PP Is Closed Under Intersection

Home , Boolean hierarchy

PP is Closed Under Intersection

z y y

Daniel Spielman Nick Reingold Richard Beigel

Yale College Yale University Yale University

Abstract

In his seminal pap er on probabilistic Turing machines Gill asked

whether the class PP is closed under intersection and union We give

a p ositive answer to this question We also show that PP is closed un

der a variety of p olynomialtime truthtable reductions Consequences in

complexity theory include the denite collapse and assuming P 6 PP

separation of certain query hierarchies over PP

Similar techniques allow us to combine several threshold gates into a

single threshold gate Consequences in the study of circuits include the

simulation of circuits with a small numb er of threshold gates by circuits

having only a single threshold gate at the ro ot p erceptrons and a lower

b ound on the numb er of threshold gates needed to compute the parity

function

Intro duction

The class PP was dened in by John Gill and indep endently by Janos

Simon in PP is the class of languages accepted by a p olynomialtime

b ounded nondeterministic Turing machine that accepts when more than half of its

paths are accepting and rejects when more than half of its paths are rejecting this

denition is from and is slightly dierent from but equivalent to the usual

denition see Section Gill noted that PP is closed under complementation

but stated that it was not known if PP is closed under intersection and union

Since Gills pap er PP and related counting classes have b een studied exten

sively by numerous researchers though few closure

prop erties have b een shown for the class In Russo showed that the sym

metric dierence of two sets in PP is also in PP and in Beigel Hemachandra

The authors may b e reached by writing to Department of Computer Science PO Box

New Haven CT or by sending electronic mail to lastnamerstnamecsyaleedu

Supp orted in part by NSF grants CCR and CCR

Supp orted in part by NSF grant CCR under an REU supplement

and Wechsung showed that PP is closed under p olynomialtime parity reduc

tions Gills question remained op en however and it was widely conjectured that

PP was not closed under intersection or union

We prove that PP is in fact closed under intersection and union and even

under p olynomialtime conjunctive and disjunctive reductions Consequently

PP is closed under p olynomialtime truthtable reductions in which the truth

table predicate is computed by a b oundeddepth Bo olean formula and hence

under p olynomialtime Turing reductions that make O log n queries That is

P PP Relative to oracles this collapse cannot b e extended to a larger

O log n T

numb er of queries For functions computed with a b ounded numb er of queries the

PP X

b ehavior is quite dierent PF PF for any oracle X unless P PP

k tt k T

Our strongest closure prop erty is that PP is closed under p olynomialtime

truthtable reductions in which the truth predicate is computed by an explicitly

pro duced multilinear p olynomial this includes all symmetric functions as a sp e

cial case The techniques presented here have b een extended by Fortnow and

Reingold to show that PP is closed under general p olynomialtime truthtable

reductions

The technique for combining PP machines can also b e applied to threshold

gates with p olynomialsized weights For example we show how to compute

the AND of k threshold gates as the threshold of ANDs We also show that

any constant depth circuit with AND OR NOT and threshold gates can b e

simulated by a circuit with a single threshold gate at the ro ot with depth

greater by a constant and only a limited increase in size If the original circuit

p olylog n

has size and only O log log n threshold gates then the new circuit still

p olylog n

has size

As an application we prove that no constant depth circuit with olog n

AND OR and NOT gates in arbitrary p ositions and threshold gates

wires can compute parity This is the rst natural example of a function

that is known to require more than a constant numb er of threshold gates in such

a circuit Previous lower b ounds had b een obtained for circuits consisting en

tirely of threshold gates Ha jnal et al have shown that inner pro duct mo d

cannot b e computed by any p olynomial size depth circuit of threshold gates

Paturi and Saks have shown that a depth circuit of threshold gates which

computes the parity on n inputs requires n log n threshold gates Siu et

al have shown that a depthd circuit of threshold gates which com

putes the parity on n inputs requires dn log n threshold gates Recently

Beigel extending our techniques has shown that that no constant depth cir

o o

o n n

cuit with n threshold gates AND OR and NOT gates and wires

can compute parity

The remainder of the pap er is organized as follows In Section we dene

PrTIMEtn and PP and we show how to combine nondeterministic Turing

machines according to a sequence of rational functions In Section we con

struct the rational functions appropriate for our closure prop erties The closure

prop erties are proved in Section In Section we show how the techniques of

Section can b e mo died to apply to threshold circuits and in Section we ap

ply these techniques to obtain the parity lower b ound mentioned ab ove Finally

in Section we consider query hierarchies over PP

Notation

Throughout this pap er we will use X rather than the customary x to denote

an input to a Turing machine We will use x to denote a variable ranging over

the reals We will use jxj to mean the absolute value of the real numb er x To

avoid confusion we will never use jX j to denote the length of the input X All

logarithms are base two logarithms

Building Turing machines from rational func

tions

Beigel and Gill and Gundermann Nasser and Wechsung have used p oly

nomials to prove closure prop erties of various counting classes In this section

we extend the techniques of where they used a single p olynomial we use a

sequence of rational functions These new twists app ear to b e crucial to obtaining

our closure prop erties for PP

Fenner et al provide a convenient notation for studying counting classes

like PP

Denition For a nondeterministic Turing machine N and input X let

Gap N X denote the numb er of accepting paths of N on input X minus the

numb er of rejecting paths of N on input X

Denition A language L is in PrTIMEtn if there exists a tntime b ounded

nondeterministic Turing machine N such that for all inputs X

X L Gap N X

Gill shows that NTIMEtn PrTIMEtn

It should b e noted that in our denition of PrTIMEtn all accepting reject

ing paths are counted equally regardless of length as in Other denitions

of PrTIMEtn either insist that all paths have the same length or weight

the paths according to length For timeconstructible tn these denitions

are equivalent

Denition PP PrTIMEn

For any nondeterministic Turing machine N let the complement machine

N b e the machine which runs N and then rejects if N accepts and denoted

accepts if N rejects Clearly Gap N X Gap N X Let N and N

b e two nondeterministic Turing machines Consider the Turing machine N

which nondeterministically cho oses to run either N or N It is easy to see

that for all inputs X Gap N X Gap N X GapN X Let N b e a

nondeterministic Turing machine which runs as follows First run N if N

accepts then run N otherwise run N It is not hard to verify that for all

X GapN X GapN X Gap N X Combining these observations we

obtain Lemma b elow

Denition A sequence of p olynomials fp x x g is snuniform if each

n k

co ecient of each p is an integer and sn is a b ound on the time needed to

compute the degree of p or to compute the co ecient of any monomial in p

n n

Lemma Let N N be tntime bounded nondeterministic Turing ma

chines Let fp x x g be an snuniform sequence of polynomials Sup

n k

pose p has degree d and each coecient of p is bounded in absolute value

n n n

by M Then there exists a nondeterministic Turing machine N that runs in

d k

time O log M d tn sn such that for al l X Gap N X

n n

p y y where n is the length of X and y is Gap N X

n k i i

x Pro of N rst nondeterministically cho oses a monomial of p say cx

d k d k

n n

with c Since p has at most monomials this requires time O log sn

k k

Then N nondeterministically cho oses one of c branches computes the pro duct

as describ ed ab ove and complements if necessary This takes an additional time

O log M d tn sn Figure shows the computation tree for machine

n n

Note that in the denition of PrTIMEtn only the sign of the Gap is essen

tial Although we cannot directly apply Lemma to a rational function since

we cannot divide we can build a Turing machine such that the sign of the Gap

is given by the sign of a rational function We dene the degree of a rational

function to b e the maximum of the degrees of its numerator and denominator

A sequence fr x x g of rational functions is snuniform if b oth the se

n k

quence of numerators and the sequence of denominators are snuniform

Lemma Let N N be tntime bounded nondeterministic Turing ma

chines Let fr x x g be an snuniform sequence of rational functions

n k

where the degree of r is d and both the numerator and denominator of r have

n n n

integer coecients bounded in absolute value by M Then there exists a nonde

d k

d tn sn terministic Turing machine N that runs in time O log M

n n

such that Gap N X and r y y have the same sign for al l X where the

n k

latter is dened where n is the length of X and y is Gap N X

i i

time sn

d k

time dlog e

time sn

time dlog M e

N N

time d tn

Figure The computation tree for a nondeterministic Turing machine N with

Gap N X p GapN X GapN X In the rst stage N computes

n k

the degree of p In the second stage N nondeterministically cho oses a monomial

In the third stage N computes the co ecient on the monomial say c for c

In the fourth stage N makes a cway branch In the last stage N runs the

machines corresp onding to the monomial complementing if necessary In this

gure the leftmost monomial is x and the rightmost monomial is x x x

Pro of If r a b where a and b are p olynomials with integral co e

n n n n n

cients then apply Lemma with p a b

n n n

Rational Approximation

In this section we dene the rational functions A x y which are the key to

proving that PP is closed under intersection The key prop erty of A x y is that

for jxj jy j A x y is p ositive if and only if b oth x and y are p ositive In

his seminal pap er on rational approximation Newman constructed rational

functions that closely approximate jxj for x Our A s will b e dened

in terms of similar rational functions S x that closely approximate sign x for

jxj The function sign x is if x if x and if x

Dene

x P x x

P x P x

n n

S x

P x P x

n n

A x y S x S y

n n n

The lemma b elow shows that S x is a go o d approximation to sign x for

appropriate values of x In fact the approximation is much b etter than indicated

here but these estimates will suce

Lemma For n

i If x then P x P x

n n

ii If x then S x

iii If x then S x

Pro of

i Since x we have that P x Clearly x x and x

i k k k

x for i n Also if x then x

k k k

x Together these imply that P x P x

n n

ii If P x then S x If P x then we can write

n n n

S x

P x

Simple algebra and part i yield the desired result

iii This follows from ii and the fact that S x is an odd function ie

S x S x

n n

Lemma Let x and y satisfy jxj jy j

i The degree of A is O n

O n

ii Each coecient of A has absolute value

iii If x and y then A x y

iv If x or y then A x y

Pro of The assertions ab out the degree and size of the co ecients are easily

veried The last two assertions follow immediately from Lemma iiiii

We now consider analogous rational functions in many variables Let hk b e

the least o dd integer greater than or equal to log k We dene

hk hk

P x P x

n n

x S

hk hk

P x P x

n n

k k k

x k x S x x S A

k k

n n n

is O n log k Lemma i The degree of S

O n log k k

has absolute value bounded by ii Each coecient of S

n k

iii If x then S x k

n k

x iv If x then k S

Pro of The b ounds on the degree and co ecient sizes are easily veried Sup

hk hk

p ose that x By Lemma i k P x P x

n n

We now pro ceed as in the pro of of Lemma ii Part iv follows from iii and

the fact that S x is an o dd function

Lemma Let x satisfy jx j for i k

i i

i The degree of A is O nk log k

O n k log k k

has absolute value bounded by ii Each coecient of A

iii If every x then A x x

i k

iv If some x then A x x

i k

Pro of The rst two assertions are easily veried The last two follow from

Lemma just as Lemma iiiiv follow from Lemma

Closure Prop erties for PP

It has b een shown that PP is closed under complementation under symmetric

dierence and under p olynomialtime parity reductions In this section

we use the rational functions built in the previous section to prove closure under

intersection and several sp ecial cases of closure under p olynomialtime truth

table reductions The techniques of this section have b een extended by Fortnow

and Reingold who show that PP is closed under general p olynomialtime

truthtable reductions

Theorem The intersection of nitely many PrTIMEtn languages is in

PrTIMEtn

Pro of Let L L b e languages in PrTIMEtn and let N N b e

k k

nondeterministic Turing machines such that for i k

X L Gap N X

i i

X L Gap N X

i i

Also let r x x A x x By Lemma the degree of r

n k k n

is O tnk log k and every co ecient of r is b ounded in absolute value by

O tn k log k

By Lemma there is a nondeterministic Turing machine N whose

running time is O tn and such that GapN X and r y y have the

n k

same sign where n is the length of X and for i k y Gap N X

i i

Since jy j Lemma yields that

L every y r y y Gap N X X

i i n k

L some y r y y Gap N X X

i i n k

L PrTIMEtn so that

Denition A p olynomialtime conjunctive reduction is a p olynomialtime

truthtable reduction in which the truthtable predicate is a p olynomial size con

junction AND A p olynomialtime disjunctive reduction is dened similarly

Toran asked whether PP is closed under conjunctive reductions and

he noted that an armative answer would imply closure under O log nTuring

reductions dened b elow

Theorem The class PP is closed under polynomialtime conjunctive reduc

tions and disjunctive reductions

Pro of The pro of is similar to the pro of of Theorem For p olynomialtime

conjunctive reductions we use the rational functions A x x as b efore

although k now dep ends on n with k n The details are as in the previous

pro of Closure under p olynomialtime disjunctive reductions is obtained by using

the rational functions A x x

Denition A p olynomialtime boundeddepth Boolean formula reduction is

a p olynomialtime truthtable reduction in which the truthtable predicate is

computed by a b oundeddepth Bo olean formula that is explicitly pro duced by

the reduction b efore any queries are made

Theorem PP is closed under polynomialtime boundeddepth Boolean for

mula reductions

Pro of An easy induction on the depth of the formula

Denition A p olynomialtime f nTuring reduction is a p olynomialtime

denote the class Turing reduction that makes at most f n queries Let P

f nT

of languages p olynomialtime f nTuring reducible to a set A

Theorem PP is closed under polynomialtime O log nTuring reductions

That is P PP

O log nT

Pro of Every p olynomialtime O log nTuring reduction can b e converted to

a p olynomialtime depth Bo olean formula reduction write the reduction as a

CNF or DNF formula over the query answers

C P

PP It was previously known that P PP and that P

O log n T

O log nT

A language L is in C P if there is a nondeterministic Turing machine N such

that for all inputs X X L if and only if Gap N X

Denition A p olynomialtime threshold reduction is a p olynomialtime truth

table reduction in which the truthtable predicate is true if and only if at least

half of its inputs are true

Denition A p olynomialtime symmetric reduction is a truthtable reduc

tion in which the truthtable predicate is a symmetric function ie a function

that dep ends only on the numb er of inputs that are true

Theorem PP is closed under polynomialtime threshold reductions

Pro of Dene

k k k

T x x S x S x

k k

n n n

n k

Assume that jx j for i k Then T x x is a rational

i k

function that is p ositive if at least half of the x s are p ositive and negative

otherwise The degree of T is O nk log k and the absolute value of each of its

O n k log k

co ecients is b ounded by The result now follows from Lemma

Since every symmetric function can b e computed as a threshold of thresh

olds this immediately implies

Corollary PP is closed under polynomialtime symmetric reductions

Lemma Suppose P x x is a multilinear polynomial in k variables

with integer coecients bounded in absolute value by M Then there exists a

rational function U x x of degree O k nk log M with coecients

n k

O k n k log M

such that for jx j bounded in absolute value by

jP sign x sign x U x x j

k n k

x and h Pro of Let U x x b e P y y where y S

i n k k i

k M The verication of the b ounds on the degree and co ecients is

straightforward

The error b etween corresp onding monomials of P sign x sign x and

P y y is at most

h h

M sign x sign x S x S x

k k

n n

which by Lemma is at most M h Since P x x has at

k k k

most monomials the total error is at most M h

Let expx e We will make use of the inequality exp y y

which can b e derived from the well known fact that x expx for x by

substituting x y The total error is therefore

k k

jP sign x sign x U x x j M h

k n k

M expk h

M exp

Although seemingly stronger results may b e stated note that every p olyno

mial over variables in f g is equal to a multilinear p olynomial over those

variables

Denition A p olynomialtime multilinear reduction is a p olynomialtime

truthtable reduction in which the truthtable predicate is computed by a multi

linear p olynomial that is explicitly pro duced by the reduction b efore any queries

are made

By Lemma we now have

Theorem PP is closed under multilinear reductions

We note that all previous pro ofs of containment in PP hinge on the existence

of certain p olynomials having xed degree in fact or less in each variable

In our pro ofs we implicitly use a sequence of p olynomials For nonzero integral

n n

values of x and y within the square centered at the origin the rational

function A x y of Section is p ositive if and only if x and y are b oth p ositive

By clearing denominators we can construct a p olynomial of degree n which

takes on the correct sign for such x and y

If there were a single p olynomial P x y with integral co ecients such that

for all nonzero integers x y P x y if and only if b oth x and y are p ositive

then a simpler pro of of Theorem could have b een given We would have

no need for the rational approximations to the sign function and no need for

Lemma Minsky and Pap ert have shown however that no such P x y

can exist Careful analysis of their pro of shows that a p olynomial in two variables

of degree d with each nonzero co ecient having absolute value b etween and

n n

M cannot take on the correct sign in a square centered at the origin

when n log M d log d

The approach taken in this pap er building such a sequence of p olynomi

als by using an approximation to the sign function cannot b e carried out

using p olynomial approximations to sign By a simple application of Markovs

inequality see no sequence of p olynomials P x where the degree of P

n n

is p olynomial in n can satisfy

x P x

for a xed

Our rational function A x y has a sp ecial form namely Qx Qy

where Q is a rational function It is p ossible to prove that any degree n

p olynomial Ax y of this form satisfying the conclusions of Lemma can b e

directly converted to a rational function S of degree O n which satises the

conditions of Lemma iiiii That is S is a go o d approximation to the sign

function

We sp eculate that the need for p olynomials whose degree varies with the

numerical range of the inputs and the need to consider rational approximations

are the main reasons why the questions answered in this pap er remained op en

until now

Threshold Gates and Perceptrons

In the next two sections we will consider circuits containing threshold gates For

convenience of exp osition we will use to represent false and to represent true

in our circuits a slight departure from standard practice By a threshold gate we

mean a b o olean function of n inputs x x with weights w w f g

n n

x w t and outputs false otherwise and threshold t which outputs true if

i i

If we call a numb er i accepting if x w and rejecting if x w then

i i i i

a threshold gate with t acts like a PP computation in that the gate outputs

true if more is are accepting than rejecting and outputs false if more is are

rejecting than accepting For this reason our techniques for manipulating PP

computations have analogues for threshold gates

By a perceptron we mean a circuit with a single threshold gate at the output

and with constantdepth Bo olean circuits as inputs to the threshold gate All

gates have unb ounded fanin We dene the size of a p erceptron to b e the numb er

of wires in the circuit The depth of a p erceptron is dened as for general circuits

The top fanin of a p erceptron is the fanin of the threshold gate

The following two lemmas are analogous to the lemmas of Section

Lemma For i k j f let C be an ANDOR circuit of

size s and depth D Let c denote the output of circuit C and let s

ij ij i

c Let px x be a polynomial of degree d whose coecients are

ij k

j f

integers bounded in absolute value by M Then there exists a perceptron whose

inputs are the union of the inputs to al l the C with top fanin at most M f

d d

k f s and depth D that outputs true size at most M d f

if and only if ps s is positive

Pro of Expand ps s using the distributive law to obtain a sum of

monomials each of degree monomials over the c s There are at most f

The usual denition of threshold gate has no restriction on the size of the weights though

in the denition of TC see all weights are assumed to b e p olynomial sized If some

input to a threshold gate had an arbitrary integral weight w it could b e replaced by jw j copies

of the input each of weight if w is p ositive or if w is negative Therefore the techniques

presented in this pap er will work for p olynomial b ounded weights as well

at most d The value of a monomial with co ecient is the XOR of the various

c s o ccurring in it

We now build a circuit which determines whether ps s The cir

cuit consists of a threshold of XORs We use one XOR for each monomial o ccur

ring in the expansion of p Each XOR has the correct numb er of wires weighted

or leading to the threshold gate so that the total weight of the XOR is

the co ecient of the corresp onding monomial The inputs to the XORs are the

ANDOR circuits computing the appropriate c s The fanin of the threshold

gate is at most M f

Each XOR can b e computed by a depth Bo olean formula over the c s in

disjunctive normal form having size d If we replace each XOR by the

appropriate OR of ANDs then the resulting circuit is the desired depthD

p erceptron The top fanin is at most M f as indicated ab ove The AND

OR circuits computing the c s contribute k f s to the size of the p erceptron

Each ANDOR circuit for an XOR contributes at most M d wires

d d

Thus the total size is at most M d f k f s

Note It is also p ossible to build a depthD p erceptron which output true

if and only if ps s Assume that true is represented by and false by

In this case the c s are or so in the expansion of p we must replace each

c by c Call the resulting p olynomial q The degree of q is the same as

ij ij

the degree of p and the co ecients of q have absolute value at most M Since

a pro duct of the c s can now b e computed as the AND of the c s the resulting

ij ij

d d

p erceptron has depth D The top fanin is at most M f the fanin of

d d

the AND gates is d and the size of the p erceptron is d M f k f s

Lemma For i k j f let C be an ANDOR circuit having size

c s and depth D Let c denote the output of circuit C Let s

ij ij ij i

j f

Let r x x be a rational function of degree d whose coecients are integers

bounded in absolute value by M Then there exists a perceptron whose inputs are

the union of the inputs to al l the C with top fanin at most M f size

dk dk

d d

and depth D such that when d f at most k f s M

k k

r s s is dened the perceptron outputs true if and only if r s s is

k k

positive

Pro of This follows from the ab ove lemma just as Lemma follows from

Lemma

Applications to Threshold Circuits

By a threshold circuit we mean a circuit with any numb er of threshold AND

OR and NOT gates In this section we prove some simulation results for such

AND

Threshold Threshold

C C C C C C

11 12 13 21 22 23

Threshold

XOR XOR XOR XOR XOR XOR XOR XOR XOR

C C C C C C

11 12 13 21 22 23

Figure The AND of two thresholds as the threshold of XORs In the b ottom

circuit we have used the p olynomial x x x x which works for this

small case The heavier lines which go directly from the c s or to the

threshold gate represent a pair of wires

circuits We rst apply the techniques of the previous section to show that

the AND of k p erceptrons can b e computed by a single p erceptron of slightly

larger size and depth Next we show how to simulate a circuit containing many

threshold gates by a p erceptron of only slightly higher depth

Siu et al have shown that a depthd circuit consisting entirely of

threshold gates which computes parity must have dn log n threshold gates

Combining our simulation result with Fred Greens lower b ound on the fanin of a

p erceptron which computes parity we show that a constant depth circuit having

size and only olog n threshold gates cannot compute parity answering

a question that arose during discussions with Russell Impagliazzo Recently

Beigel has extended our techniques to show that no constant depth circuit of

n o

size and even n threshold gates can compute parity

Theorem Consider k perceptrons having top fanin f size s and depth D

The AND of these k perceptrons can be computed via a perceptron having top

O k log k log f O k log k log f

k f s and depth D size fanin

Pro of This follows from Lemma and Lemma with n dlog f e

Corollary Consider k perceptrons having top fanin f size s and depth D

The OR of these k perceptrons can be computed via a perceptron having top fanin

O k log k log f O k log k log f

k f s and depth D size

Lemma Consider any threshold circuit C having size s depth D and only

O k log k log s

k threshold gates There is a perceptron having top fanin size

O k log k log s

and depth D which computes the same function as C

Pro of Numb er C s threshold gates k Let C b b b e the result of

replacing the ith threshold gate of C by the bit b for every i and then evaluating

the resulting thresholdfree circuit see Figure Let Ab b b e a circuit

that veries that the result of the ith threshold gate in C is b for every i using

the parameters to A as the output values for any threshold gates b elow gate i

see Figure a The output of C can b e computed by taking the OR over all

sequences b b of the AND of Ab b and C b b see Figure

k k k

Negations can b e pushed to the leaves so Ab b can b e evaluated as

the AND of k p erceptrons Since each of these p erceptrons has fanin b ounded

by s Theorem implies that Ab b can b e computed by a p erceptron

O k log k log s O k log k log s

k s and depth D see size having top fanin

Figure b

The AND of C b b and the p erceptron which computes Ab b

k k

O k log k log s O k log k log s

size can b e computed by a p erceptron having top fanin

k s and depth D since C b b do es not involve any threshold

Circuit C

Figure The circuit C b b is obtained from the circuit C by replacing

each threshold gate T by the bit b

i i

AND AND AND

A C A C A C

Figure The circuit C can b e computed as the OR over all bit sequences

b b of the AND of Ab b and C b b

k k k

Threshold A

AND

NOT NOT

AND AND AND

T T

1 3

T T

2 k

a b

Figure The circuit Ab b a Ab b outputs true if threshold

k k

gate T outputs b for i k The inputs to each T are ANDOR circuits

i i i

b Ab b is converted to a p erceptron by pushing any negations past the

threshold gates and then applying Theorem

gates This can b e converted by Corollary to a p erceptron with top fanin

k k

O k log k log s O k log k log s

size k s and depth D

p olylog n

For circuits of size our b est result is for O log log n threshold gates

p olylog n

Corollary Consider any threshold circuit C having size depth D

p olylog n

and O log log n threshold gates There is a perceptron having top fanin

p olylog n

size and depth D which computes the same function as C

To obtain our lower b ound for the numb er of threshold gates in a threshold

circuit which computes parity we will make use of the following theorem of Fred

Green

Theorem F Green For any D there exists a constant c such that

the fol lowing is true Consider any perceptron with top fanin f with depth D

and with subcircuits each of size If the circuit computes parity of n

variables correctly then

D D

Theorem Let C be a threshold circuit having size depth O and

only olog n threshold gates Then C does not compute the parity function of n

inputs

Pro of By Lemma C can b e simulated by a p erceptron having top fanin

o o

n n

and depth O By F Greens theorem such a p erceptron size

cannot compute parity of n inputs

A multilinear gate evaluates a multilinear p olynomial of its inputs represented

as and and outputs true if and only if the result is p ositive Note by

Lagranges interp olation formula that every symmetric gate is a multilinear gate

Lemmas and yield the following simulation result

Theorem If g is computed by a depthD circuit with a multilinear gate at

p olylog n

the output p olylog n threshold gates at the next level AND OR and

p olylog n

NOT gates at the remaining levels and wires then g is computed by a

p olylog n

perceptron that has size and depth D

Query Hierarchies over PP

We have shown that O log n queries to a PP oracle are no b etter than one when

solving decision problems In this section we present two results that contrast

with that one First in relativized worlds we show that if f n O log n then

f n queries to a PP oracle are b etter than one when solving decision problems

Here and for the remainder of the section f denotes a p olynomialtime com

putable function Second if P PP we show that k queries to a PP oracle

are b etter than k when computing functions

The class of languages p olynomialtime reducible to a set A with at most

f n adaptive queries is denoted P The class of languages p olynomialtime

f nT

reducible to a set A with at most f n nonadaptive queries is denoted P

f ntt

A A

The analogous classes of functions are PF and PF By varying the

f nT f ntt

b ound f we obtain the query hierarchies over A

In Section we showed that P PP We wonder whether that result

O log n T

is tight or whether P PP for some function f n O log n Since

f nT

our pro of techniques are valid for computation relative to an oracle we turn to

relativized complexity for insight There is an oracle A for which P PP

f nT

B B

if and only if f n O log n Since NP PP for all B with that same

A we have

A PP

PP f n O log n P

f nT

This is circumstantial evidence that our collapse is the b est p ossible which is

surprising b ecause we have come to exp ect that collapses translate upward For

NP then PH NP On the other hand it is op en whether example if P

NP NP NP NP

P P P P

T T

O log n T

Similar results hold relative to random oracles It was shown in that

R B PP

for all oracles PARITYP PP for almost all R Since PARITY P P

B it follows that P PP for almost all R In fact from the tight lower

b ounds of for approximating parity it follows that if f n O log n then

P PP for almost R Since the pro of of Theorem relativizes we

f nT

conclude that P PP for almost all R if and only if f n O log n

f nT

By combining the results of this pap er with some previously known lower

b ounds we can prove relativized separations all the way up the query hierarchies

Theorem Assume that log n f n n and g n O f n Then

there is an oracle A such that

A A

NP PP

P P

f nT g nT

Pro of We take the following denitions from

CHUNKA n f is the set containing the lexicographically rst f n strings

starting from that b elong to A

maxB is the lexicographically greatest element of B if it exists the empty

string otherwise

ODD MAXELEMENT is f maxCHUNKA n f ends in a g

ODD MAXBIT is the set of all strings over f g whose rightmost is

in an o ddnumb ered p osition ie the set of strings of the form x where

the length of x is even

Note that ODD MAXELEMENT b elongs to P via a binarysearch

f n

f nT

algorithm

Supp ose that for every oracle A ODD MAXELEMENT b elongs to

f n

PP f n

P Then by standard diagonalization techniques of bit in

g nT

stances of ODDMAXBIT can b e decided by depth stratied circuits having

g n

the following form the ro ot is an ORgate having fanin the second level

consists of ANDgates having fanin g n the b ottom level consists of threshold

gates having fanin n

If such a circuit is converted to a p erceptron by the techniques of the note

following Lemma the resulting circuit has the following form the ro ot is a

O O g n

threshold gate having fanin the remaining level consists of AND

O O g n

gates having fanin n However it was shown in that such a circuit

deciding mbit instances of ODD MAXBIT requires ANDgates with fanin m

Therefore

O O g n f n

In particular if f n log n then g n f n

Combining the parity lower b ounds of with our work one can prove a

separation relative to almost all oracles

Theorem Assume that log n f n n and g n O f n Then for

almost al l oracles R

R R

PP PP

P P

g nT f nT

The b ehavior of the query hierarchies over PP is quite dierent when it comes

to functions

X PP

PF Theorem If P PP then X PF

k T k tt

Pro of It is known that if A is a selfreducible set that is not in P and if there

A A

then A must b e psup erterse P exists i such that for all j i we have P

i T j T

A X

ie for all k and all sets X we must have PF PF Let A b e one

k tt k T

of the standard complete sets for PP which are known to b e selfreducible

A X

A A

Then P PP P so PF must not b e contained in PF for any k

j T T k tt k T

and X unless P PP

PP PP

In particular PF PF This is the opp osite of what happ ens to the

k T k T

query hierarchy for decision problems

Higher levels in the query hierarchy of functions over PP are distinct assuming

that PP PHp oly The following results are immediate from

Let f b e a nondecreasing function such that f n O log n If

PP NPp oly coNPp oly then

PP X

X PF PF

f ntt f nT

Let b e a p ositive real numb er and let g b e a nondecreasing function such

that g n n If PP p oly then

PP X

X PF PF

g ntt g nT

Note that by To das theorem and the KarpLipton theorem each of

the hyp otheses ab ove follows from the assumption that the p olynomial hierarchy

do es not collapse

Concluding Remarks

Paturi and Saks have also used rational approximations in their study of

threshold circuits We are grateful to them for sharing with us a preprint of

their pap er in which we discovered Newmans theorem That theorem was the

inspiration for our pro ofs that PP is closed under symmetric reductions and under

multilinear reductions and the corresp onding circuit simulations

Our interest in the current research topic was inspired by discussions with

Fred Green of the query hierarchies over PP The idea of lo oking at multivariate

p olynomials germinated during a visit from Gerd Wechsung

We would like to thank the following p eople for helpful comments advice and

suggestions Mike Fischer Lance Fortnow Bill Gasarch John Gill Fred Green

Johan Hastad Russell Impagliazzo Simon Kasif Rao Kosara ju Ramamohan

Paturi Steven Rudich Roman Smolensky Martin Tompa Gerd Wechsung and

Andy Yao

We thank Bill Gasarch for pro ofreading b eyond the call of duty We also

thank the referees for their very helpful comments

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