If you

were

right for the invitation.

I want to talk today about this
notion of what is this something that

for those of you who want.

I want to talk about randomness and

in particular viewpoint often
taken by computer complexity and

namely the point of view is that
people using the randomness.

You know have limited resources in
particular with a focus on time but

then we'll talk about some other things.

So after the plan of
the talk with the full.

We're going to count out time for

I'm going to sort of bring
everyone up to speed about

computational complexity in a few
slides talk about official times and

what we mean by the problems and
the people centric problems.

And then we start talking about randomness
and there will be like two parts in

the stock one sort of demonstrating
the power of randomness and

then over the demonstrating it's RICK NESS
and you guys will have to decide what to

believe in particular in this
second part we'll have to

actually sort of define give a definition
for randomness or so the randomness and

we'll see this connection and this how
that's rough this randomness paradigms of

connection between computer difficulty and
randomness and

then depending on time with
hopefully go into other settings

where randomness is relevant in trying to
save space instead of time and in systems.

So let me start.

Please stop me if something is not clear.

So I want to just high level view of
what are how that is a problem for

our standpoint there are just
the most cited by example.

So an example of an easy problem
is multiplication if you have

two numbers you're supposed to multiply
them then it's an easy problem.

You know it's easy because rely on it in
elementary school actually in second or

third grade and the algorithm we learn is.

You know clearly if we have very
very large inputs sort of the great

quadratically with the number
of with the size of the data so

quadratic is a polynomial polynomial
time out of the returns record easy for

this stuff namely efficient
algorithms as an example of an easy

problem What's an example of a hard
problem they involve a problem.

Factoring into jobs I give you and
its edge and ask for it still

ask for its factors for this problem.

The best known algorithm
is exponential time.

So the best running time something like
exponential in the lot then is that and

digits in the input.

And so it's a hard problem well
on the one hand we don't know

maybe two more somebody will come up
with a much faster car but if if that's

the best kind of on a time of financial
then we'll call it a heart problem.

I just want to point out that all of
us think it's a health problem or

everyone who ever uses a credit card
by shopping online believe it's a hard

problem because the Internet security
electronic commerce is protected by

basically rests on this
assumption The only is how any

way from this size was remember polynomial
time is easy exponential time is hard.

I just want to give another example
of how that is the problems and

for those who are not in this field.

It's sort of also in some senses and
there are variations of why it's so

hard to tell whether the problem is hard.

Similarly looking problems can have vast

computer is not a difficulty and
let's look at the following problem

the problem of color maps of the process
you have and as an input to a problem.

I map and with the end countries and
you want to color it's a coloring

means that adjacent countries should
get different colors actually a canal

several questions about computational
questions about this input for

example you can ask Is it true
caliber some maps are some maps are so

it's a good question Is it a hard or

is the problem right.

It's easy because basically the grid the
algorithm works I mean you if you color.

This country read it and

forces all its neighbors to be blue which
enforces all the neighbors be red and so

if you have a get stuck in a conflict
that you know it's not true.

How about as far as we know and
if they are good.

It does.
World for sure.

And as far as we know it's the best way
we know how to solve it is expansion of

time so it looks.

We're not sure but
it does look very hard and

I could ask another problem is it
is it a hard or is it problem.

It's not about that better than
is it sort of a trivial problem

when you can answer a question without
looking at their input it's very very

similar very similar looking problems
can be can have vast difference.

Different computers not be
behavioral difficulties so

it's hard to tell if it's a problem.

It's hard but sometimes.

So we see here another problem
which seems hard but we

don't really know sometimes in countries
that are complex or we don't know whether

the problem is hard or easy but sometimes
we can relate and so for example.

This problem has a very very special
feature that if you have never

seen it should be somewhat shocking.

If you have a fast method for
solving the current problems of maps.

You also have automatically a fast
matter of fact the integer and

this is comes from this
basic understanding

of that and to complete problems I'm
not going to explain what they are but

this is one of them free
coloring is one of them and

because of this we have this
consequence as fast algorithms for

color and light fast algorithms for.

Factoring for example and
there are many such problems across

science this is a very important
phenomenon and all I'll say about it is

that the problem is that first of all you
are now that the way if you never had it.

You have now a way to tell your friends
what it is it's just a question.

Just a question.

Whether the Caliber has the coloring
has a fast algorithm but

in fact it's captures
something very deep and

captures the best in some sense whether
mathematical creativity can be automated.

So this is not the topic of the cause and
effect of the lecture it's exam later to

the topic of the lecture but still we will
connect this to undermine as let me just

summarize this part the complexity part
by just asking this first fundamental

question which is probably the most
important question of computer science.

There are these problems that we think are
hard does not solve problems like facts or

Intel and many others I list that they
are satisfied because you're from without

traveling salesman story important your
equations computing strategies and

games and so
on the best way we can solve them.

Today is by Panel time I'll do it and then
we have no method of showing that they

require more than when our time is this is

what we think is it's exponential
time our response our size

they know how the necessary for at least
some of them that they then to complete.

Once that's what everybody believes
that is a conjecture that's what

everybody believes there's no way
around exponential search for

some of these problems but we don't know.

So now you'll have to just
remember this bitter and

will come back to it in
about fifteen minutes.

OK question so far about this.

All right so
now we are going to handle my thought.

I want as I said the first part is that
the power Fantomex in computation and

the way you demonstrate this is just by
showing examples of lots of publicity

allegory times.

For which this publisher
calculations are extremely fast.

For nominal time but it's.

As we know there are now that
I mean if you can a loss.

If you try to solve these problems that
are mystically they seem to require

exponential time.

So there is an exponential gap in the
theming power fund on that for these and

many other cases are good for that.

Let me just make sure that this concept
of a public scalability MISKELLY of these

are just algorithms computer programs
that are allowed to toss a coin to make

decisions randomly so they have available
to them is an unlimited sequence

of independent unbiased random bits
which they can use in the situations and

of course then I'll go with them you know
is there is useful and there is out there

on the algorithms we'll consider is our
voice and the rich answer correctly.

Whatever they're supposed to do
with a high probability that

the ninety nine percent.

OK so probably to calculate and he's
one with whatever it's supposed to do.

Does it.

Maybe it may has some
ever about the negative.

OK it's a good question why why
should anybody tolerate this.

I mean why should review and
consider publish a calculator.

But of course there are many
good answers to this question

one that's probably best is that we
tolerate the holes everywhere else.

Why not give them

this part of life but
actually errors in public categories and

behave better than I think
any uncertainty in life.

I mean in allegory terms you could
reduce they're all just by repeating

the algorithm many times taken majority
of all the uncertainty of that you will

be alive while trying to cross the street
does not increase with a petition for

and of course then he leaves and

is that we can do things with
is that we cannot do without it.

So we tolerate for that.

OK so let's see examples of probably
six algorithms actually the best

the prime example we had for
the gap of public sector Mr Galbraith and

was taken away from us a few years ago
it was this problem given any thought

at the end they just in for
just a terror with a primer on and

in the seventy's people discovered
a fossil basic algorithm probably and

it was important for many reasons first
of all it's really I mean there were

published calculations before but
this is the stuff that the big time

the field of publish the calculator and
the other thing is the connection to cook.

Thought of it again because the thirty
five primes is absolutely essential for

people graphics system so
that was a very big result and

the question of finding a similar
the terminated algorithm was open for

a long time but
it was a result about a decade ago.

And so.

So that's a polynomial
time algorithm also and so

this is not an example of the power form.

However if you slightly change
the problem you get a very good example.

So consider the following
problem about primes.

I give you an eye thousand and I asked.

Give me a thousand digits and
such are just finding that out spines.

Probably Stickley if it is that you pick

our stars and then integers at random.

OK.

That's the meter for primality and

I claim one of them will be a prime with
at least ninety nine percent probability.

Just because the plans
are dense in the center and

that's the we can do it using this
always exists doesn't matter.

However if you as far as we
know the best arboretum we

know today to find a large integer and
Bijan integer takes exponential time then.

More or less mass a lot of them and so

that's an exponential gap
nobody knows any better.

OK Let's move to another clip show
a few more examples like this.

So let's see something from a different
mathematical field algebra.

So that the term answer of the so

called random of the matrix here is
a product of valuables Here's a product of

differences of pairs of are about
what this ask is whether this

part is the same as this
part it's probably not.

Now is it true this identity for
some people know of their heads.

So it's good that

they are not random on the so
that's not the real

question whether it's still on I was
rather quick question is how did you know.

Well I mean OK or are all
mathematicians you know how in your

proof that and
actually it's not just not a difficult

process more of an exercise to do
it by induction but more generally.

I mean mathematics is full of algebraic
identities and the people opposing them.

Usually you know it would
be nice of them to know

that what they're working some sometimes
years to prove actually is true or

not so often there are some intuition and
so on but for

this type of problems are you can you
can do better than just intuition.

So let me explain what type of problems.

Am I talking about.

So we have some a polynomial here
which is given implicitly given by in

this case you know by a formula.

So you are given polynomial in and

variables of some of the greedy and
you are asked whether it's identically

zero not that identity you want to prove
and the question is can you have a.

A way to know whether it's thrown
a quick way to know whether or not.

What a polynomial is zero exactly when
all these questions are zero for you.

Why don't you expand the formula and
find out whether it's now but

that's of course a stupid way because and

then via the ranks of
the polynomial will be exponential.

So that's not a good idea one
really nice way to do it publicly.

And that's all that simple allegory
tonight is almost the first thing

you think about you want
to know whether it's on or

just plug random values to the input and
evaluate it.

You can evaluate it because
you have a formula for it.

So what happens.

And in fact what the but

it's true is even if you plug random
values from a small integer range.

If the problem here is identically
zero no matter what you plug into it

you'll get zero and
it turns out that it's not much harder to

prove there's a problem there is
not identically zero then almost

everything you plug into it
will give you a non-zero value.

Basically because the variety of zeroes of
this polygon it has measures the one space

that's basically for integers you have to
work a little harder but that's basically

it so it will give you the correct
result with very high probability so

it's a fast are published a category time
for this problem in general if you want to

do is determine if the best way we know
how to do it is exponents of time more or

less if they're expanding it all
trying exponentially many value.

So again an exponential gap
in the power of probably six

versus deterministic algorithm.

And that's actually also very
applicable algorithm but

let's not talk about Mali examples.

So now this one is sort of an analytical
example about computing for you for

your question.

Suppose I give you.

As an import some description again maybe
of the formula and implicit description

of a function on the blank you're OK
on to the end and I also give you

some fresh upon it and what I asked
you is to find all that out for

your questions of the function of
the high modes of this function.

Right.

Exponentially many so
it's not clear how to go about it.

That's what I said accusations about you
know that not too many can be allowed just

by parts of our right so you can have more
search least in terms of the output size

you know that it's OK but how to find them
right turns out the theory of the not so

obvious the taught how to find
them even publicly at least

in the previous two examples there was the
not so I'll probably stick math out but

there is an algorithm published
algorithm levying that I'm not

going to describe it's quite sophisticated
has some adaptive something in it but

anyway it's a good publicist algorithm.

It's a very important result
because it's being used in

basically all over the place and
people are extended it's not only for

this group but basically for
any other billion.

So it's a very basic result very important
published a calculator man again you would

like to have a deterministic one.

But again I'm repeating myself to make
this point clear the best that there

are mistakes are great and we have all
this problem is also exponential time

more or less running for
potential for question.

OK last example people in the audience
some people in the OF THIS know it and

I'm probably thought of the other
people in the audience but

let me tell you about is
do matter if example our.

So you're given a contract bought.

It's a poly top so you are given it
by the bounty hyperplane by the Linn

are inequalities and you're asked to
estimate the volume of this body so

again it's in high dimension and
why am I asking to estimate

the volume other than to compute it
which is a monitor our QUESTION Well we

know that computing exactly is hard
computing volume is how the of them.

This and people obviously talked about so

let's talk about estimating the volume
let's say within a ten percent of one

percent something like that our body but
imagine this is not into them and

since about in the Thousand there
mentions How will how can we do it.

So this is absolutely beautiful are going
to die off is uncannily falling.

We are supposed to estimate the volume so

might as well put a fine grade in space
and estimate the number of integer points

inside this body that's obviously
the same the greatest find in our But

how are we going to do that how we're
going to count this number of points.

Well then I think the very
fundamental observation

is that approximately counting a discrete.

That is essentially the same problem
as almost uniformly something from it.

So we want is a random point
inside our right to do that.

Well now we can just walk around
only on the inside the body so

we start somewhere and
just start taking random walk right.

Remember Ganesan either mentioned but
we can just implement it and

each time cause outside
the body which we can share.

We immediately retract.

So it's a mark of change
here now it's easy to

check that this Markov chain is basically
the metropolis are going to Mater converge

a straight uniform distribution
on this set of points we want.

So in the limit around.

Point with age is a uniformly random
point but how long does it take.

I mean that exponentially many points and

the deep thing is to actually
arises Markov chain a spec for

the gap is to prove that it converges
in polynomial time there's another

very beautiful and powerful probably
cargo term of a different nature that

in polynomial time efficiently returned
the answer to this very basic problem.

And again if you want
to do it the time and

if you create a best method
with no excess sponsors OK So

this is my set of examples I could give
many more but I think you got the point.

So here's a first second fundamental
question is what we are seeing reality

a figment of our imagination or rather of
a figment of our inability to find good

that I mean is it out of all the problems
that have published a polynomial time ago.

It does not have know the time and
it is a power real or imagined.

And these examples and maybe many
others I could support a conjecture.

Yes people have tried to do it.

And so that's a start of
this problem of randomness

rest as the does it help
to save time on and

that's a good time to remind you
of the previous conjecture that we

talked about fifteen minutes
ago about whether N.P.

requires or whether the just the type
of space you don't care about and

B. the type of problems we talked about
in the beginning I think coloring and

solving equations and
probably does at this one of them require

exponential time mind you
that's also there of the sort

of that's what I'll believe but
people really believe things.

Almost every one is
that the answer is yes.

So OK so
we have a two fundamental questions and

we have what about seem to be
obvious conjecture for each of them

they don't seem to be very related to each
other and so for those who haven't seen it

before it will come I think pretty much
as a surprise for a sea of about them.

So one of the conjectures is for
Again they are related.

They're very strongly related.

If the answer for one of them is yes
then the answer to the other is not

to make sure this is clear.

We'll go into this in a second.

I want to just say it again

and this is an understanding
that was developed over

the course of about twenty years and
that's the connection

I promise to talk about between
computational difficulty and randomness.

So intuitively what about
the film says direction

that everybody believes So
if you ask me which one of them is

true of course both can be false but
if one of them is true is this one.

OK So the way it goes.

Is it says that if there are natural
health problems this natural problems

again probably sells man.

Strategies in chess and so on.

If any of them is exponentially
higher any of them any one

of them is that financially hard.

It means that randomness can
be officially eliminated.

So all the examples riffs in what
we had a published a calculator.

We can get a deterministic are greater
which is as efficient namely also

polynomial time maybe not exactly as
efficient but of a polynomial time.

Families are several
ways to phrase it but.

Perhaps the strongest result
of this type is that if and

be required to explain itself I
suck at how to a requirement.

It's exponential for the said
Robin says man then every Probably

not a great time has a deterministic
counterpart randomness is when

I'm going to prove the theory
on to you in a minute

let me check that it's clear
what the statements are but

nobody asked questions in
georgia tech in general yes OK

thank you have run me.

Right.
I said exactly what what I still myself

that they are going to
access to an unlimited

supply of genuine independent
unbiased clients offered here.

I didn't talk about where you got them.

Yeah.

Now we could talk about this is
a very interesting topic and

this is all so far so

far I don't know what
the polarizer uniform distribution.

OK good.

So I'll let me just say
something about this.

So with proof of this in a second.

I want to say.

So that's already a connection
that's already a connection between

how the thing randomness which goes
in one direction if you have a health

problem you can only move around and
from other GO IT times and

I just want to say that
the results that actually.

Go the other way.

If you can remove randomness
from algo it and then you can

prove low bond you can prove how you
can provide for much of our problems.

The results are so far partial and
we are still working and understanding it.

But this connection how that's where
sound on this is actually two ways.

We'll talk on the about this one way you

know not finding that you know that
something is not forms of the difference

between assuming that circuits
are large and time is along.

Is this non-uniformity aspect.

There are analogs of this theory or

AM us on their uniform assumption but
I won't talk about.

OK we have them.

So we are going to prove it.

Well of course they were
not going to prove it but

I'm going to show you they
censor ideas in this.

OK OK So

because of that we understand and

they find this notion of computational So
the randomness good enough randomness.

OK So we want to prove that

using hardness we can remove
the randomness from and

that I mean if any publisher
calculates not just our loitered

as I showed you and not just there were
times I didn't show you that we know but

also from algorithms that
were never invented there.

How can we how can we are you such and
make such an argument how can we prove

a statement about any publisher Calgary's
them including those we haven't seen so

we have to argue generic the generic
algorithm is just a computer program

official computer program polynomial
time that takes two kinds of inputs

a real import about which it wants to know
something like whether it's a plan on and

this you want to form
distribution this random me both.

And then it outputs something some of
course that's a distribution here and

what we want to do is to be able to take
it take such any such allegories them and

make do without the random and
that's a tough.

That's what we promise and then your
algorithm should be century as official

and absolutely key to doing it is

to removing it is to understand
what this algorithm DARS

on other distributions other
than the uniform distribution.

So let's just imagine this
imagine this algorithm

being run under a different distribution
just arbitrary distribution.

Maybe by the currency of our buyers and

they planned that them now or do we can
just feed it and see what what happened.

See how what's the ratio between this and
this.

And now you can make a definition and

that's the causal definition mixed made in
the context of cryptography by a group of

us and
McCully just core such a distribution.

So the random if you know it looks
random to fall back to cup purpose so

if success probability here was ninety
nine percent maybe success probability

here will be ninety eight percent or
that would be the only difference.

OK so that's a pseudo random distribution
out of sort of one of the submissions.

Of course there are the runs of
one the uniform one is OK and

think statistically close to it are.

But we are trying to eliminate randomness
altogether so we should be interested

in such distributions which
are sort of on them and

have very little entropy like zero.

OK that's what we really want.

So how can we get so I mean do
they do such things exist what do

we really want to do we want to start with
you know bit you know on the base at all.

And somehow.

The Tara mystically and

efficiently want to generate
a distribution which is sort of on them.

I did have around Zone One.

Maybe we can't and we don't care about.

I mean so the random is as good
as a now this is not quite

possible if you want an entropy here
because we know that entropy just cannot

be generated deterministically But
let's give ourself a break and

start with a few under a few truly
random bits of distribution on K.

which will be some constant times log and
random bits.

Give me this.

And later we'll see why it's a justify
the assumption but so we are allowed.

When we are doing this deterministically
officially generating a sort of random

distribution to start from a local
rate McCleary long random C.

OK And if we have this so
that's what we want to do is sort of

want to generate a distribution obviously
which will have just logarithmic entropy.

Nevertheless we'll be sort of on the if
we have this we call it a sort of random

generator that's what So the end of the
writer is for the purpose of the story.

So how can we do that so that me show you.

OK Well somewhere we have
to use our assumptions

somewhere we have to use some so
that we have a heart function.

OK so

that's how we will see what the first
we want to generate exponentially more.

So the random bits and we have random bits

but the how we do is actually
not going to show you how to

generate exponentially many more I'll
just show you how to generate one more so

than the missile given to K.
probably random bits will generate K.

plus one that cannot be distinguished
from random by efficient procedure and

using this sort of device.

Not in the sun and I like that.

It turns out that if you can.

One All you can do exponents
are the many more.

So that's why I'm not
going to explain this but

let's go out to do one more once you
are down there sort of almost obvious

what you should do because that's what
you have and that's what you want and

you have Kerry why don't you take them OK
that is probably you get the last one and

you're the thermistor if there's
nothing you can no more and more and

you want this to be sort of for and
on even if you are unpredictable.

Given the previous ones.

If you have the time and if that means
you're just kind of tired function.

That's all you can do which function
the heart function we have so

that's what we are going to do we are
applying the health function we have and

it's hard but
on the longer it makes the small

c it's still so
this will still be polynomial time.

So you can apply.

Now this is our function by sort of
hard maybe tolerance and it's hard but

only on very few in fourth what we want
this to be a half hour want a function

that completely unpredictable
to any afficionado of time and

this is one of the another gadget
with called the other certificate and

that takes a function which
is hard in the worst case and

converts it to a functional that on
average I actually becomes unpredictable.

So that the cells that idea as a go
it will construct in the generator.

Now let me explain why we could
assume that we have so many there.

So if you.

Why do I require

both Can you repeat which
which to find your.

No no I didn't fully why Rice just said
I did not require this I said this

is arbitrary.

For example you can think of
it often you know bias or

it's just an arbitrary distribution.

I don't not require it.

I want you know I don't want to assume
anything about it in particular

it could be.

I don't want.

So again I don't want any I'm just saying

little an arbitrary distribution
forget what's written here just

an arbitrary distribution it's not
a requirement I want to do it now.

Any distribution I just
classified a pseudo random.

If it makes There are greater behave
just as if there is not the requirement.

So it's just meant that it can be as
bad as you know by a dependent by them.

OK So

why why does it allow us this so
the random generator to learn eliminate

the Undernet from every officials out of
North and well because we are going to do

the following We know that with this
distribution the algorithm will be full.

But this is a business
has very small support.

I mean it just starts with
a log very quickly long seed so

what we can do is just try all
possibilities of the longer it makes

it right so there are just probably
not marry many possibilities.

Each of them generates a sequence.

We run the algorithm and we know that on
the distribution on the small collection

of random in inputs the algorithm
will be correct with probability

ninety percent so ninety eight percent
of them will give the correct answer.

Take a majority vote to get
the correct answer for.

So OK that question

about the Terminator simulation of any.

Algorithm now.

Yes OK What do you have
an end to the say exactly.

That's right yes and you know.

So you know that if this is a run down
it induces a distribution on sequences

here on this and for the say sequences and
you know that under this distribution.

It's a pseudo random distribution
because of this algorithm will

behave as if it was looking at
the borders to be a uniform distribution

algorithm gave the correct probability
with ninety nine percent success.

This one will give probability
ninety eight six percent success

which means that of this polynomial
in many imports ninety eight

percent of them will give the correct
answer either more take a majority.

OK OK Now

this is this idea of the most

randomness from every algorithm but if
there's an assumption it's yours as this

assumption that we have this real have how
it functions but I want to point out is

this idea of pseudo random generation so
the randomness actually allows you in some

cases to remove one donors from particular
other great terms without any assumptions

primality testing result I mentioned of
Agarwal care and success of this famous

the term mystical way something that's
biology works exactly in the spirit.

This guy's designed their own
published a calculator for

primality that they could understand well
enough the randomness requirements off and

then they designed a pseudo
random generator for it.

Which floors it only but
without an assumption.

That's how it came to be and I'll show you
one another one that has to do with space.

OK so now it's a good time.

So I have fifteen minutes.

Yeah right.

So I think I'm leaving my
way we can stop anytime

I want to tell you about other contexts
in which randomness may be powerful or

maybe not so let's talk about space
complexity here is a very basic problem.

It's just finding your way in an unknown.

It's a very uncertain problem.

Hope everybody remembers the knowledge

it's a very modern problem you know
you get to a place you don't know and

you have to get to Atlanta for the first
time and you have to find out that.

OK what if you have a map it
is about you don't have a map.

You don't have a map you just
sort of a local view you can

go to an intersection and right or
left and you don't in fact have enough

space to draw a map if you have enough
space to do a map you just explore and

but you have only you know you
can't write down anything based.

OK So what do those are formally
this is like having a lock space

you just know where you are the name
of a Veldt X. in your graph.

What do you do to find your great
idea is again Georgia Tech.

I don't have proof that it's
a good idea as a random or

I saw this wonderful are going to do the
following You know before you go look for

them out there go to a pub
have a few of the beers and

then you're start going out
sort of randomly in town and

what they say I'm saying is that not
only will you find your daughter.

Quickly but in fact you will find
all of those in film and about and

so on and that is graph you will
reach every corner in this town.

OK with a very high probability.

So that's a really beautiful result and

the natural question is
Well how about the time.

Instantly the time mystically for
forget there is no party and

this was a very long open problem
which was resolved just very few years

ago by on my right grog beautiful
way that cost guides editor Mr Cool.

That is as good as a random or

it is a polynomial can be
generated in a lot space and

it's text lost any draft any
city in polynomial time and

doesn't remember it it doesn't
it's best to write down and

also here it's a different organization
based on so the randomness.

But the through the randomness
the software so

the one that still is very different and
it has to do with the expander graphs and

particle construction of expander graphs
using what's called this exact product.

Yes something very good.

So this is yes as you can tell by

the picture there are no one way
streets it's a great question

that we don't know how to answer what
to do if the city has one way street.

So remember we were blinking so
we were not driving.

So we could walk both red but if you.

Yeah it's a great open question

about a similar result for directed graph.

OK Last topic.

OK very good maybe even
five minutes about systems.

So again what I saw from.

So far I depressed you know that
randomness is never helpful.

But I want to show you at least one
setting where randomness is not just is

sort of magically powerful and
that's idea of course that's a set in.

Of course systems that want to explain
what published it's supposed to

stands out.

I'm sure that this is not that you
don't normally learn about them in

mathematics but what are published it's
about let's see what the tell me I suppose

systems are first of all what they are for
that for finding out the thoughts about

various mathematical statements and
the way they operate.

I mean what's the process that my post is
that is always a very fire somebody for

creation process and I'll go into them and
efficient algorithm that has

that takes three and
puts a claim in an argument that satisfies

the users think that the logic of
the satisfy when you have a poor flight or

if the claim is true that at
least some argument which

convinces this very fire that they had to
case what we call the climate here and

we call that argument the proof that
completeness side on the other hand

if the claim is false that nobody
can convince the very far right no

argument will be convincing
it will reject any attempt.

That's the soundness that's And
all not opposed to them

but the public support system
Well we have an algorithm here.

Let's allow it to be published a letter
published it post this time of course

now we can have again can have Also let's
say what we demand incompleteness and

soundness completeness we just say
that if the claim is true then

there is an argument that convinces
this verify with probability one but

if the claim is false.

Let's allow a tiny bit of
OK let's say one percent.

There are so
you can convince this very fire.

With probability at most one
percent over is random choices.

Of course every five can
reduce it by a petition so

whatever you're happy with so
that's a problem system.

What can we do with it.

Of course you know I think mathematicians
may may shudder to think of

situations where there is
a mistake in the post but

something arms are not as serious
as other feelings and effect and

I'm serious about this lots
of mathematical claims

are happening every day on your
computer as it particularly in these

situations where you are convincing
something someone else that you

are doing the right thing when you
are implementing some published reports or

it's a matter of the magical claim like
I chose to primes and multiply them.

But of course I'm not telling you what the
primes are so that's a mathematical claim.

How do we convince somebody
else that what we do.

OK.

This is our publicist proof systems
are very interesting and important so

let me tell you about true properties
remarkable properties you can have for

publicly posted something which
you cannot without this once and

I have a slide on it.

So what is the first published
eclipse checkable poor.

Probably sickly talk about poor again
you can think of the claim being some

mathematical client prover
claims to have an argument for

it and the very file just wants
to know whether it's wrong.

So just checking correctness and it's
best to think of this guy as a referee.

So he or she does not have time to waste
on this first course of the game and

I thought that they still
believe me I've met a friend.

So I don't have to deal with this.

So what is a piece of peer
publicly checkable post this time

it's a public system which
simply forbids the very.

Fired more than one hundred
bits of the four B.

So you look at this is they kill us.

I mean this cannot have both soundness and

completeness Harvard bits of a long
post cannot find bugs in it.

If that's what you think you are wrong and

that's a very very important
basic results of precipitately or

I am certain that every post every
mathematical post can be converted

officially disappear into
a post that can very files

by a random project every server

post can be verified this way of course
it's still an official forward for

reading up as there is at the bridge
application of this understanding for

proving how the next of
approximation algorithms.

So it's very important.

So let's run one magic
that can happen with with

around on that fundamentals can
allowed us to check both by looking

at them at all that resolves
the concern of the verify

let's look at the concern
that might have processed and

this is your knowledge post and for
those still we are in the same setting.

Somebody has a proof of some
famous mathematical conjecture.

What's to convince somebody else but here
the concern is of the poor for you and

I is going to tell it to the other guy and
the other guys will claim that my post and

publish this is basically
a copyright issue

that's what they wanted
to pour looks ridiculous.

Yes like the previous one.

But your source of before
that went something to look

at it looks ridiculous probably do and
this story is doable.

Again not only are there always posts
possible the universe of every.

Every mathematical proof can be converted
into a pool which convinces the others

without a villain anything about it except
a cork not unlike the previous one.

This comes with an assumption and it's
basically a cryptographic result if you

wanted to film the basic to the graphic
assumption that practical use out of that

you have one way function but with it you
can you can have that on our support and

that's a major use of
it is into programming.

So again something that you cannot
do deterministically So summing up.

We talked about looking at random.

That's one of its authors are limited.

This is a game we play not
just with randomness but

with a lot of other basic notions
learning and knowledge and proof and and

such are they all cases rely on
something when we look at them for

compensation they are things
that dies from randomness we saw

that in some sense it's in the eye only
to the computer power of the beholder.

And this sort of under Mr finish and
that how they're having their problems

can generate good enough wonderfulness
to fall publish to calculate times.

And so they are not as
powerful as they may seem and

in some cases we saw that this
can be put without assumption but

thank you thank you.

And

yes.

Yes I completely agree.

So so when I said weeks that's all I
said is that having random bits or

not having them does not affect much
the computer complexity of the problem but

you're absolutely right that conceptualise
are very powerful tool because many

algorithms are discovered first by many
deterministic outgrowth of by now it's

sort of a little in the way you felt
discover published a calculator and

then you know you remove the randomness
from it without the some sense and

the other advantage is that you know I
was obviously cheating in the in saying

that polynomial time is sufficient
in all these cases in essentially

all these cases that we know the publicity
algorithm is still much much faster so

that the counterpart that
the term is the counterpart

in this image is general
simulation if we started with and

published a cargo of time and implemented
this sort of hands on the idea based on

the hardness of traveling salesman where
maybe it would be and to the two hundred.

So that's not that's great but
even a concrete case of life primality

testing the best are going to the public
algorithm runs about and people time.

OK and that the time is the one around
the best Verizon I guess runs about and

to the fifth time.

Well guess what.

Nobody is using that on this one for

various reasons because this is just
part if they are of the time and there.

So it is also a practicality fish and
see importance despite this is a good

point there are not

the not so there's absolutely no
relationship that I see at all.

The P.C. is basically an information
theoretic result a complex

information theoretic result
which is analogous to code but

it basically says if you can take a poor

sort of eval collect it so that if there
was a said I will collection with that.

Quality you can take a bow and
arrow corrected construct for

me to another poor so that if
the region are headed back somewhere.

It will be distributed everywhere so
that but it's information theoretic

zero knowledge is
the difference is several

aspects first of all it's assumed
something so it's a complete result

it assumes that you know the problems like
factoring that the guy you are talking

to your convincing some somebody else
your post cannot fact or integer.

So there is an assumption there is
a computer always on the other that

specified but
the system in this case is interactive so

I don't just do a poor like in the P.C.P.
but I actually interact with you.

You can ask me questions and so on.

At the end of the day you
are convinced and I reveal nothing but

it's an interactive.

So that's another I don't see any
connection between between the two

are very very serious

from our questions also

gave me a new wave.

Well it's hard to hear the thing for
you OK I

know that

that's a number that.