But the right really really
really really just pretty.

I mean.

What are they really right.

You are right.

And we're supposed to.

Yeah but the one.

You hear is all of us.

If you think a lot of people

program which is really something you're
just not going to say much about this

particular talk but
I know you've used your.

Other problem before right.

Talk about these recent workplace you
know sort of the best paper right.

Thanks.

Thanks for the introduction and

the so that part of this talk is
meant to suggest a kind of duality.

So we're going to talk about two problems.

That are both basic in
the computer science.

One is to try to design algorithms that
efficiently analyze circuits in some way.

And the other is the problem of trying
to design efficient circuits for

any official algorithms or

prove that there aren't efficient
circuits for inefficient algorithms.

So it's really a kind of duality
jested between algorithms so

proving upper bounds for
algorithms and proving lower bounds.

So limitations on our and well the nice
things that come out of this work is that.

In many cases.

This is sufficient to actually find
an efficient algorithm in order to get

a lower bound which runs counter
to all of our intuitions that

we publish lots of papers on algorithms.

Every year but you don't see
too many lower bound papers but

in fact rhythms in certain situations
can imply a lower bounds and

sometimes our things actually exist.

So you get the lower bounds.

So yeah yeah.

So efficient algorithm and
certain settings for

analyzing circuits imply limitations
against circuits and things.

Everything will be made clear and talk.

It sounds.

Mystical but it will be made clear.

So the first thing I'm going to do
is introduce you to these problems.

So what is this problem circuits for
algorithms what is ours for circuits.

And so that will take some time and
then we'll look at finding faster.

Our thumbs and how they imply a lower
bound so it's a new direction in

which just finding slightly faster
islands for certain problems improving on

exhaustive search will get you
surprising the strong lower bounds.

And then we'll see a particular example of
that I won't go into too much detail but.

We'll see this highlights of the proof
that not interested exponential

time doesn't have E.C.C. circuits whatever
that means it will be defined later.

And then we'll talk about
where we can go from here.

OK So what do I mean by
circuits versus algorithms.

So with algorithms we did because
the Imagine having a single algorithm it

gets inputs of arbitrary length and
it produces some answer.

With circuits were allowed much more so
this is so

called non-uniform computation we get.

Distinct program a Suburban that
we can run on inputs of Linked In.

And the point is that.

We have a separate program for
each input link in this.

So the program link the entire
program link the union of

all the circuits could be infinite
in itself infinite in description.

So we run a substantial.

Inputs of link then and so on and
the size of these programs can grow.

So one particular class of study a lot
is piece last poly which is the class of

problems solved with one of these infinite
families of programs where the program

size is a fixed ball Gnomeo and in the
running time of that program on inputs of

Linked In is a fixed polynomial on what
does this have to do with circuits.

Well there's an equivalent form you
ation of this where we allow a different

logical circuit a suburban and
we run all inputs of Linked In So

we have a small circuit for inputs
of length one and it gets bigger and

bigger as inputs get bigger and bigger.

But it doesn't grow more than polynomially
So probably is a in this framework.

The problem solver with
the circuit family a sub and

with a number of gates they
sit in a movable Gates

going here in a logical circuit is
in moves to fix polynomial and it.

OK.

So a huge open problem in complexity
theory is to show that in P.

problems like satisfiability
are even in peace last poly

This is a harder question in Peter's in
piece if you go to work on this question.

Good luck.

But what's interesting about proving
something like this is that.

Because we are allowing a separate
logical circuit on inputs of links in

a proof this kind of thing
would presumably entailed.

Coming up with a relationship between
how the size of a program must grow

as the inputs grow so.

So we would hope from the.

Moving stuff like an N.P.C.
poly We got some sort of almost every type

lower bound we could show something like
every logical circuit sewing all sentences

is on ten of the eight clauses
requires at least say to sixty eight.

OK So
this these numbers are totally made up

you know this is something that would
be extracted from such a proof.

If we could prove it.

So what's the basic problem
of algorithms for circuits.

Well we take some generic computational
problem where circuits or inputs.

So we're given a small circuit
see if we wanted term and

whether See satisfy some property.

Or can the property be could
be many different things.

One famous property is circuit
set is there an imput that makes

a circuit print one or not.

Another problem is counting circuit set.

So is a number of inputs
make the circuit print

one at least half all inputs
greater than half of all inputs.

Then there are some trivial
problems like zero said if I

give this thing all
zeroes does it print one.

OK so the the properties can be trivial.

They can be Sharpy complete
it could be N.P. complete.

So such problems are almost always
hard to solve rhythmically.

And these problems of this kind can
typically be solved in two to in times

time exhaustive search time where N.
is the number of inputs to the circuit and

S. is the size of the circuit so
just evaluation all possible inputs.

And the question is can we do
better than this trivial exhaustive

search are there faster algorithms for
circuit analysis problems.

That's the algorithms for
circuits question in general.

So the kind of circuit analysis problem
will look at is the set problem.

So it's in a sense very simple because for
any.

Particular class of boolean circuit
you might have you can define some C.

Set problem for that class of circuit C..

And it's just I give you a circuit case
from that class is their assignment and

makes the circuit print one is the circuit

trivial is that a trivial
function is that all zeroes.

You know equivalently.

But negating things is that all ones.

So do you just checking on whether or
not it's a trivial circuit.

And as I mentioned earlier this.

Problem is N.P. complete for
essentially all interesting class A C. and

of course we can solve it
in exhaustive search time.

So what has been studied
about this problem.

Well for simple enough circuits if we
make the circuits really restricted

we know a faster so
the three set problem can be seen

as very very restrictive
circuits where we have and or

as of three variables which
are either positive or negated right.

And in that case we have all rhythms
the form to the in minus and

over C. times K. were C.
is some fixed constant and for K.

set in general this vein in
here will grow like a and

OK but is K. grows our improvement
over to the in gets weaker and weaker.

So we get some kind of improvement but

it's still an exponential time proven
in still some experimental time.

Right.

And there's been a long line of
work in the most recent one is

that I have to moan her plea
in Fox a couple weeks ago

we show that three set can be solved in
one point three zero eight to the end.

So this steady progress.

On these kinds of things.

Now also for

counting the number in satisfying
assignments to a case you know formula.

You also have faster our thumbs.

Now we can generalize things a bit and
say OK.

What if instead of just debt through
circuits we look at constant debt servants

say ANS of or
that ends with knots at the bottom.

And this is a c's erode set.

Now it's known even there you get
some kind of improvements but

the improvements get worse as the depth
increases and as a size increases so

this is some recent work of impious of
Matthews and material that will appear and

so to use the phrase easier said we
can get slightly better are things.

And for this class A C.

C. which I'll just describe very briefly
here and go into more detail later.

It's a constant that with and
Gates or gates not in some funny

model in gate which is basically computing
divisibility by some integer him.

And there we can also get some
small improvement over to the end.

We can also count the number of satisfy
assignments and he see in this time.

And so moving up to stronger things.

What about generic formulas over AND
OR and NOT.

So their roles on thought him as
a result where he shows that that

formula set you can count
the number of science science.

Again and slightly faster than two in time
but this only works for formulas of size

seat times and so it only works for
sufficiently small science formulas.

OK Now what about circuit set.

We have just generic circuits over and
or not.

Let's say their particular size and
the number of inputs.

Where we know they're nothing as
far as I know so we we don't know

of any improvement over just an exhaustive
search trying all possibilities.

Not even an asymptotic improvement and
will see that this question.

So it seems like a very simple just
improve slightly exhaustive search

that actually deeply related
to the problem of proving.

Limitations on generic circuits over and
we're not.

So just advertise where this work
comes from the topic of algorithms or

circuits at concerts a very small part of
this area of exact algorithms for N.P.R.

problems where you want to solve the
problem you don't believe be equals N.P.

but you want the fastest hour of
them anyway so what's a fast car and

you can get sometimes you
can get to the Square event.

Better than through the end.

Sometimes you can only
get one point ninety and

sometimes you don't know anything so
understanding.

What we can do is the main
goal of this area.

OK so that's circuits.

Now what about circuits for Allison's
though again recall that piece poly is

a set of problems solvable with
one of the circuit families.

Where the number of gates is
polynomial in the input size.

So most boolean functions
require huge circuits and so

most functions that you would come across.

If you were to pick random functions.

They need enormous circuit so
most functions are not going to most

languages just take random language
is not going to fall in peace.

But we'd like to know
specific language is so

we'd like to know what
can be done in piece.

So what sorts of our thumbs can be
simulated with small size circuits

when I give you the ability to have
a separate program for each input length.

So so what sorts of traditional uniform
ours can be similarly in this way

that's the circuits for our sins problem.

The problem is that not uniformly can be
very very powerful we can have extremely

simple circuit families that
compute undecidable languages so

trying to understand what
can be simulated in pieces.

Paul and what cannot is quite difficult.

So for example one big open question
is whether or not a term a sick.

Excellent.

Time is in peace poly.

So just in English can all problems
that have exponentially long solutions.

So in N.P. you get Palami along
solutions you can verify here you have

exponential long solutions you verify.

Can these problems be solved with
polynomial size circuit families

seems kind of ridiculous it would
mean that in a sense every.

Exponential long solution can
be compressed down to a poly or

size circuit and we know that exponentials
are much bigger than Paul mules.

So this should work.

Another way of saying is give an infinite
pre-processing time to set up

a different program for each input length.

Can we then solve in X.
problems with some very small programs.

I mean seems strange with
this kind of thing is open.

We really don't have good
ways of attacking it.

Well maybe we do it depends on whether or
not you believe the rest of my talk.

So one think that's really
nice about the circuits for

algorithms question is that they have
interesting consequences regardless of how

they get result if if
exponential time has pong

your size circuits that's that
imply something interesting.

We know that.
Thanks Michel.

Time doesn't have power and size or has
that implies another interesting thing.

So a result that.

Back to the famous paper of corp and
Lipton so

that if exponential time is everything
computed an exponential time actually had

Pono side circuits we
would have a proof that.

And slight strengthening of this do
the I think it's usually pretty common.

It's a knock off is that if every problem
solver want to do the in time has circuit

smaller than say one point
nine nine to the in.

Friendly many input links you can still
prove that P. is different from and B.

So just getting some so
if not you know for

many if having a different circuit for
each input link helps even.

Point zero zero one in the exponent.

We can still separate
people on the other hand

it's known that if exponential
time is not in pieces poly.

This implies forms of
pseudo random generators.

So this means that.

For randomized algorithms we can
come up with a terminus equivalence.

They don't run too much and

they don't have to do exhaustive search
over all random sees some significant

speed up over exhaustive search in
the simulation of the randomized.

So famous theory of impious when
bigger sun is that is some problem and

say to the nth time the circuits larger
than one point nine nine to the end for

almost all input links in peak was B.P.
So randomization for

Paul the meal time is just the same
as a term instict long time.

So either way understanding.

This question.

We know that it gives really
interesting consequences.

When it was OK and
even if we show that in X.

weren't in pieces Poly that would give
us some non-trivial the randomization

of Merlin Arthur games basically
the probably stick version of M.P.

inside of the terminal stick
version of N.P. in P..

You

mean as far as circuits go.

We don't really know anything better
than say take the best our grip and

you know and rewrite it as a circuit
with say logarithmic overhead.

In whatever the running time
was logarithmic overhead.

Yeah yeah I mean for

B.P. of course we know it has no necessity
but like you know for things like factory.

So this evidence or

circuits in the phrase is trying to design
a faster circuit analysis aren't so

given a service input analyze it more
efficiently the circuits for algorithms

problem is the problem of trying to design
small circuits to simulate complicated

ones running and not in terms of
exponential time or exponential time.

So we'll mostly study
the negation of this.

We don't believe the circuits are there.

We don't believe that say exponential
time things can be simulated by palm

sized objects.

Even if I give you a different one for
each input link so mostly study this and

the big question is can we use one of
these kinds of tests to inform the other

can it be that interesting
circuit analysis our thumbs.

Tell us something about
the limitations of circuit so

if I can analyze a circuit
faster than exhaustive search.

What does this say about
the limits of that circuit

say something about the inability
of that circuit to do something.

OK so
this question has quite a history and.

You have some of the highlights
that history here.

So if you take the contrapositive
of what I said a few slides back.

If P. equals N.P.
If we have perfect circuit analysis for

circuits at Parliament
time errands to get set

then we have that special time of
course exponential size circuits.

OK So in that case perfect perfect circuit
analysis gives us a circuit lower bound.

But there are other things like
the natural proofs barrier for

us were often routes can
be seen in this way.

So informally what they say show is that

if your circuit lower bound proof
constructs upon a time hour with them

they can distinguish part of functions
with respect to your circuit class

from easy functions respect that sort of
class then in fact this is too much and

so your circuits are actually too
weak to even support strong suit of

random functions and because we believe
that pseudo random functions exist.

We don't believe.

That such proofs will be able to give us
the like separations of P. from and B..

So again this is a kind of circuit and

Alice's out of them
showing weakness circuit.

Then there's another.

Hearing from the ninety's called
the satisfied really coating them but

to repeal like insane where they
show a particular image says

basically casing their formulas with few
satisfying assignments actually have

a very compressible satisfying assignment.

So like by setting just a few variables
you can recover the rest of the Simon and.

This little structural limit was
very interesting in that they gave

a faster arm for case at using a lemma and

they also proved circuit sizes were bounds
using the same limit in a different way.

So having some nice structural
lemma got algorithms and

they got lower bounds as well.

So another example is from so
yes so they showed that.

So they show that the PIRA the function
they should really tight circular bounds

of the period function and they nailed
it down to even the leading polynomial.

So we know that from its results
that parity isn't in a C.Z.

or with to the square root of it.

The nail down up to the polynomial
in the in the front of that.

So another example is from cabinets and
also they show that some expansion time

mountains were part of your identity
testing also give us circuit size lower

bounds in a certain sense but
here the SO expense of time has to

be of the form to the end of epsilon So
it has to be really good circuit analysis.

And finally example is
this minimum size problem.

Where you're given it to the end bit
truth table of a boolean function and

you give an integer K..

In just one though.

Does this function have
a circuit of size in most cases.

So this is in fact a problem and in M.P.
because the input is of linked to the N.

so guessing the circuit for it won't
take more than order to the in bits and

then verify all possible inputs it takes
and we're more than Paul Gnomeo into

the end it was actually N.P. problem and
they show that if this problem.

Turn out to be implemented time then in
fact you could get new maximum circuit

complexity lower bounds but also you would
get that there are no strong suit of M.

generator so so

maybe this thing doesn't hold depends on
your beliefs about a sort of Ram Singh.

OK So that's just an overview of circuits
for algorithms and I don't see circuits

now talk about a new way in which
our thems can imply lower bounds.

So suppose we have an hour of them for
the circuit set problem

where the circuit has into the K.
gates for a fixed constant and inputs.

Suppose for all constants K.
we can get out when that runs not into

the IN TIMES into the K. but something
like To the end over end of the ten K.

here ten is just some small constant
maybe make it four or five.

So this is like a very small
improvement over exam so

it's not like one point nine to
the In my case at our thumbs.

So in fact this kind of circuit analysis
has already imply that in X.'s and

pieces falling.

So even getting a tiny improvement
over exhaustive search and

since of like it's come to
nominator would already imply.

That X. can have places.

So in fact this thing can be generalized
to different circuit classes.

So I suppose you could solve formulas set
in such a fashion into the end time then

that would entail that in X. isn't in non
uniform in C. one basically says an extra.

And have palm oil science
formulas even one uniformly.

And A C C set.

Not if it's in less than through the in
time we also get a lower bound for A C C.

And moreover it doesn't have to be
an impi complete problem that we solve

in order to get these lower bounds.

So suppose I give you a circuit C.

It's either unset or has at least
half its assignment satisfying and

you just want to turn which is the case
and I want to do it not using randomness.

So the user end of this problem is trivial
to distinguish yet so I want you to do it.

Through the end over in
the tenth time deterministically

if you can do this in turn a sickly
In fact you still get extras and

pieces probably so in fact very very weak.

DRAM as Asians.

Already get circuit lower bound.

So out of the insights into the analyzing
circuits translate directly into

limitations and circuits.

So why my something like this be true.

I'm saying that you get an algorithm
in actually implies a lower bound.

So why is something like this hold.

Well OK.

Suppose you can inspect the circuit.

And determine whether or not it prints
a bit and do it in less than two in time

then for that same class of circuits
we get the circuit lower bound now.

So we're turning positives.

Into negatives.

OK So what is this thing
saying saying well a faster.

C. set out with them is uncovering a
weakness in representing computations with

these C. circuits the C. circuit cannot
hide input from me and makes it plain.

When it's not like a black box if it were
a black box it would take to in right.

So I can beat the black box by
looking inspecting the circuit.

And so.

So somehow I'm showing that C. set that C.
circuits are weak by solving C.

set faster into the N..

So I'm showing that Caesar gets a week.

And another way of looking at is that
a faster C. said our of them is uncovering

a strength in our rooms at which one
in Lesson two that in and telling you

something that can be done and listen to
the end you didn't think could be done.

Namely finding input which
makes the circuit print one.

So just very very informally showing
that the circuits are weak and

that lets an exponential time a strong.

So how I'm going to separate the two and

show you that there is a problem and next
doesn't have C. circuits following size

is that again this is like you know
some really massive handwaving.

But that's something to which
an as to why this thing.

Might work and why it might be true.

So let's look at the sort of vaguely
go over how the argument works.

So assume that in X.
has polar Sysiphus from a class Q.

and this Q. set problem for this class.

Q Can we saw them listen to the N.

who want to take these two assumptions and
derive a contradiction.

So that if we actually get the Q.
set out with him in X.

can't have coincide seconds.

That's the basic idea.

And.

In fact you can take
the argument of corp Lipton and

Meyer from nine hundred eighty and put
it in this kind of framework as well so

assume that people's M.P.
an exponential time as part of the size

circuits you want to derive
a contradiction from this.

So the first observation is that the
expansion of time isn't peace lost Polly.

Then.

For every exponential time machine.

So it runs in say to the nth time
there's a small circuit see if

it goes a computation
history of this thing is so

enclosed like a whole computation
Tablo of that machine running.

So more formally for every string X..

If I go.

The circuit.

X. and I and J. will print the content and

say the Jets sell the Jets register
of that machine at the ITE step.

And it's appalling the size of the so
this is not too hard to show.

Now the behavior of this machine.

Given this.

This circuit exists and
can be simulated in Sigma two P..

Namely by just guessing the circuit that's
in coding this computation history and

then trying all possible steps
in all possible cells J.

we can check the circuit is making
consistent claims about the cells around

J In the steps around
this is just the usual So

many proofs of the eleven theorem
work like this you may have seen.

So here we're just say guessing a circuit
that encoding is Tablo then checking

all these claims.

Now the point is that this inner
part is a koan P. predicate.

So this is some universal guessing and

this is a problem real time check
because C. is just a bonus.

I circuit.

So just checking that the thing is
consistent with the transition table

of the machine is something
that's easy to check.

So the point is that because the peak was
empty because we're assuming people's M.P.

we can replace this inner part with
a poly all time compute will predicate.

OK.

And now this is an empty predicate.

So we can replace this whole thing
with something in polymer time.

So we take an exponential time machine and
simulate Impala real time that contradicts

the time so it can't be both people's M.P.
and X. is in peace and.

So that's how that argument works.

So we use this assumption to get small
objects and coding large computations.

And then use this to simulate.

The large computation
faster than it should be.

So the basic principle
is used in our proof.

So we have to start from a different.

Kind of connection by impossible
comments and vigorously from a one.

So they show that if not in
terms of exponential time

as Palmer size Q. circuits.

Then in fact their Q. circuits see again
in coding tell blows and this means

that for every non-deterministic
to the end time machine and

every string this circuit C. is going to
print the G eight zero at the step for

some accepting path of the computation.

Now that this is actually a very
non-trivial theorem getting

a circuit that actually encode something
for a single accepting computation path

is not easy to do in fact the proof goes
through results and the randomization.

So.

So it's very nice result.

Now the behavior of this machine can then
again be simulated in Segment two Pete.

Now just guessing the circuits in going
the Tablo and just doing as we did before.

Now the question we have now
is we'd like to use our Q.

set our of them instead of the Palm
real time hour and for set.

So we need to somehow express
this thing efficiently

as a queue set instance somehow.

And provided we can do that then we
could replace this thing with say a to

the end over in the ten time predicate so
this an inner part and then we'd have

this machine which ran into the in time
can now be implemented to run into the end

over end of the tenth time and that
will contradict the non-terminal stick.

That's that's the idea at a high level.

Now the.

The big catch is expressing this
thing efficiently as a queue set and

that's that takes several steps.

So how much time.

OK.

Right.
So I'll just briefly go over

what you need to do here.

There are basically two problems with the.

There are basically two
problems one is that.

The number of bits being fed
to the circuit set in stone so

you can think of this is a circuit set and
since the number of bits being fed to this

thing is like two in not in so the number
of inputs is too large to go from so

here you get like forty in here if you
grant through the rest of first problem.

The second problem is that OK this is a Q.
circuit C.

is a Q. circuit but to check that C. is
making consistent claims about everything

that may not be describable
as a circuit from this class.

Q So there are two things to worry about.

So just very briefly talk about them.

There's another way to encode efficient
computation so or efficiently encode.

Exponential time computations where
instead of having a huge table.

You just on the step print order
in bits of Compu of content

representing that thing in step by and
this thing lets

us reduce the number of inputs the circuit
set instance down to something like him.

Plus five log.

So we get the number of employees down and
then the second step is

that we need to use the assumption that an
excess power Messias or get some class Q.

so we're assuming that Hugh circuits can
do tons of things exponential time things.

So what we can do is we can get help or
circuits from this class Q.

they're going to help us check
different parts the computation and

the check those different
parts are correct.

We used a cue set our rhythm to check to
say this part is equivalent to that part

and by checking whether or not.

This you know say the two
circuits with an equal gate above

it is in a knot is unset say.

So we use the cue set to check that
our guesses are going to work out.

Anyway that's just very very high
level idea of how the thing goes.

Yes So.

So there are there are certain.

Operators that need to be satisfied.

You know in order for
the the proof to work out.

Yes So depending on how so
they're different versions of the proof.

So one.

So one version of proof means that say the
class is close in a composition circuits.

So like if I take one circuit family and
there is circuit family and

I think their composition.

Let's say then this is also
in the surrogate family.

But they may not be true for

circuits that they have a special
output gate at the top so.

So there are certain properties that
have to be satisfied in order to.

It doesn't go through for every class Q.
but but most classes.

Yeah yeah it could be that like if they
you insist on having certain get to

the bottom with like bounded Fane
in there and it's only like that.

Three.
But

then you compose of that
six as another thing.

So that's just

a very broad overview
of that now I'm going

to show you how we actually get
a lower bound from this connection.

So how can we prove that an X.
is not in this class a C.C..

So this was the connection that we
showed and in fact we're going to give

explicitly a faster satisfy Billy
hour than for these eighty circuits.

So now it's time to define what is
a C C Before you go any further.

So he circuit family has
the following property so

as we saw earlier the circuit
takes in bits of input and

outputs a bit the gates of this
circuit can be and or not and

Madea and they have valid fan for
each gate and the modern gate.

So given the X. one two X. T.

is input it prints want if and only if
the sum of the bits is divisible by M..

So for.

Arp.

Services equal six was already
like wide open territory.

So thank him equal six it was six and
more with the circuit has

constant depth in the pin of in so you can
take the constant to be equal to three.

So I think that the group
of three equals a six.

Understanding this classes was over.

So here's just an example circuit so and

is eleven the size the number of gates
this five and the depth is three.

So just some remarks the default
size of the circuit.

When I forget to tell you.

Maybe is pong only one in OK and
more this is again a non

uniform model of computation so you can in
fact compute some undecidable languages.

Even in this class but
at the same time we don't think eighty

circuits are very strong so they can be
efficiently simulated by constant that

threshold circuits constantly
are neural networks.

So we don't think they can do very much.

Even though we get to design
a separate circuit future.

So where does AC come from.

Yeah.

Yes this slide will answer your question.

So yes well you can

if you didn't raise your hand and
was like OK thanks.

So so where does a C. come from.

It looks like some weird.

And so Terry thing like what is
mob stuff about so well it started

from people trying to prove and that
doesn't have all of the science circuits.

And so they started with
constant depth and I try first.

Sex and
sips are showed in the early eighty's and

the parity of in bits is not in a C.Z.
row.

And this is size as you see but
without the mod him.

So and or is in knots constant up.

Was both and
Smolinski extended this to show.

Suppose you want to
compute three function.

OK so they show you can't do that and
easier if you have two gates.

So if I give you the parity function for
free as a primitive then what can you do.

Well you can't do much three.

And if I give you month three.

You can't do much to.

And in general if being Q.
distinct primes then you can't do P.

Even if I give you my Q.
and or not constant.

K. so.

So the next question is OK what if I
give you three and mod two for free.

Six.

It's equivalent to months six so
Berrington suggested this A.C.C.

is the next step and he conjectured
that the majority function and

bits are just computing the most
popular bit is not in A.C.C. and

we believe this is true but
some time went by and we couldn't show it.

So in the early ninety's people
said OK find a function in M.P..

That is not in A.C.C. so prove the clique
function does not have depth three.

So it gets a modest six.

Some time passed and not much happened.

And so in the late ninety's
it was conjectured in X.

is not in A.C.C. so if I find like
something with experts here long

solutions that can be done
with that three months six.

Yes laugh.

Laughter is the appropriate
response to this really I mean it's

like over time this thing ballooned into
the most embarrassing conjecture and

complexity theory which.

You know stories also where is all
that I mean yes so we can prove it

and it goes through this satisfiability
our rooms imply a lower bounce connection.

OK so how does this new algorithm work.

Well uses some fairly old ingredients.

So one is a known representation of A C C.

So a representation due to being
on Tyree is that every A.C.C.

function can be efficiently
put in the following form so

it can be to the compose into a G. of h's.

Where H is some sparse multilinear palm.

So it doesn't have any squares in it and
as I evaluated over all this

possible Cyrus It takes a value between
zero and K. where K. is not too large.

So it's quasi polynomial
in whatever the A.C.C.

circuit sizes underlying and
by sparse here.

I mean it hasn't most K..

On Wheels and it.

OK.

That's age and what is G is just
some arbitrary look up table.

So it takes something from zero to K.
in Princes single bit.

OK now because this is quasi polynomial
we could just write the thing down.

We could just represent it
with quasi polymer you size.

This is a zero one table.

So we have this known
representation of A.C.C. and

it was derived as far as I know for the
purpose of trying to prove lower bounds.

Now the second ingredient we use
is a fast for a transform for

multivariate multilinear polynomials.

Which lets us quickly evaluate such
an age when all that's possible.

Simon's.

Now idea is we're going to combine
these two ingredients together.

To get our faster set of.

So the fast Free Transform thing is
just conversion from the coefficient

representation to the point representation
say in the bullion hypercube so

that doesn't depend on the sparsity
of the sparsity is necessary for

getting a set algorithm anyway.

For reasons that we'll see.

So there's some there's some other tricks
being used in the set written which would

get us there so just to show you

how simple this thing is so

that there is given a multilayer
Paul Gnomeo and Co version representation.

We can get the point representations on
the bullying hypercube so we can compute

evaluate effects on all these points in
Paul Gnomeo M. or ties time per point.

OK So
the idea is very simple we can just write

a multilinear polynomial in this
kind of form where these are all

the one on wheels at the pin on X.
one and these are those that don't and

we want to the table that contains
a value of age and altered in points.

So algorithm is extremely simple.

So if the number of inputs is
one we just return zero and one.

OK.

Otherwise we are cursedly
compute the table.

T. one for each one in the table T two for
H. two OK.

We get back the wanted to do and
then we can catch the table.

T. to with the point why
some of the wanted to do.

That's it.

So we're using the fact that
these are polynomials we're

working with that we can pull out
variables and things are community.

OK so
those running time as this recurrence.

We call this thing twice
on to the minus one.

And that things also true in terms of
poly and so it's really that simple.

Now.

How does a city set of them go.

So we can.

So just precisely for
all that there's an epsilon So the A.C.C.

set with the inputs into
the epsilon science can be solved.

And to the end minus about
in the epsilon time.

So recall that we just needed
to end over end of the ten.

So here are getting a nominee or
two to the salon so that's good.

So the proof is by picture we start
with some some exponential size

circuit see A.T.C. It has in input.

OK so we're going to do
the following input size trade off.

We're going to do a partial brute
force search for satisfy really

we take the first in the epsilon inputs
and we try all possible assignments for

them that induces some copy of
a circuit and we take there or.

So so
we get a circuit which is a bit larger.

But it has fewer inputs
because we're essentially

forcing sat on into the salon input.

So this has two properties.

The simple reduction one.

If this thing is a C C C had that
these say this thing is that the post.

One is still a C C or is contained
in the operations you can use and.

The second observation is that
this is satisfiable if and

only if this is satisfiable.

If there is some way to set this thing
to make the rest print one so that in

the epsilon then there is some copy here
which will print one when I give it.

OK.

It was very simple trick.

But now because this thing is as you see
it has fewer him puts we can apply this

transformation I told you about earlier
we take this thing in represented as a G.

of his Ok now that blows things
up by a quasi polynomial but

a quasi polynomial of this means
we incur a constant factor.

You know and.

So we get some look up
table gee of this size and

we get Paul Gnomeo with
this many one only else and

it's only over in minus in
an absolute input variables.

Yeah yeah yeah I forgot to mention Yeah so
it weren't quasi punctual time.

So the transformation was incredibly.

You know.

OK So then the final step is just use this
fast evaluation argument from the previous

slide.

We just evaluate this in all this
possible Simon some polymer ties time and

we just look each of them up see
if any of them print one and

even print one we report satisfied.

Otherwise.

And so I said OK.

And so it's really exploring this
connection that was known for

a long time between A C C in polynomials.

The problem was that there was this
composition going on of the poem it was

something else and that's what
the Serafin was getting around.

OK so I want to clue with some
thoughts about future work.

There's a lot of directions for
future work.

The main two directions order a place in
next with simpler complexity classes.

OK so can replace an exponential time.

B.P. space.

Maybe M.P. now the natural
idea is that OK to do that we

may have to improve on exhaustive search
for more complicated circuit analysis

problems means that we did the thing is
a hard problem but when you look at it in

the circuit analysis perspective you're
just checking whether it's all zeroes or

not but you could do other things you
count the number of satisfy assignments.

You could you know solve some Q.B. F.

where the circuit is a predicate
some point five billion for

me with the support of it and
you could do other kinds of things.

So for example one can show that
if you can count the number

of satisfying assignments to a three C.

in a formula and substantial time
then you would have that extra.

Not in pieces by and
we don't know whether or

not such a thing is true if
you remove the counting part.

So this goes through a connection

between the circuit lower bounds and
the randomization.

So another direction is to

find stronger circuits which you can
prove in X. isn't in that circuit class.

So try to get faster sat rhythms for

formulas or for Threshold circuits.

Or maybe just arbitrary Paul middle sized
circuits and using the P.C.P. theorem.

We know that we can weaken
the hypothesis so the very strength and

version of it is used to show
that this promised problem

of distinguishing between unset
circuits and at least half.

Satisfying circuits is still
enough to get lower bounds.

So we know that.

In fact if I give you a formula and
I say this formula unset or

half as assignment satisfy determine
which is the case and lesson to the end.

Solving that would give us an exposition
in C. one should be a big advance.

So one thing we can show along these lines

is something about the Quantified
boolean formula problem.

So we saw earlier that case set can be
solved in about say one point nine of

the in for sufficiently small K. so for
the second we saw one point three zero

eight and suppose we could extend that
to not satisfy Billy but for quantified

things where instead of having existential
quantifier as in all of variables.

We get to put universal quantifiers
here and there in the variables.

If we could do that.

In the not one point
nine of them to the N.

minus into the Point One
time is much much weaker.

Then we could actually get
the separation of an X. from N.C. one.

So there's something about quantified

putting quantifiers on formulas
that make some seeming.

Much much harder to solve or
maybe this hypothesis is true.

And finally just in general I think
there are lots of other connections

between finding better our things
improving lower bounds that are out there

trying to analyze our thumbs and turning
into lower bounds taking lower bounds and

using them to guaranteed
another hour in works.

So just the idea of discovering
further connections is something

I think it's worth exploring.

Thank you.

Thank you.

Yeah yeah

yeah.

This is yes.

So this definitely would
imply that stuff is false.

Like this is much stronger than that.

Yeah so.

So this is I don't know how believable
this is this is quite believable.

Yes I should say that carefully this.

I'm not sure.

Like maybe you could design such
accounting set algorithm assuming already

the exponential time has Panos I circuits.

That's allowed if you want to prove
that expense of time is in peace.

I'd probably.

In fact you can go pretty

far trying to design our than to
assume you have the circuits already.

But I don't know how to get
a contradiction to what you

know that would be the idea
behind using that result.

If you

need games as false as that
imply circular bounce or

circuit anything could be.

I don't know I don't
know of any connections.

There's an algorithm I don't
know what it gives us.

I don't know it's a good question.

I don't

think.