,740
OK, now what this
corollary shows you

716
00:36:34,740 --> 00:36:37,950
that there is some language out
there, which is not decidable.

717
00:36:37,950 --> 00:36:40,200
What we're going to
show next is that there

718
00:36:40,200 --> 00:36:41,610
is a specific language--

719
00:36:41,610 --> 00:36:44,690
A,TM-- which is not decidable.

720
00:36:44,690 --> 00:36:47,930
And but first, I think we
have a check-in coming up.

721
00:36:47,930 --> 00:36:50,573
And let me give you a
little bit of background

722
00:36:50,573 --> 00:36:52,490
here because this is
relevant to this question

723
00:36:52,490 --> 00:36:55,550
that I got about
intermediate size sets.

724
00:36:55,550 --> 00:36:58,100
So the question of
whether there is

725
00:36:58,100 --> 00:37:01,370
a set of size between
the natural numbers

726
00:37:01,370 --> 00:37:04,580
and the real numbers
strictly in between--

727
00:37:04,580 --> 00:37:07,100
so bigger than the natural
numbers, not the same size

728
00:37:07,100 --> 00:37:09,320
as the natural numbers,
but not the same size

729
00:37:09,320 --> 00:37:14,990
as the real numbers either,
but in between in size.

730
00:37:14,990 --> 00:37:18,500
That was Hilbert's question
number 1 out of his list of 23

731
00:37:18,500 --> 00:37:21,777
that I talked about
a few lectures back.

732
00:37:21,777 --> 00:37:24,110
It's interesting that he put
it as number 1 in that very

733
00:37:24,110 --> 00:37:28,970
privileged special place because
I know Hilbert was very--

734
00:37:28,970 --> 00:37:30,920
he felt that the
understanding infinity

735
00:37:30,920 --> 00:37:33,168
was a really central
issue in mathematics.

736
00:37:33,168 --> 00:37:35,210
And that if we can't answer
a question like this,

737
00:37:35,210 --> 00:37:38,408
we don't really
understand infinity.

738
00:37:38,408 --> 00:37:40,700
You want to understand what
kind of sizes of infinities

739
00:37:40,700 --> 00:37:41,452
are there.

740
00:37:41,452 --> 00:37:43,160
We know there's the
real number is bigger

741
00:37:43,160 --> 00:37:44,960
than the natural numbers.

742
00:37:44,960 --> 00:37:46,340
Is there something in between?

743
00:37:46,340 --> 00:37:48,800
So fundamental, really.

744
00:37:48,800 --> 00:37:52,250
But it was shown during the
course of the 20th century,

745
00:37:52,250 --> 00:37:54,980
really, in two separate steps--
one in the 1930s by Godel,

746
00:37:54,980 --> 00:37:57,740
one in the early
1970s by Cohen--

747
00:37:57,740 --> 00:37:59,840
that we can't
answer this question

748
00:37:59,840 --> 00:38:04,440
by using the standard
axioms of mathematics.

749
00:38:04,440 --> 00:38:07,750
The answer can go either way.

750
00:38:07,750 --> 00:38:11,320
And both of them are consistent
with the axes mathematics.

751
00:38:11,320 --> 00:38:13,260
So you're never going
to be able to prove

752
00:38:13,260 --> 00:38:18,660
that there is a set
whose size is in between

753
00:38:18,660 --> 00:38:20,340
or that there is no such set.

754
00:38:20,340 --> 00:38:23,885
It's just impossible to
prove either way using

755
00:38:23,885 --> 00:38:26,010
the standard axiom of
mathematics, which, actually,

756
00:38:26,010 --> 00:38:29,147
is kind of remarkable.

757
00:38:29,147 --> 00:38:31,230
And so my question for you
is-- and this is really

758
00:38:31,230 --> 00:38:35,300
a philosophical question,
not one that is directly you

759
00:38:35,300 --> 00:38:37,050
need to know about
material in the course.

760
00:38:37,050 --> 00:38:39,538
I think it's just a matter
of your own interest.

761
00:38:39,538 --> 00:38:40,830
I hope you find it interesting.

762
00:38:40,830 --> 00:38:44,790
If you don't, you can
just answer it randomly.

763
00:38:44,790 --> 00:38:50,260
But what's going on here
that we can't answer

764
00:38:50,260 --> 00:38:52,150
that question
about whether there

765
00:38:52,150 --> 00:38:54,040
is a set of intermediate size?

766
00:38:54,040 --> 00:38:57,970
Is it because our axioms for
mathematics are inadequate?

767
00:38:57,970 --> 00:39:02,470
Or maybe there is no such thing
as a mathematical reality.

768
00:39:02,470 --> 00:39:05,420
You can talk about
what's real here?

769
00:39:05,420 --> 00:39:07,120
What's the reality?

770
00:39:07,120 --> 00:39:09,905
Either there is a set
in between or not.

771
00:39:09,905 --> 00:39:11,530
If you can imagine,
all of these things

772
00:39:11,530 --> 00:39:14,200
have their own reality
to them, well, then,

773
00:39:14,200 --> 00:39:15,467
there's going to be an answer.

774
00:39:15,467 --> 00:39:17,050
And then you would
expect, well, maybe

775
00:39:17,050 --> 00:39:18,967
we can find better axioms,
which will actually

776
00:39:18,967 --> 00:39:19,840
give us that answer.

777
00:39:19,840 --> 00:39:22,540
Or you can say, well,
there is no reality.

778
00:39:22,540 --> 00:39:27,220
And infinite sets are
kind of human constructs

779
00:39:27,220 --> 00:39:30,130
anyway, so we can make them kind
of play out any way we like.

780
00:39:30,130 --> 00:39:34,570
Mathematicians argue
about that to this day.

781
00:39:34,570 --> 00:39:38,290
And it is, as I say, really,
it's a philosophical question.

782
00:39:38,290 --> 00:39:42,160
But just out of curiosity,
let's see how you guys end up

783
00:39:42,160 --> 00:39:44,270
deciding on that one.

784
00:39:44,270 --> 00:39:46,310
So here is the poll.

785
00:39:46,310 --> 00:39:47,450
5 seconds to go.

786
00:39:47,450 --> 00:39:48,930
Please vote.

787
00:39:48,930 --> 00:39:52,370
And we're going to end
the polling, 1, 2, 3, now.

788
00:39:55,200 --> 00:39:57,070
All right, here we go.

789
00:39:57,070 --> 00:40:00,220
So there's no right answer.

790
00:40:00,220 --> 00:40:05,220
I think if most mathematicians
were to, I think,

791
00:40:05,220 --> 00:40:07,830
the instinct of most
logicians, especially, I'm

792
00:40:07,830 --> 00:40:10,230
not sure if general
mathematicians really even care

793
00:40:10,230 --> 00:40:13,500
about this question, but
logicians would probably

794
00:40:13,500 --> 00:40:17,050
have an instinct that, probably,
there are sets in between.

795
00:40:17,050 --> 00:40:20,070
There's no reason that
there shouldn't be.

796
00:40:20,070 --> 00:40:21,840
It seems kind of
strange that there

797
00:40:21,840 --> 00:40:26,280
should be this sort of jump
from the natural numbers

798
00:40:26,280 --> 00:40:31,090
to the real numbers and
why nothing in between?

799
00:40:31,090 --> 00:40:36,595
But I don't think that
question is really settled.

800
00:40:39,962 --> 00:40:41,170
All right, let's continue on.

801
00:40:41,170 --> 00:40:43,630
I think we are--

802
00:40:43,630 --> 00:40:44,710
OK, so we're going to--

803
00:40:47,872 --> 00:40:50,080
our coffee break is coming,
in case you're wondering.

804
00:40:50,080 --> 00:40:51,790
So this is my last
slide before then.

805
00:40:51,790 --> 00:40:57,880
But this is an important
slide, so please hold out.

806
00:40:57,880 --> 00:41:02,170
So here is our promised
theorem of the day.

807
00:41:02,170 --> 00:41:04,510
I'm going to show that
A,TM is not decidable,

808
00:41:04,510 --> 00:41:07,280
the acceptance problem
for Turing machines.

809
00:41:07,280 --> 00:41:10,157
And it's all going to be
contained on this one slide.

810
00:41:10,157 --> 00:41:11,990
We're going to give a
proof by contradiction

811
00:41:11,990 --> 00:41:14,990
using diagonalization.

812
00:41:14,990 --> 00:41:18,620
And we're going to assume
some Turing machine, H,

813
00:41:18,620 --> 00:41:21,360
decides A,TM.

814
00:41:21,360 --> 00:41:24,300
And we're going to
get a contradiction.

815
00:41:24,300 --> 00:41:25,800
So let's, first of
all, make sure we

816
00:41:25,800 --> 00:41:28,900
understand what H is doing.

817
00:41:28,900 --> 00:41:30,340
So H gets an input--

818
00:41:30,340 --> 00:41:33,910
a Turing machine and an input.

819
00:41:33,910 --> 00:41:37,180
And H is going to be a
decider, so it always

820
00:41:37,180 --> 00:41:40,830
halts with an
accept or a reject.

821
00:41:40,830 --> 00:41:45,830
It's supposed to accept if
M accepts w and reject if it

822
00:41:45,830 --> 00:41:46,740
doesn't.

823
00:41:46,740 --> 00:41:48,620
So in other words,
it's going to reject

824
00:41:48,620 --> 00:41:53,330
if M rejects w either by
halting or by looping.

825
00:41:56,340 --> 00:42:02,530
That's the job of H. And
we're assuming we can do that.

826
00:42:02,530 --> 00:42:04,140
But we will see a
contradiction occur.

827
00:42:06,900 --> 00:42:11,450
So the way we're going to do
that is really kind of just one

828
00:42:11,450 --> 00:42:13,220
step here in a way.

829
00:42:13,220 --> 00:42:16,910
And we're going to use H
to construct another Turing

830
00:42:16,910 --> 00:42:20,840
machine D. H is going to be a
subroutine to D. We've already

831
00:42:20,840 --> 00:42:22,490
seen us doing that in the past.

832
00:42:25,170 --> 00:42:27,830
D is going to do something
a little strange,

833
00:42:27,830 --> 00:42:31,050
just to warn you.

834
00:42:31,050 --> 00:42:33,350
D's input is just
a Turing machine--

835
00:42:33,350 --> 00:42:36,170
no w.

836
00:42:36,170 --> 00:42:41,630
And what D is going to
do using its subroutine

837
00:42:41,630 --> 00:42:48,220
H is going to simulate H on
input M, comma, the description

838
00:42:48,220 --> 00:42:51,860
of M. Now what is that?

839
00:42:51,860 --> 00:42:55,310
Well, the description of
M is just some string.

840
00:42:55,310 --> 00:42:56,920
So what H is trying--

841
00:42:56,920 --> 00:43:01,000
what it's asking H to
tell, to answer is,

842
00:43:01,000 --> 00:43:07,010
does M accept the
string representing

843
00:43:07,010 --> 00:43:10,560
M's own description?

844
00:43:10,560 --> 00:43:14,140
It's as if we're
feeding M into itself,

845
00:43:14,140 --> 00:43:20,810
which seems like a totally
twisted thing to do,

846
00:43:20,810 --> 00:43:21,520
you might say.

847
00:43:25,120 --> 00:43:27,760
Why would you ever feed
a program into itself?

848
00:43:30,370 --> 00:43:32,710
Somebody has written
cannibalism here--

849
00:43:32,710 --> 00:43:34,480
yeah, kind of.

850
00:43:34,480 --> 00:43:38,050
I'd say it's worse
[LAUGHS] because it's not

851
00:43:38,050 --> 00:43:39,010
eating somebody else.

852
00:43:39,010 --> 00:43:41,900
It's eating yourself.

853
00:43:41,900 --> 00:43:49,760
OK, but I claim that there are
actual cases in practice where

854
00:43:49,760 --> 00:43:51,940
we do this.

855
00:43:51,940 --> 00:43:56,500
We feed programs
into themselves.

856
00:43:56,500 --> 00:44:00,700
And the example that I
know of where this is done

857
00:44:00,700 --> 00:44:04,030
is when you're
making a compiler.

858
00:44:04,030 --> 00:44:06,010
You might want to
make a compiler

859
00:44:06,010 --> 00:44:09,280
and then written in the same
language that it's compiling.

860
00:44:09,280 --> 00:44:12,940
And then you feed the
compiler into itself.

861
00:44:12,940 --> 00:44:15,370
You may say, why even
bother because it's already,

862
00:44:15,370 --> 00:44:17,000
obviously, if it,
once it's running,

863
00:44:17,000 --> 00:44:21,840
you don't need to
compile it again.

864
00:44:21,840 --> 00:44:25,260
But actually, an example
where this was really used

865
00:44:25,260 --> 00:44:27,960
was when there was an
optimizing compiler, I think,

866
00:44:27,960 --> 00:44:31,940
for C written on
early Unix machines.

867
00:44:31,940 --> 00:44:37,020
And the optimizing compiler
for C was written in C.

868
00:44:37,020 --> 00:44:41,280
So you would feed the
optimizing compiler

869
00:44:41,280 --> 00:44:43,950
into the regular
compiler, first of all.

870
00:44:43,950 --> 00:44:48,970
Now, you had the
compiler running,

871
00:44:48,970 --> 00:44:51,340
but it was an optimized.

872
00:44:51,340 --> 00:44:54,460
So but now that it's the
optimizing compiler is running,

873
00:44:54,460 --> 00:44:56,290
you can feed the
optimizing compiler

874
00:44:56,290 --> 00:44:58,040
into that, which is itself.

875
00:44:58,040 --> 00:45:01,290
Now you have an optimized
optimizing compiler.

876
00:45:01,290 --> 00:45:03,360
So it really makes some--

877
00:45:03,360 --> 00:45:06,330
there is at least one case
where this has actually

878
00:45:06,330 --> 00:45:09,620
been done in practice.

879
00:45:09,620 --> 00:45:10,730
Not that we really care.

880
00:45:10,730 --> 00:45:14,900
This is a theory class-- but
just for fun to observe that.

881
00:45:14,900 --> 00:45:18,880
So here, H is trying to
say, well, does in a M end

882
00:45:18,880 --> 00:45:23,167
up accepting when it's fed
the description of itself?

883
00:45:23,167 --> 00:45:25,000
You know, at least,
mathematically speaking,

884
00:45:25,000 --> 00:45:27,610
that's a reasonable
thing to ask.

885
00:45:27,610 --> 00:45:31,330
And then what D is going to do
when it gets the answer back

886
00:45:31,330 --> 00:45:32,440
from H--

887
00:45:32,440 --> 00:45:34,240
H is a decider, don't forget--

888
00:45:34,240 --> 00:45:36,980
is D's going to do the
opposite of whatever H does.

889
00:45:36,980 --> 00:45:38,470
It's going to
accept if H rejects

890
00:45:38,470 --> 00:45:42,030
and reject if H accepts.

891
00:45:42,030 --> 00:45:45,050
So let's, in summary,
let's pull this together,

892
00:45:45,050 --> 00:45:51,230
so it's easy to understand,
in the end, what is D doing?

893
00:45:51,230 --> 00:45:53,240
D is going to accept.

894
00:45:53,240 --> 00:45:55,010
D is also going to be
decider, by the way.

895
00:45:55,010 --> 00:45:57,750
So D is always going to
either accept or reject--

896
00:45:57,750 --> 00:45:58,250
Just.

897
00:45:58,250 --> 00:46:00,150
The opposite of what
H tells it to do.

898
00:46:00,150 --> 00:46:06,360
So D is going to accept M
exactly when M doesn't accept M

899
00:46:06,360 --> 00:46:12,430
because when M doesn't accept
M, H is going to reject,

900
00:46:12,430 --> 00:46:14,350
and so then D is
going to accept.

901
00:46:14,350 --> 00:46:17,490
So D accepts M if and
only if M doesn't accept

902
00:46:17,490 --> 00:46:20,130
M. That's exactly the
condition in which D accepts

903
00:46:20,130 --> 00:46:24,250
M. I think it's important to
just step back and make sure

904
00:46:24,250 --> 00:46:26,680
you understand this line
because we have only one line

905
00:46:26,680 --> 00:46:31,660
left to get our contradiction.

906
00:46:31,660 --> 00:46:32,160
Right?

907
00:46:32,160 --> 00:46:32,850
Are we together?

908
00:46:32,850 --> 00:46:35,070
D accepts M if and
only if M doesn't

909
00:46:35,070 --> 00:46:38,770
accept M. That's just by the
way we've defined setup D.

910
00:46:38,770 --> 00:46:43,790
Now what we're going to do is
feed in, instead of M to D,

911
00:46:43,790 --> 00:46:46,900
and not some arbitrary feed,
we're going to use feed in D

912
00:46:46,900 --> 00:46:50,760
into D. And that is going
to be our contradiction

913
00:46:50,760 --> 00:46:53,250
because D is now
going to accept D if

914
00:46:53,250 --> 00:46:55,260
and only if D doesn't
accept D, and that's

915
00:46:55,260 --> 00:46:56,675
certainly impossible.

916
00:46:59,520 --> 00:47:03,450
That's our contradiction which
proves that H cannot exist.

917
00:47:03,450 --> 00:47:07,270
And therefore, A,TM
is undecidable.

918
00:47:07,270 --> 00:47:10,060
OK, so we're done,
except for the one

919
00:47:10,060 --> 00:47:16,090
point, which is that why
is this a diagonalization?

920
00:47:16,090 --> 00:47:20,800
And I think you can get that
from the following picture.

921
00:47:20,800 --> 00:47:24,280
If you imagine here writing
down all possible times--

922
00:47:24,280 --> 00:47:25,610
I'm going to make a table here.

923
00:47:25,610 --> 00:47:30,490
Here is the list of
all Turing machines,

924
00:47:30,490 --> 00:47:32,860
including D, which
is a machine which

925
00:47:32,860 --> 00:47:38,320
I built under the
assumption that H exists.

926
00:47:38,320 --> 00:47:39,790
So D appears here somewhere.

927
00:47:39,790 --> 00:47:41,590
But here are all the
other Turing machines.

928
00:47:41,590 --> 00:47:43,757
And here are all of these
descriptions of the Turing

929
00:47:43,757 --> 00:47:47,860
machines along the labeling
all of the columns.

930
00:47:47,860 --> 00:47:49,390
OK, so these are
the rows labeled.

931
00:47:49,390 --> 00:47:50,710
These are the column labels.

932
00:47:50,710 --> 00:47:54,100
And inside, I'm
going to tell you

933
00:47:54,100 --> 00:48:01,120
the answer for whether a given
machine accepts a given input.

934
00:48:01,120 --> 00:48:04,630
So for example, M1 accepts
its own description

935
00:48:04,630 --> 00:48:07,000
but rejects the
description of M2,

936
00:48:07,000 --> 00:48:08,920
but accepts the
description of M3.

937
00:48:08,920 --> 00:48:09,760
I don't you know.

938
00:48:09,760 --> 00:48:10,960
I'm obviously I'm
making this up.

939
00:48:10,960 --> 00:48:11,960
I don't know what M1 is.

940
00:48:11,960 --> 00:48:16,112
But just hypothetically, that's
what the machine M1 does.

941
00:48:16,112 --> 00:48:17,570
So I'm just filling
out this table.

942
00:48:21,632 --> 00:48:24,500
H could get the answer to any
of the elements in this table

943
00:48:24,500 --> 00:48:30,440
because it can test whether M4
accepts the description of M3.

944
00:48:30,440 --> 00:48:33,020
So H could fill out this table.

945
00:48:33,020 --> 00:48:35,720
So maybe M2 is a machine
that always rejects.

946
00:48:35,720 --> 00:48:38,570
It's a very unfriendly,
rejecting machine.

947
00:48:38,570 --> 00:48:40,050
M3 is a very friendly machine.

948
00:48:40,050 --> 00:48:41,900
It accepts all inputs.

949
00:48:41,900 --> 00:48:46,395
M4 rejects some and accepts
others, dot, dot, dot.

950
00:48:46,395 --> 00:48:51,110
Now, I want to look and
see, what does D do?

951
00:48:51,110 --> 00:48:53,840
Now based on the information
that I've already given you,

952
00:48:53,840 --> 00:48:57,240
we can understand what D does.

953
00:48:57,240 --> 00:48:59,430
So for example, what
does D do when I

954
00:48:59,430 --> 00:49:03,940
feed it the description of M1?

955
00:49:03,940 --> 00:49:05,600
What does D do?

956
00:49:05,600 --> 00:49:06,610
Well, we can look here.

957
00:49:06,610 --> 00:49:10,210
D accepts M if and only
if M doesn't accept M. So

958
00:49:10,210 --> 00:49:18,090
D is going to accept M1 if and
only if M1 does not accept M1.

959
00:49:18,090 --> 00:49:20,490
Well, M1 does accept M1.

960
00:49:20,490 --> 00:49:22,980
So D does the opposite.

961
00:49:22,980 --> 00:49:24,240
D rejects.

962
00:49:24,240 --> 00:49:26,250
So OK, I'm highlighting here.

963
00:49:26,250 --> 00:49:29,250
D rejects because
D is going to do

964
00:49:29,250 --> 00:49:32,790
the opposite of what the machine
does on its own it on input.

965
00:49:35,370 --> 00:49:39,150
So D on M2, you have to
look what M2 does on M2.

966
00:49:39,150 --> 00:49:41,560
It rejects, so D does
the opposite of that.

967
00:49:41,560 --> 00:49:43,220
It accepts.

968
00:49:43,220 --> 00:49:49,100
And similarly, each
one of these things--

969
00:49:49,100 --> 00:49:52,060
D's answer is going to
be the opposite of what

970
00:49:52,060 --> 00:49:54,820
the machine does on
its own description,

971
00:49:54,820 --> 00:50:00,050
just by virtue of the
definition of D. OK, and so on--

972
00:50:00,050 --> 00:50:01,540
so far, so good.

973
00:50:01,540 --> 00:50:06,400
But the problem is, what
happens when D is fed itself?

974
00:50:06,400 --> 00:50:11,590
Because, as you can see,
we're heading for trouble

975
00:50:11,590 --> 00:50:13,990
because this is a
diagonal down here.

976
00:50:13,990 --> 00:50:15,340
D is just one of the rows.

977
00:50:15,340 --> 00:50:17,890
That diagonal is going
to intersect that row

978
00:50:17,890 --> 00:50:19,300
at this point.

979
00:50:19,300 --> 00:50:22,772
And D is defined to be going to
be doing the opposite of what

980
00:50:22,772 --> 00:50:25,105
that element is, but it can't
be the opposite of itself.

981
00:50:27,980 --> 00:50:31,300
And so that's the contradiction.

982
00:50:31,300 --> 00:50:34,740
So I think we're--

983
00:50:34,740 --> 00:50:39,190
I'm going to call us, give
us a little break here.

984
00:50:39,190 --> 00:50:41,200
And then you can also
text me in the meantime.

985
00:50:41,200 --> 00:50:43,860
I'll be happy to answer
some questions during that.

986
00:50:43,860 --> 00:50:46,170
A little over 4 minutes
to go-- so let's see.

987
00:50:46,170 --> 00:50:49,770
Let me see if I can
answer your questions.

988
00:50:49,770 --> 00:50:52,810
OK, what's so special about
A,TM that enables us to do this?

989
00:50:52,810 --> 00:50:57,020
Why can't we do this
on ADFA, for example?

990
00:50:57,020 --> 00:50:59,490
That's a good question.

991
00:50:59,490 --> 00:51:14,890
And the answer is that, in a
sense, we can do this on DFA.

992
00:51:14,890 --> 00:51:17,920
I mean, I think this is,
perhaps, a bit of a stretch.

993
00:51:17,920 --> 00:51:22,360
But DFAs could not answer ADFA.

994
00:51:22,360 --> 00:51:26,698
I mean, we could prove that in
other ways as well by just--

995
00:51:26,698 --> 00:51:28,490
we could use things
like the pumping lemma,

996
00:51:28,490 --> 00:51:31,210
and it's not clear, even
how you'd formulate ADFA.

997
00:51:31,210 --> 00:51:36,940
But what's important
here is that it's really

998
00:51:36,940 --> 00:51:40,270
the model talking about
itself that really

999
00:51:40,270 --> 00:51:42,710
is where the problem comes up.

1000
00:51:42,710 --> 00:51:45,490
So if you try to push
this argument through

1001
00:51:45,490 --> 00:51:49,390
to show that ADFA is not
decidable by Turing machines,

1002
00:51:49,390 --> 00:51:54,690
you're going to fail
because we're starting off

1003
00:51:54,690 --> 00:51:56,130
with a Turing machine.

1004
00:51:56,130 --> 00:52:00,060
And I think I'm going
to confuse myself

1005
00:52:00,060 --> 00:52:01,570
if I try to just repeat it.

1006
00:52:01,570 --> 00:52:04,350
But you won't get
a contradiction

1007
00:52:04,350 --> 00:52:07,108
because the Turing machine
is not a finite automaton.

1008
00:52:10,245 --> 00:52:10,745
OK.

1009
00:52:13,870 --> 00:52:15,880
Will this argument
get into self-loops?

1010
00:52:15,880 --> 00:52:19,060
I don't see why it would--
there is some self-reference,

1011
00:52:19,060 --> 00:52:19,693
perhaps.

1012
00:52:19,693 --> 00:52:21,610
We're going to talk about
that a little later.

1013
00:52:21,610 --> 00:52:26,800
So we're going to come back and
revisit this argument in a week

1014
00:52:26,800 --> 00:52:29,170
or so when we talk about
the recursion theorem which

1015
00:52:29,170 --> 00:52:32,740
talks about machines that
can refer to themselves.

1016
00:52:32,740 --> 00:52:35,375
But this is a little
bit of a head--

1017
00:52:35,375 --> 00:52:36,500
getting ahead of ourselves.

1018
00:52:36,500 --> 00:52:40,510
So somebody's commenting
on this reminding them

1019
00:52:40,510 --> 00:52:43,150
of the barber paradox,
if you remember that,

1020
00:52:43,150 --> 00:52:48,122
which has some similarity.

1021
00:52:48,122 --> 00:52:50,880
There is a town
in which there was

1022
00:52:50,880 --> 00:52:54,580
a barber which shaves every man
who doesn't shave themselves.

1023
00:52:54,580 --> 00:52:56,370
It seems he's a
very good barber.

1024
00:52:56,370 --> 00:52:58,710
So there are some people
who shave themselves.

1025
00:52:58,710 --> 00:53:01,950
And all the rest,
the barber shaves.

1026
00:53:01,950 --> 00:53:04,560
The question is, does
the barber shave himself?

1027
00:53:04,560 --> 00:53:07,650
Because he shaves only those
men who don't shave themselves.

1028
00:53:07,650 --> 00:53:09,890
So you've got a same
kind of a contradiction.

1029
00:53:09,890 --> 00:53:11,580
There is a relationship there.

1030
00:53:14,560 --> 00:53:17,400
So someone wants to know, where
did we use the decidability?

1031
00:53:17,400 --> 00:53:23,210
So we used the decidability
to come up with H.

1032
00:53:23,210 --> 00:53:25,220
Once we know that
A,TM is decidable,

1033
00:53:25,220 --> 00:53:29,600
then we have that H function,
and then we can build D.

1034
00:53:29,600 --> 00:53:35,550
So that's the chain of reason.

1035
00:53:35,550 --> 00:53:37,370
So you assume A,TM is decidable.

1036
00:53:37,370 --> 00:53:40,580
Then you have the
decider called H,

1037
00:53:40,580 --> 00:53:44,250
and then you can build
D. Somebody wants

1038
00:53:44,250 --> 00:53:47,350
to see the previous slide.

1039
00:53:47,350 --> 00:53:49,150
What part of the
slide do you want?

1040
00:53:52,880 --> 00:53:55,228
So I'll leave that up there.

1041
00:53:55,228 --> 00:53:57,520
Why can we apply the proof
that all Turing machines are

1042
00:53:57,520 --> 00:53:59,020
accountable to all languages?

1043
00:53:59,020 --> 00:54:02,200
Well, because Turing
machines have descriptions.

1044
00:54:02,200 --> 00:54:04,840
General languages don't
have descriptions.

1045
00:54:04,840 --> 00:54:06,640
And so that's why.

1046
00:54:06,640 --> 00:54:09,850
OK, the candle has
burnt to the bottom,

1047
00:54:09,850 --> 00:54:11,470
and it's time to move on.

1048
00:54:14,200 --> 00:54:17,370
So now, let's look at
the acceptance problem

1049
00:54:17,370 --> 00:54:20,380
for queue automata.

1050
00:54:20,380 --> 00:54:24,040
I'll give you a queue automaton
on input, and I want to know,

1051
00:54:24,040 --> 00:54:25,450
does it accept the input?

1052
00:54:25,450 --> 00:54:27,970
Is that going to be decidable?

1053
00:54:27,970 --> 00:54:29,095
And you have your choices.

1054
00:54:29,095 --> 00:54:32,830
It's either yes, it is
decidable because these

1055
00:54:32,830 --> 00:54:35,200
are similar to pushdown
automata and APDA is decidable,

1056
00:54:35,200 --> 00:54:40,730
or no because yes
contradicts results

1057
00:54:40,730 --> 00:54:42,560
that we know at
this point, or we

1058
00:54:42,560 --> 00:54:45,320
don't have enough information
to answer the question.

1059
00:54:45,320 --> 00:54:46,520
OK, let's put that up.

1060
00:54:49,100 --> 00:54:55,120
One second-- [LAUGHS]
all right, that's it.

1061
00:54:55,120 --> 00:54:56,110
Ready, going, gone.

1062
00:55:00,180 --> 00:55:04,260
So yes, the answer is, well, no.

1063
00:55:04,260 --> 00:55:06,900
[LAUGHS] The answer
is, indeed, the answer

1064
00:55:06,900 --> 00:55:12,563
is B. True, that
queue automata are

1065
00:55:12,563 --> 00:55:14,730
similar to pushdown automata,
but all these automata

1066
00:55:14,730 --> 00:55:16,740
are similar to each
other, and that's not

1067
00:55:16,740 --> 00:55:17,740
going to be good enough.

1068
00:55:20,670 --> 00:55:22,550
What the homework
has asked you to do

1069
00:55:22,550 --> 00:55:24,260
is to show that you
can simulate Turing

1070
00:55:24,260 --> 00:55:26,330
machines with queue automata.

1071
00:55:26,330 --> 00:55:27,800
So if you can
answer the question

1072
00:55:27,800 --> 00:55:31,310
about whether the queue automata
will accept their input,

1073
00:55:31,310 --> 00:55:33,470
that would allow you to be
able to answer questions

1074
00:55:33,470 --> 00:55:35,810
about whether Turing
machines accept their input.

1075
00:55:35,810 --> 00:55:38,720
And we just proved
that's not possible.

1076
00:55:38,720 --> 00:55:47,120
So it would be a contradiction
if we could answer--

1077
00:55:47,120 --> 00:55:50,480
if we could decide A, queue, A.

1078
00:55:50,480 --> 00:55:52,820
Now, we have an example of
an undecidable language.

1079
00:55:52,820 --> 00:55:56,210
Let's look at an example of
an unrecognizable language.

1080
00:55:56,210 --> 00:56:08,560
Now A,TM is not going to serve
that purpose because A,TM is

1081
00:56:08,560 --> 00:56:10,900
Turing-recognizable, as we
pointed out by the universal

1082
00:56:10,900 --> 00:56:12,470
Turing machine.

1083
00:56:12,470 --> 00:56:15,340
So A,TM is undecidable, however.

1084
00:56:15,340 --> 00:56:18,520
How about an
unrecognizable language?

1085
00:56:18,520 --> 00:56:24,020
For that, we will see that the
complement of A,TM will serve.

1086
00:56:24,020 --> 00:56:28,630
So the complement of A,TM is
neither decidable nor even

1087
00:56:28,630 --> 00:56:30,700
recognizable.

1088
00:56:30,700 --> 00:56:32,830
Now it's not
Turing-recognizable.

1089
00:56:32,830 --> 00:56:35,830
And that's going to follow from
a pretty basic theorem that

1090
00:56:35,830 --> 00:56:39,130
connects recognizability and
decidability that I've put up

1091
00:56:39,130 --> 00:56:41,620
here on the screen,
which is that if you have

1092
00:56:41,620 --> 00:56:44,170
a language where it
and its complement

1093
00:56:44,170 --> 00:56:47,470
are both recognizable, then
the language turnout turns out

1094
00:56:47,470 --> 00:56:48,220
to be decidable.

1095
00:56:48,220 --> 00:56:50,460
In fact, a language and its
complement are decidable.

1096
00:56:50,460 --> 00:56:53,020
But being decidable is
closed under complement,

1097
00:56:53,020 --> 00:56:56,220
so that's something
you should be aware.

1098
00:56:56,220 --> 00:56:59,720
But being-- OK.

1099
00:56:59,720 --> 00:57:00,970
We'll get to that in a minute.

1100
00:57:00,970 --> 00:57:07,600
But if-- so anyway, so
if you have a language

1101
00:57:07,600 --> 00:57:09,550
and it's complement
both recognizable,

1102
00:57:09,550 --> 00:57:12,460
how do we know the
language is decidable?

1103
00:57:12,460 --> 00:57:14,890
So first of all, let's take
the two Turing machines,

1104
00:57:14,890 --> 00:57:20,380
M1 and M2 that recognizes
A and A-complement.

1105
00:57:20,380 --> 00:57:22,270
And we're going to
put those together

1106
00:57:22,270 --> 00:57:28,580
to get a decider for A. And
that's going to work like this.

1107
00:57:28,580 --> 00:57:31,740
It's going to be
called T. So T says,

1108
00:57:31,740 --> 00:57:34,020
on input w, what
it's going to do,

1109
00:57:34,020 --> 00:57:37,380
it's going to feed w
into M1 and M2 both.

1110
00:57:40,582 --> 00:57:42,040
A is the language,
by the way, yes.

1111
00:57:42,040 --> 00:57:45,280
A is-- when I say it's
Turing-recognizable,

1112
00:57:45,280 --> 00:57:48,190
you know, Turing-recognizable
only applies to languages.

1113
00:57:48,190 --> 00:57:52,300
So yes, A is often this symbol
I'm going to use for languages,

1114
00:57:52,300 --> 00:57:53,650
sometimes for an automaton.

1115
00:57:53,650 --> 00:57:57,530
But A is typically
going to be a language.

1116
00:57:57,530 --> 00:58:06,978
So now, I'm trying to
make T be a decider for A

1117
00:58:06,978 --> 00:58:10,230
from the recognizers
for A and A-complement.

1118
00:58:10,230 --> 00:58:13,920
So I'm going to
take an input to T

1119
00:58:13,920 --> 00:58:17,165
and feed it into both
recognizers, M1 and M2.

1120
00:58:20,960 --> 00:58:24,230
OK, I'm going to run
them in parallel.

1121
00:58:24,230 --> 00:58:31,940
What's nice is that
because M2 recognizes

1122
00:58:31,940 --> 00:58:34,700
the complement of
what M1 recognizes,

1123
00:58:34,700 --> 00:58:40,070
every string is going to be
accepted either by M1 or by M2

1124
00:58:40,070 --> 00:58:43,880
because every string is either
an A or an A-complement.

1125
00:58:43,880 --> 00:58:48,790
So if I run M1 and M2 on w until
one of the halts or one of them

1126
00:58:48,790 --> 00:58:51,400
accepts, I know I'm not
going to run forever

1127
00:58:51,400 --> 00:58:55,860
because, eventually, one or
the other one have to accept.

1128
00:58:55,860 --> 00:58:58,607
So and then I got my answer
because if M1 accepts,

1129
00:58:58,607 --> 00:58:59,940
then I know I'm in the language.

1130
00:58:59,940 --> 00:59:03,300
But if M2 accepts, I know I'm in
the complement of the language,

1131
00:59:03,300 --> 00:59:04,720
so I'm out of the language.

1132
00:59:04,720 --> 00:59:07,140
So if M1 accepts,
then T should accept.

1133
00:59:07,140 --> 00:59:14,640
But if M2 accepts,
then T should reject. ,

1134
00:59:14,640 --> 00:59:20,340
So that proves that nice little
theorem written at the top

1135
00:59:20,340 --> 00:59:22,580
in blue.

1136
00:59:22,580 --> 00:59:25,630
So I got my decider
for A built out

1137
00:59:25,630 --> 00:59:28,900
of the recognizers for
A and A-complement.

1138
00:59:28,900 --> 00:59:34,210
Now immediately, it follows that
the complement of A,TM is not

1139
00:59:34,210 --> 00:59:41,650
Turing-recognizable because
we know that A,TM itself is

1140
00:59:41,650 --> 00:59:44,460
recognizable, but
it's undecidable.

1141
00:59:47,410 --> 00:59:50,230
If the complement was
also recognizable,

1142
00:59:50,230 --> 00:59:54,250
then A,TM would be
decidable, but it isn't.

1143
00:59:54,250 --> 00:59:58,240
So when something is decidable,
either it or its complement

1144
00:59:58,240 --> 00:59:59,683
have to be unrecognizable.

1145
00:59:59,683 --> 01:00:02,350
And in the case for A,TM, it has
to be the complement because we

1146
01:00:02,350 --> 01:00:05,530
already showed that it is
itself is recognizable.

1147
01:00:09,130 --> 01:00:10,610
So that's the proof of that.

1148
01:00:10,610 --> 01:00:12,340
So here is a little
picture of the way

1149
01:00:12,340 --> 01:00:14,890
the world looks right now if
you have here, in the middle are

1150
01:00:14,890 --> 01:00:17,230
the decidable languages--
so these are all

1151
01:00:17,230 --> 01:00:19,810
languages, this Venn
diagram of languages.

1152
01:00:19,810 --> 01:00:23,740
We showed earlier, the regular,
the context-free, decidable,

1153
01:00:23,740 --> 01:00:26,080
recognizable.

1154
01:00:26,080 --> 01:00:29,170
Here, I've got the recognizable
and what I'm calling

1155
01:00:29,170 --> 01:00:33,290
the co-Turing recognizable.

1156
01:00:33,290 --> 01:00:36,350
This is the collection
of all complements

1157
01:00:36,350 --> 01:00:37,940
of recognizable languages.

1158
01:00:37,940 --> 01:00:41,540
So A,TM bar, A,TM
complement is the complement

1159
01:00:41,540 --> 01:00:43,130
of a recognizable language.

1160
01:00:43,130 --> 01:00:48,290
This region here are
all the complements

1161
01:00:48,290 --> 01:00:53,030
of the recognizable languages
or the so-called co-Turing

1162
01:00:53,030 --> 01:00:57,110
recognizable languages--
complement of.

1163
01:00:57,110 --> 01:00:58,880
So A,TM is on this side.

1164
01:00:58,880 --> 01:01:00,950
A,TM-complement is on that side.

1165
01:01:00,950 --> 01:01:04,610
But if something's in both, by
virtue of this theorem here,

1166
01:01:04,610 --> 01:01:05,925
then it's decidable.

1167
01:01:08,600 --> 01:01:12,050
OK, last check-in for the day.

1168
01:01:12,050 --> 01:01:15,950
From what we learned so far,
which closure properties

1169
01:01:15,950 --> 01:01:18,200
can we prove for the
class of deterring

1170
01:01:18,200 --> 01:01:20,720
recognizable languages?

1171
01:01:20,720 --> 01:01:21,710
Choose all that apply.

1172
01:01:24,730 --> 01:01:28,080
Well, as I say, you don't
have to get it right.

1173
01:01:28,080 --> 01:01:30,250
Let's not spend too
much more time on this

1174
01:01:30,250 --> 01:01:34,740
because we'll talk a
little bit about it.

1175
01:01:34,740 --> 01:01:36,997
Almost all-- its closed
under almost all of them,

1176
01:01:36,997 --> 01:01:37,830
but not all of them.

1177
01:01:45,380 --> 01:01:49,190
Because-- are we done here?

1178
01:01:49,190 --> 01:01:51,430
I think we're done--

1179
01:01:51,430 --> 01:01:51,955
5 seconds.

1180
01:01:54,700 --> 01:01:56,125
OK, here we go, ending polling.

1181
01:02:00,160 --> 01:02:03,940
I'm not sure what the meaning
of [LAUGHS] deleting your answer

1182
01:02:03,940 --> 01:02:05,320
is here.

1183
01:02:05,320 --> 01:02:07,240
Everybody likes union, I guess.

1184
01:02:07,240 --> 01:02:09,130
They're closed under
all of these operations

1185
01:02:09,130 --> 01:02:10,490
except complement.

1186
01:02:13,753 --> 01:02:15,920
But we just proved it's not
closed under complement,

1187
01:02:15,920 --> 01:02:20,760
so I'm a little puzzled by
why we have so many votes

1188
01:02:20,760 --> 01:02:23,490
for closure under complement.

1189
01:02:23,490 --> 01:02:27,090
We have here, A,TM is
Turing-recognizable,

1190
01:02:27,090 --> 01:02:29,055
but A,TM-complement is
not Turing-recognizable.

1191
01:02:29,055 --> 01:02:32,270
it's right here on the slide.

1192
01:02:32,270 --> 01:02:33,770
I'm not trying to
make you feel bad,

1193
01:02:33,770 --> 01:02:41,820
but I'm trying to just point
out that you think, please.

1194
01:02:41,820 --> 01:02:48,020
So now closure under union
and intersection, I mean,

1195
01:02:48,020 --> 01:02:49,580
you could kind of
get those answers

1196
01:02:49,580 --> 01:02:51,710
just by running things
in parallel the way

1197
01:02:51,710 --> 01:02:52,880
we did the proof here.

1198
01:02:52,880 --> 01:02:56,240
You just run both machines.

1199
01:02:56,240 --> 01:03:02,043
And if they both give--

1200
01:03:02,043 --> 01:03:03,710
I mean, it's a little
tricky, I suppose.

1201
01:03:03,710 --> 01:03:08,770
If either one of them
accept, then you can accept.

1202
01:03:08,770 --> 01:03:10,360
Or if they both
accept, you just wait

1203
01:03:10,360 --> 01:03:11,610
until they both have accepted.

1204
01:03:11,610 --> 01:03:13,460
Otherwise, you
just keep running.

1205
01:03:13,460 --> 01:03:15,963
So the first two are
pretty straightforward.

1206
01:03:15,963 --> 01:03:17,380
Closure under
concatenation-- this

1207
01:03:17,380 --> 01:03:18,590
is also going to be similar.

1208
01:03:18,590 --> 01:03:21,820
You just try every possible
way of cutting the string up

1209
01:03:21,820 --> 01:03:24,440
into two pieces and
run in parallel on.

1210
01:03:24,440 --> 01:03:26,920
And if you ever find a
way of cutting it up,

1211
01:03:26,920 --> 01:03:29,540
and you run those two, and put
those two sides in parallel,

1212
01:03:29,540 --> 01:03:34,510
and if they both accept,
then you can accept.

1213
01:03:34,510 --> 01:03:36,770
And star is, again,
very similar.

1214
01:03:36,770 --> 01:03:39,340
So these are not too bad.

1215
01:03:39,340 --> 01:03:41,800
But I admit, you know, it's
not a whole lot of time

1216
01:03:41,800 --> 01:03:44,740
to have to contend with
something that you're just

1217
01:03:44,740 --> 01:03:46,990
getting used to.

1218
01:03:46,990 --> 01:03:53,270
So let's talk about the very
last topic of the day, which

1219
01:03:53,270 --> 01:03:57,050
is really going to be setting
ourselves up for Tuesday's

1220
01:03:57,050 --> 01:03:58,910
lecture next week.

1221
01:03:58,910 --> 01:04:03,230
And that's how we are going to
be showing other languages are

1222
01:04:03,230 --> 01:04:07,490
undecidable, which is something
that I'm going to be expecting

1223
01:04:07,490 --> 01:04:10,880
you guys to be able to do.

1224
01:04:10,880 --> 01:04:15,950
This is the standard
procedure for showing

1225
01:04:15,950 --> 01:04:18,620
languages are
undecidable using what's

1226
01:04:18,620 --> 01:04:21,390
called the reducibility method.

1227
01:04:21,390 --> 01:04:25,250
And what that does is it
takes, as a starting point,

1228
01:04:25,250 --> 01:04:29,090
a language that we already
know is undecidable--

1229
01:04:29,090 --> 01:04:32,630
typically, A,TM-- or it
could be another one that

1230
01:04:32,630 --> 01:04:34,760
you've previously shown
to be undecidable--

1231
01:04:34,760 --> 01:04:38,060
and leverages that information
to show other languages are

1232
01:04:38,060 --> 01:04:40,040
undecidable.

1233
01:04:40,040 --> 01:04:44,730
And it's using what's
called reducibility.

1234
01:04:44,730 --> 01:04:48,570
We're going to go into this
more carefully next time.

1235
01:04:48,570 --> 01:04:50,820
But basically,
reducibility is a way

1236
01:04:50,820 --> 01:04:54,750
of using one problem to
solve another problem.

1237
01:04:54,750 --> 01:04:58,130
And so we are going to
show, for example, let's

1238
01:04:58,130 --> 01:05:00,380
take a look at
the problem called

1239
01:05:00,380 --> 01:05:02,600
the halting problem, which
is like the famous problem

1240
01:05:02,600 --> 01:05:05,570
for Turing machines.

1241
01:05:05,570 --> 01:05:08,240
You just want to know whether
it halts, not necessarily

1242
01:05:08,240 --> 01:05:09,740
whether it accepts.

1243
01:05:09,740 --> 01:05:11,840
So it's very similar,
but not exactly the same.

1244
01:05:14,640 --> 01:05:18,680
And we're going to show
that this halting problem is

1245
01:05:18,680 --> 01:05:20,090
similarly undecidable.

1246
01:05:20,090 --> 01:05:22,520
Now we could go back and do
the whole diagonalization,

1247
01:05:22,520 --> 01:05:23,960
but that seems like--

1248
01:05:23,960 --> 01:05:26,540
well, that's more work than
necessary now that we already

1249
01:05:26,540 --> 01:05:30,830
know A,TM is undecidable because
we're going to show that we can

1250
01:05:30,830 --> 01:05:36,830
reduce the A,TM problem
to the halting problem.

1251
01:05:36,830 --> 01:05:39,510
And we'll explain what
that means again later.

1252
01:05:39,510 --> 01:05:44,060
But the idea is-- and as
we'll show in an illustration

1253
01:05:44,060 --> 01:05:45,500
shortly--

1254
01:05:45,500 --> 01:05:53,700
that by proving by contradiction
if the if HALT,TM were

1255
01:05:53,700 --> 01:05:57,490
decidable, then A,TM
would be decided.

1256
01:05:57,490 --> 01:05:59,450
And we know A,TM
is not decidable.

1257
01:05:59,450 --> 01:06:04,610
And so that's our contradiction.

1258
01:06:04,610 --> 01:06:12,590
Now the way we're going to show
that if HALT,TM is decidable,

1259
01:06:12,590 --> 01:06:18,290
then A,TM is decidable is use
a decider for HALT,TM to decide

1260
01:06:18,290 --> 01:06:28,920
A,TM with a suitable
modification.

1261
01:06:28,920 --> 01:06:35,620
So basically, we want to turn
a HALT,TM decider into an A,TM

1262
01:06:35,620 --> 01:06:36,640
decider.

1263
01:06:36,640 --> 01:06:41,020
And that's how we're going to
reduce the problem of solving

1264
01:06:41,020 --> 01:06:44,140
A,TM to the problem
of solving HALT,TM.

1265
01:06:44,140 --> 01:06:45,972
Let's just do an example.

1266
01:06:45,972 --> 01:06:47,680
If you've seen it
before, obviously, this

1267
01:06:47,680 --> 01:06:49,040
is not going to be hard.

1268
01:06:49,040 --> 01:06:52,950
But for the many of you who
have not seen it before,

1269
01:06:52,950 --> 01:06:55,470
I'm partly doing it this time
just so we can do it again

1270
01:06:55,470 --> 01:06:56,320
next time.

1271
01:06:56,320 --> 01:06:58,980
And maybe it'll sink in
by virtue of repetition.

1272
01:07:03,520 --> 01:07:06,040
So again, so as I just
said, we're going to assume

1273
01:07:06,040 --> 01:07:09,190
the HALT,TM problem is decidable
and use that to show that A,TM

1274
01:07:09,190 --> 01:07:11,170
is decidable, which
we know is false.

1275
01:07:11,170 --> 01:07:14,460
We showed it just
earlier that it's not.

1276
01:07:14,460 --> 01:07:16,480
So assume we have a
decider for HALT,TM.

1277
01:07:16,480 --> 01:07:24,000
We'll call it R. And we're going
to construct from R a decider

1278
01:07:24,000 --> 01:07:28,430
for A,TM we'll call S. OK?

1279
01:07:28,430 --> 01:07:30,650
So we're, again, typical
proof by contradiction.

1280
01:07:30,650 --> 01:07:34,100
We're assuming the opposite
of what we're trying to prove.

1281
01:07:34,100 --> 01:07:37,640
And then we're going
to get something crazy.

1282
01:07:37,640 --> 01:07:41,930
OK, so here, my job now,
is I'm assuming I have R,

1283
01:07:41,930 --> 01:07:43,350
which is a HALT,TM decider.

1284
01:07:43,350 --> 01:07:49,670
So now, I'm assuming
I know how to decide

1285
01:07:49,670 --> 01:07:54,000
if a Turing machine and an
input eventually halts--

1286
01:07:54,000 --> 01:07:54,500
Not.

1287
01:07:54,500 --> 01:07:58,390
Necessarily would it accept,
just whether it halts.

1288
01:07:58,390 --> 01:07:59,460
It's conceivable.

1289
01:07:59,460 --> 01:08:01,320
You have to bear with me here.

1290
01:08:01,320 --> 01:08:03,660
It's conceivable that
you could find a way

1291
01:08:03,660 --> 01:08:07,440
to test whether Turing
machines halt on their input,

1292
01:08:07,440 --> 01:08:10,050
even though we now know that
testing whether they accept

1293
01:08:10,050 --> 01:08:13,084
their input is not decidable.

1294
01:08:15,870 --> 01:08:18,420
So you have to be open-minded
to the possibility

1295
01:08:18,420 --> 01:08:21,840
that the HALT
problem is decidable,

1296
01:08:21,840 --> 01:08:23,819
and we're going to show
that that's can't be.

1297
01:08:23,819 --> 01:08:27,120
So we're going to show that
if we could decide the halting

1298
01:08:27,120 --> 01:08:32,479
problem, then we can use that to
decide the acceptance problem.

1299
01:08:32,479 --> 01:08:34,590
OK, so how are we
going to do that?

1300
01:08:34,590 --> 01:08:37,080
So imagine how we can
solve the halting problem.

1301
01:08:37,080 --> 01:08:43,340
So to solve the A,TM, which
is what my job is to do,

1302
01:08:43,340 --> 01:08:46,240
so S is supposed to solve A,TM.

1303
01:08:46,240 --> 01:08:49,840
I'm constructing a Turing
machine as decide A,TM.

1304
01:08:49,840 --> 01:08:51,069
I'm going to use first--

1305
01:08:51,069 --> 01:08:52,390
I'm giving it M and w.

1306
01:08:52,390 --> 01:08:55,830
I'm going to feed it into R
since that's really all I got.

1307
01:08:55,830 --> 01:09:00,810
See if R tells me what happens.

1308
01:09:00,810 --> 01:09:04,680
Does M on w at least halt?

1309
01:09:04,680 --> 01:09:08,399
Well, if R says,
no, it doesn't halt,

1310
01:09:08,399 --> 01:09:12,120
well then I'm actually done
because if M doesn't even

1311
01:09:12,120 --> 01:09:14,370
halt on w, then it
couldn't be accepting w.

1312
01:09:14,370 --> 01:09:17,399
So at that point, I know
that M doesn't accept w,

1313
01:09:17,399 --> 01:09:20,840
and I can reject right off.

1314
01:09:20,840 --> 01:09:23,899
So R, you can see how it
could potentially be helpful.

1315
01:09:23,899 --> 01:09:25,729
But it's going to be
helpful in either way

1316
01:09:25,729 --> 01:09:30,800
because if R says M
does hold, well, then,

1317
01:09:30,800 --> 01:09:34,020
I'm also good because I
don't know the answer yet,

1318
01:09:34,020 --> 01:09:38,689
but what I do know is I
can now simulate M on w

1319
01:09:38,689 --> 01:09:42,140
until it halts because
R has told me it halts.

1320
01:09:42,140 --> 01:09:47,160
So I don't have to worry
about getting into a loop.

1321
01:09:47,160 --> 01:09:50,630
So S can be confident in being
a decider for whatever it's

1322
01:09:50,630 --> 01:09:57,270
doing because I'm running
now M on w with a guarantee

1323
01:09:57,270 --> 01:10:00,380
that it halts.

1324
01:10:00,380 --> 01:10:02,960
And now, that's going to
tell me-- now eventually,

1325
01:10:02,960 --> 01:10:05,720
the simulation of M on w is
going to end up at an accept

1326
01:10:05,720 --> 01:10:06,220
or reject.

1327
01:10:06,220 --> 01:10:07,928
And that's going to
be the answer I need.

1328
01:10:07,928 --> 01:10:09,320
So if M is accepted,
then accept.

1329
01:10:09,320 --> 01:10:13,100
And if M is rejected,
then reject.

1330
01:10:13,100 --> 01:10:19,990
And that's how S solves A,TM
using R, which solves HALT,TM.

1331
01:10:19,990 --> 01:10:22,540
But S can't exist.

1332
01:10:22,540 --> 01:10:24,130
And so therefore, R can't exist.

1333
01:10:24,130 --> 01:10:28,460
And therefore, HALT,TM
can't be decidable.

1334
01:10:28,460 --> 01:10:29,170
OK?

1335
01:10:29,170 --> 01:10:30,295
So that quickly--

1336
01:10:35,630 --> 01:10:39,870
OK, I'm not sure which
diagram you wanted me to show.

1337
01:10:39,870 --> 01:10:41,490
But anyway, maybe
we can do that.

1338
01:10:41,490 --> 01:10:44,310
We're basically at the end
of the hour or end of the 90

1339
01:10:44,310 --> 01:10:44,920
minutes.

1340
01:10:44,920 --> 01:10:46,620
So let's do a quick review.

1341
01:10:46,620 --> 01:10:48,915
And if you stick around, I'm
happy to go back and look

1342
01:10:48,915 --> 01:10:51,540
at any of the other slides that
you might have missed something

1343
01:10:51,540 --> 01:10:52,490
on.

1344
01:10:52,490 --> 01:10:57,950
OK, so just to recap, we
showed that the natural numbers

1345
01:10:57,950 --> 01:11:00,950
and the real numbers are not the
same size using that definition

1346
01:11:00,950 --> 01:11:03,740
of one-to-one correspondence to
introduce the diagonalization

1347
01:11:03,740 --> 01:11:04,490
method.

1348
01:11:04,490 --> 01:11:06,950
We used the diagonalization
method to show that A,TM is

1349
01:11:06,950 --> 01:11:08,240
undecidable.

1350
01:11:08,240 --> 01:11:10,305
We also showed
that little theorem

1351
01:11:10,305 --> 01:11:11,930
that if the language
and its complement

1352
01:11:11,930 --> 01:11:15,290
are recognizable, then
the language is decidable.

1353
01:11:15,290 --> 01:11:18,950
And from that, we concluded
that A,TM complement is not

1354
01:11:18,950 --> 01:11:20,210
recognizable.

1355
01:11:20,210 --> 01:11:23,390
And then we showed, at least
by virtue of an introduction

1356
01:11:23,390 --> 01:11:26,390
to the method, the reducibility
method to show that HALT,TM is

1357
01:11:26,390 --> 01:11:27,770
undecidable.

1358
01:11:27,770 --> 01:11:29,660
And that was today's lecture.

1359
01:11:29,660 --> 01:11:32,880
And we're and we're at
the end of the hour.

1360
01:11:32,880 --> 01:11:35,900
So why don't I--

1361
01:11:35,900 --> 01:11:36,890
we are finished.

1362
01:11:36,890 --> 01:11:39,260
You can log out.

1363
01:11:39,260 --> 01:11:43,300
And if you want, I
will stick around.

1364
01:11:43,300 --> 01:11:46,510
OK, OK, this is kind of
a good question here.

1365
01:11:50,130 --> 01:12:00,790
So I'm getting a question about
the A,TM-complement, which is--

1366
01:12:00,790 --> 01:12:06,960
since we have a
recognizer for A,TM,

1367
01:12:06,960 --> 01:12:12,150
if I'm doing justice
to this question,

1368
01:12:12,150 --> 01:12:14,940
we have a recogniser for A,TM,
so why can't we just invert

1369
01:12:14,940 --> 01:12:17,950
the answer?

1370
01:12:17,950 --> 01:12:23,310
Flip the answer around, and
now, we have a recognizer

1371
01:12:23,310 --> 01:12:27,900
for the complement of
A,TM, A,TM-complement.

1372
01:12:27,900 --> 01:12:30,530
So why doesn't that work?

1373
01:12:30,530 --> 01:12:36,170
Well, the reason that doesn't
work is because the recognizer

1374
01:12:36,170 --> 01:12:42,080
for A,TM might be rejecting
some things by looping.

1375
01:12:42,080 --> 01:12:48,470
And now, if you just flip
the accepting and rejecting,

1376
01:12:48,470 --> 01:12:51,410
when it hits one of
those halting states,

1377
01:12:51,410 --> 01:12:53,660
it's going to give
the reverse answer.

1378
01:12:53,660 --> 01:12:58,130
But when it rejects
by looping, it'll

1379
01:12:58,130 --> 01:12:59,540
continue to reject by looping.

1380
01:12:59,540 --> 01:13:02,030
So you won't get the
complementary language

1381
01:13:02,030 --> 01:13:02,600
coming out.

1382
01:13:06,560 --> 01:13:15,780
So if it would be helpful, I
can go back to that slide here,

1383
01:13:15,780 --> 01:13:18,230
which proves that
A,TM-complement is

1384
01:13:18,230 --> 01:13:26,130
unrecognizable because maybe we
should start with the bottom.

1385
01:13:26,130 --> 01:13:33,170
We know that A,TM is
recognizable and undecidable,

1386
01:13:33,170 --> 01:13:33,670
right?

1387
01:13:33,670 --> 01:13:35,230
We already proved
those two facts.

1388
01:13:35,230 --> 01:13:41,020
A,TM is recognizable from
the universal Turing machine,

1389
01:13:41,020 --> 01:13:45,360
and it's undecidable by the
diagonalization argument.

1390
01:13:45,360 --> 01:13:48,060
Those two things
together tell us

1391
01:13:48,060 --> 01:13:52,200
that the complement has
to be unrecognizable

1392
01:13:52,200 --> 01:13:55,350
because if a language
and its complement

1393
01:13:55,350 --> 01:13:57,750
are both recognizable-- and
we already know the language

1394
01:13:57,750 --> 01:13:59,490
itself is recognizable.

1395
01:13:59,490 --> 01:14:01,697
So now, if the complement
is also recognizable,

1396
01:14:01,697 --> 01:14:04,155
the language is going to be
decidable by the upper theorem.

1397
01:14:10,670 --> 01:14:12,980
So it must be the case had
either the language itself

1398
01:14:12,980 --> 01:14:16,948
is unrecognizable, or its
complement is unrecognizable.

1399
01:14:16,948 --> 01:14:18,490
We know the language
is recognizable.

1400
01:14:18,490 --> 01:14:20,560
That's what the universal
Turing machine told us.

1401
01:14:20,560 --> 01:14:22,540
So the only thing left
is for the complement

1402
01:14:22,540 --> 01:14:25,300
to be unrecognizable.

1403
01:14:25,300 --> 01:14:28,690
You should review that
if you didn't get it

1404
01:14:28,690 --> 01:14:32,560
because this is the
kind of reasoning

1405
01:14:32,560 --> 01:14:35,930
we're going to be building
on things like that.

1406
01:14:35,930 --> 01:14:40,510
So I think it's good to
make sure you understand.

1407
01:14:40,510 --> 01:14:41,010
OK.

1408
01:14:44,147 --> 01:14:45,730
OK, the diagram on
the right-- so this

1409
01:14:45,730 --> 01:14:47,350
is just a Venn diagram here.

1410
01:14:47,350 --> 01:14:49,420
I threw this in at
the last minute here.

1411
01:14:49,420 --> 01:14:53,640
I was worried about
it being confusing.

1412
01:14:53,640 --> 01:14:55,030
That part is--

1413
01:15:00,570 --> 01:15:03,240
I'm trying to show that the
three classes that we've

1414
01:15:03,240 --> 01:15:04,590
already talked about--

1415
01:15:04,590 --> 01:15:06,780
the languages which
are decidable,

1416
01:15:06,780 --> 01:15:09,210
the languages which are
Turing-recognizable,

1417
01:15:09,210 --> 01:15:11,160
and the languages
whose complements

1418
01:15:11,160 --> 01:15:12,780
are Turing-recognizable.

1419
01:15:12,780 --> 01:15:17,290
Those are three separate
classes of languages.

1420
01:15:17,290 --> 01:15:22,630
And those come up here
in those three regions.

1421
01:15:22,630 --> 01:15:24,450
These are the decidable ones.

1422
01:15:24,450 --> 01:15:27,240
Here are the recognizable ones.

1423
01:15:27,240 --> 01:15:31,930
And here are the ones whose
complements are recognizable.

1424
01:15:31,930 --> 01:15:36,780
Now if a language is in
both the recognizable,

1425
01:15:36,780 --> 01:15:38,310
and its complement
is recognizable--

1426
01:15:38,310 --> 01:15:41,460
so it's in both of these
bigger regions here--

1427
01:15:41,460 --> 01:15:45,980
then this theorem tells
you it's decidable.

1428
01:15:45,980 --> 01:15:49,600
So that's why the intersection
of these two regions

1429
01:15:49,600 --> 01:15:51,940
is marked as being
decidable because that

1430
01:15:51,940 --> 01:15:52,870
means you're in both.

1431
01:15:57,510 --> 01:15:59,580
OK?

1432
01:15:59,580 --> 01:16:03,090
But we know that A,TM is
sitting out here as recognizable

1433
01:16:03,090 --> 01:16:04,170
but not decidable.

1434
01:16:04,170 --> 01:16:06,720
So A,TM is in the
recognizable side,

1435
01:16:06,720 --> 01:16:12,450
but it's not on the complement
of recognizable, A,TM itself.

1436
01:16:12,450 --> 01:16:15,840
The complement of A,TM is the
complement of a recognizable

1437
01:16:15,840 --> 01:16:22,120
but itself is not recognizable
and not decidable.

1438
01:16:22,120 --> 01:16:26,180
So you got this side
of nice, you know--

1439
01:16:26,180 --> 01:16:28,340
I hope you think
it's nice, but it's

1440
01:16:28,340 --> 01:16:31,710
sort of a try to summarize
things in this little Venn

1441
01:16:31,710 --> 01:16:32,210
diagram.

1442
01:16:35,200 --> 01:16:40,610
So I think I'm going
to then sign off.

1443
01:16:40,610 --> 01:16:45,690
And I'll see you all on Tuesday.

1444
01:16:45,690 --> 01:16:47,100
And have a good weekend.

1445
01:16:47,100 --> 01:16:48,770
Bye bye.