Complexity classes

dened by counting quantiers

Jacob o Toran

Dept SI

Universidad Politecnica de Cataluna

Pau Gargallo

Barcelona Spain

Abstract

We study the p olynomial time counting hierarchy a hierarchy of complexity classes related

to the notion of counting We investigate some of their structural prop erties settling many

op en questions dealing with oracle characterizations closure under b o olean op erations

and relations with other complexity classes We develop a new combinatorial technique

to obtain relativized separations for some of the studied classes which imply absolute

separations for some logarithmic time b ounded complexity classes

Intro duction

One of the main goals of complexity theory is the classication of computational

problems in complexity classes according to the amount of resources these problems need

Probably the b est known complexity classes are P and NP since b oth of them capture the

complexity of many natural problems and also b ecause the long standing op en question

P NP has motivated most of the research in the area The similarities in the denitions of

the class NP and the recursiontheoretic class of the recursively enumerable sets b oth can

b e characterized by an existential quantier provoked the translation of other recursion

theoretic notions into the eld of complexity theory and the analogous concept to the

arithmetic hierarchy the p olynomial time hierarchy St was dened The idea is a

natural generalization of the class NP and provided a go o d to ol to classify more complex

problems It was taken also by many researchers as a frame for the study of structural

complexity theory and the idea b ehind the hierarchy the alternation of existential and

universal quantiers ChKoSt inuenced very much the work in the area

Nevertheless there are many natural computational problems whose complexity can

not b e mo delized in terms of existential or universal quantiers on the other hand this

This article is part of the PhD Thesis of the author Some of its results have b een

presented at the international conferences STRUCT and ICALP

complexity is captured by other complexity classes more adapted to the idea of counting

Following this motivation Simon denes in Sim the class of threshold languages

A language L is in this class if there is a p olynomial time M such that for

every input x M has at least k accepting computation paths if and only if x is in L where

k is a xed constant or fraction This class is placed b etween NP and PSPACE and it is

closely related to Valiants class P of functions that count the numb er of accepting paths

in a nondeterministic Turing machine Va It contains natural complete problems a

typical problem in this class is

SAT fF k j F is a b o olean formula with at least k satisfying assignmentsg

Simon also shows that the class of threshold languages is the same as the class PP

of languages accepted by p olynomial time probabilistic Turing machines Gi The

languages in this class are those recognized by p olynomial time b ounded Turing machines

which accept an input if and only if more than half of the computation paths accept

In order to characterize the complexity of some languages called games against na

ture Papadimitriou Pa generalizes the idea of probabilistic machine and obtains the

class PPSPACE of languages accepted by p olynomial time b ounded Turing machines which

alternate b etween nondeterministic and probabilistic congurations The class PPSPACE

turns out to b e equal to PSPACE Papadimitriou shows that in the same way as a language

L in PSPACE can b e characterized by an alternating string of existential and universal

quantiers followed by a p olynomial time predicate they can also b e formulated by alter

nating the existential quantier and a probabilistic or random quantier

p p p

x L x R x x P x x x

x n

pn

where R x P x means that there exist more than half of the strings of length pn

satisfying the predicate P

One step further is taken by Wagner Wa when he denes the counting hierarchy

CH in a similar way as the p olynomial time hierarchy PH trying to classify the com

plexity of certain combinatorial problems in which counting is involved Instead of using

the probabilistic quantier R Wagner intro duces the quantier C inspired in the idea of

pn

y P y means that there are at threshold machines As we will see more formally C

f x

least f x strings y of length pn satisfying predicate P This quantier is equivalent to

the probabilistic one R in the same way that the probabilistic machines recognize the

same languages as the threshold machines The hierarchy arises in a natural way combin

ing the counting quantier not only with the existential quantier as in Pa but also

with the universal one The counting hierarchy turns out to b e a very useful to ol to express

the complexity of many natural problems It contains the p olynomial time hierarchy and

is included in PSPACE Wagner shows that every level of CH has complete problems and

proves some other results ab out the hierarchy

As we have already mentioned many concepts in complexity theory are direct trans

lations of the same concepts in the recursive function theory to the p olynomial time case

ideas like reduction p olynomial time hierarchy oracles etc are taken from the same

ideas in the theory of recursive functions It is interesting to observe that the p olynomial

counting quantier is particular to complexity theory since the analogous concept in

cursive function theory the unb ounded counting quantier is equivalent to an unb ounded

existential quantier In our opinion the lack of a parallel concept in recursion theory has

determined the late app earence of the concept in complexity theory

As we have said the counting hierarchy has great imp ortance for the classication of

a variety of computation problems Nevertheless its structural prop erties have never b een

studied in depth and it was assumed to b ehave in a similar way as other b etter known

hierarchies like the p olynomial time hierarchy This has b een shown to b e true only to

a certain extent In this work we try to complete this knowledge investigating dierent

asp ects of CH and solving some op en problems related to the hierarchy

The article is divided into dierent sections After intro ducing notation and prelim

inaries in sections and we basically continue the work started by Wagner when he

dened the p olynomial counting hierarchy We study the b o olean prop erties of the classes

in CH showing that a class is closed under union and intersection if the rst quantier

dening it is either or and closed under complement and symmetric dierence if it

is the C quantier Classes whose characterization starts by a C quantier do not seem

to b e closed under union and intersection and this fact makes that the classes in CH

b ehave in a very dierent way than the classes in PH We also study the unb ounded

cartesian pro duct op eration which can b e considered as a certain kind of unb ounded in

tersection showing that the classes closed under unb ounded cartesian pro duct coincide

with the classes closed under intersection and also that the closure under this op eration

of a class whose characterization starts with quantier C implies certain collapse result

in CH Using these results we are able to characterize the counting hierarchy in terms of

nondeterministic and probabilistic machines with access to oracles This characterization

p p

oracle characterization of the PH and C was only known for the classes of typ e

k k

Wa Our result completes the characterization for every class

Section is motivated by the problem of whether the classes studied can b e separated

We intro duce a new combinatiorial metho d to obtain relativized separations of the count

ing classes dened in the previous sections Although counting classes have b een separated

from the p olynomial time hierarchy b efore An Ya Ha to our knowledge this

is the rst time that counting classes have b een separated from other counting classes The

technique used to obtain our results is new since the metho ds from previous relativizati ons

do not seem to work for counting complexity classes The idea is to diagonalize gathering

the numb er of accepting computation paths of the oracle Turing machines in combinatorial

formulas in which the oracle is a variable and then argue over the formulas using combina

torial techniques and the fact that our machines are p olynomial time b ounded We present

=

three relativizati ons separating NP from C exact counting NP from P and P from

=

PP As a consequence we obtain relativizat ions in which the three classes NP P and C

=

are incomparable P and PP are incomparable and NP and C are strictly contained in

PP These separations also imply a relativizati on in which PP is dierent from PSPACE

solving an op en problem prop osed by Angluin in An as well as relativized separations

of the lower levels of the counting hierarchy Another consequence of the relativizati ons

presented is the absolute separation of logtime complexity classes

We include at the end of the article a section of conclusions and further research areas

Notation and preliminaries

The notation used in this article is the standard one in stuctural complexity theory

and when new concepts are used a denition of them is included However trying to avoid

any p ossible confusion we include a short summary of notation

The sets that we will considered are languages over some xed alphab et For a

set A jjAjj will represent the cardinality of A and for a string x jxj will

A and its characteristic denote its length The complement of a set A will b e denoted

function Easily computable pairing functions are assumed and denoted by angular

A

parenthesis as in hx y i The marked union or join of two sets A and B is dened as

A B fx j x Ag fx j x B g

Our mo del of computation will b e the multitap e Turing machine If M is a Turing

machine LM represents the language accepted by M If M is a nondeterministic Turing

machine for a pair of strings x y M x represents the computation of M on input x

y

A

following computation path y if M has access to some oracle M x represents the

y

computation on x following path y and oracle A and LM A represents the language

accepted by machine M with oracle A In some parts of this work it is assumed without

explicit mention that the input tap e alphab et of Turing machines is f g

For a denition of the language classes P NP PSPACE UP LOGSPACE and the

ones in the p olynomial time hierarchy we refer the reader to the b o oks written on the

sub ject BaDiGa Sc WaWe PP Gi is the class of language recognized

by p olynomial time nondeterministic machines that accept an input x if and only if more

than one half of the computation paths of the machine accept x For a language class

K

K P represents the class of languages computed by a p olynomial time machine with

K log n

access to an oracle in K Moreover P denotes the same class of languages with

the restriction that the p olynomial time machine can only make O log n queries to the

oracle

We will also consider dierent function classes We will denote by FP the class of

A

p olynomial time computable functions and by FP the class of functions computable in

p olynomial time by a deterministic machine with access to oracle A

p

and the The p olynomial time reducibilities used are the manyone reducibility

m

p

whose denition can b e seen in the mentioned b o oks Turing reducibility

T

The counting hierarchy

As mentioned b efore the p olynomial time counting hierarchy was rst intro duced by

Wagner Wa as a to ol to classify certain combinatorial problems in which counting is

involved However many problems were left op en by his work among them many of the

structural prop erties of this hierarchy such as equivalent denitions by oracle machines

similar to the case of the p olynomial time hierarchy closure under b o olean op erations

and ability to extract information from oracle sets In this and the following section we

address these problems extending the work of Wagner

We rst study closure prop erties of the classes in the hierarchy We show that they

dep end only on the rst quantier of the string of alternating quantiers dening the class

b eing a class closed under complement and symmetric dierence if its rst quantier is

a counting one and closed under union and intersection if the rst quantier is either

existential or universal We also study the closure of the classes under unb ounded carte

sian pro duct an op eration which is needed to obtain other characterizations of CH We

show that the classes that are closed under unb ounded cartesian pro duct coincide with

the classes closed under intersection Using these results we also prove that for certain

classes K in CH a deterministic asking just one question to a set in K

can only recognize sets in K

The following denitions are taken from the article Wa

Denition The p olynomial counting quantier C is dened in the following way

for a function f IN f FP a p olynomial p and a two argument predicate Q

p

y Qx y jjfy jy j pjxj and Qx y gjj f x C

f x

If K is a language class for any set A A CK if there is a function f in FP st for

every x f x a p olynomial p and a language B K such that for any x

p

y hx y i B x A C

f x

Recall that b ounded quantiers have b een used b efore to dene complexity classes

For example in Wr a characterization of the p olynomial time hierarchy in terms of

p p

the p olynomially b ounded existencial and universal quantiers and is given

We alternate now the p olynomial counting quantier C with the existential and the

universal quantiers in order to dene the counting hierarchy

Denition The p olynomial counting hierarchy CH is the smallest family of language

classes satisfying

i PCH

p p p

ii If K CH then K K and C K CH

Since in this section we will talk only ab out quantiers ranging over strings of p oly

nomial length we drop the sup erscript p from all the quantiers Also for simplicity C

will denote the class CP and the context will make clear when we talk ab out a quantier

and when ab out a language class

The next lemma shows that the threshold function of the C quantier can b e changed

to strictly more than one half of the p ossible quantied strings As a consequence the

class C is the same as the class PP of languages accepted by probabilistic Turing machines

This fact has b een observed in Sim Wa

Lemma

i C PP

p p

K ii For any class K in CH and any function f IN IN f FP C K C

p(n)1

f

To lo cate CH with resp ect to other complexity classes we observe the following prop

erties

Lemma Wa

a Every language in CH can b e accepted in p olynomial space

b PHCH

c For every class K in CH K K CK CK CK

p

d Every class in CH is closed under reducibility

m

log

e Every class in CH has complete problems

m

Some easy closure prop erties of the classes in CH that will b e used later are

Lemma For any class K in CH any sets L L in K and any set P in P

i L L K

ii L IN K

iii L P K

=

Next we dene the exact counting quantier C which is a littl e dierent than C in

its denition but as we will see has very dierent prop erties This dierence b etween C

= =

and C will b e crucial in the pro of of our results C was also dened for the rst time in

=

Wa In section we present a relativizati on in which C C considered as language

classes

Denition For a function f IN IN f FP a p olynomial p and a two argument

predicate Q

p

=

y Qx y jjfy jy j pjxj and Qx y gjj f x C

f x

=

The denition of C K for a language class K is analogous to the denition of CK

using the new quantier Also lemma holds for any class K in CH and any sets L L

=

in C K

In order to show the closure under certain b o olean op erations of the classes in CH

we need to prove the following results which will b e improved later in corollary

Lemma For any class K in CH

=

i CK C K

=

ii CCK CC K

= = =

iii C CK C C K

=

iv C K CK CK

Pro of i Let K b e a class in CH and L a set in CK There is a function f in FP and

a set A K such that for every x

x L y C z hx y z i A

f xy

=

y v C z hx y z i A and v z

f xy

ii and iii Analogous to i

=

iv Let K b e a class in CH and L a set in C K There is a function f in FP and a

set A K such that for every x

=

x L C y hx y i A

f x

Let L fx C y hx y i Ag and L fx C y hx y i Ag

f x f x

L and L are in CK and L L L tu

We will improve the ab ove result in corollary after proving certain b o olean prop

erties of the classes in CH

=

In Wa it is mentioned that the class C is closed under intersection This can b e

=

extended to any class starting by the C quantier and to a certain kind of unb ounded

intersection the unb ounded cartesian pro duct This extension will b e necessary for the

oracle characterization of CH in the next section

Denition For any set L dene the unb ounded cartesian pro duct of L L as the

set

x Lg L fhx x x i

i k

i

=

Theorem For any class K in CH C K is closed under unb ounded cartesian pro duct

=

Pro of Let L b e a set in C K There is a set A in K a function f in FP and a p olynomial

pjxj

p such that for any x f x and

=

x L C y jy j pjxj hx y i A

f x

Given a sequence of strings hx x x i let m maxfjx j jx jg and dene

k k

pm pm k pm

g x x f x f x f x f x

k k

Notice that from g x x it is p ossible to recover the unique values of f x f x

k k

hx x x i L

k

= =

C y jy j pjx j hx y i A C y jy j pjx j hx y i A

k k k k k

f x f x

1 k

=

C hz z z i there exists an i such that z x and jz j pjx j

i i

g x x x

1 2 k

and hx z i A and jz j i pm

i

This is true b ecause we have multiplied the witnesses of x L in such a way that

i

ipm

there must b e exactly f x of them for every x It follows

i i

hx x x i L

k

=

C hz z z i hz z z i A

g x x x

1 2 k

=

b eing A in K which implies L C K tu

We prove now the main result of this section

Theorem Let K b e the class in CH characterized by the quantiers Q Q Q

k

i If Q is either or then the class K is closed under intersection and union

ii If Q is C then the class K is closed under complement and symmetric dierence

iii If Q is then coK K coK K

Fact iii is a technical prop erty needed for the pro of of the rest of the theorem

Pro of The pro of is by induction over k the minimum numb er of quantiers characterizing

the class K It is divided in dierent cases in order to cover all typ es of quantiers

Induction basis k

i If Q is either or then triviall y K is closed under intersection and union

ii If Q is C then

K is closed under complement Gi and

K is closed under symmetric dierence Ru

iii If Q is then coK and triviall y If Q is then coK and

trivially

Induction step k k

Let K b e in CH K Q K Q Q K if Q is either or then Q Q

i Q is

Intersection

a Q is

K is the class K with K in CH and characterized by k quantiers Let

L and L b e two sets in K There are two sets B B in K and a p olynomial

p such that for i and for any x

x L y jy j pjxj and hx y i B

i i

x L L hy y i hx y i hx y i B IN IN B

By lemma B IN IN B K by induction hyp othesis B IN

IN B K and L L K A similar argument works for K K

b Q is C

There are two sets B B in CK satisfying the ab ove formula By lemma

=

these two sets can b e substituted by two new ones D D in C K and combining

= =

the C version of lemma and lemma D IN IN D C K By

the third part of lemma D IN IN D is in the symmetric dierence

of two sets in CK which by induction hyp othesis is in CK It follows that

L L CK

Union

a Q is

The pro of is completely analogous to the intersection case

b Q is C

Let B B b e the sets in CK dened in the intersection case For every x

x L L hy y ai a f g and ahx y i hx y i B ININ B

by lemma B IN IN B CK and L L CK

If Q is then the pro of follows from the ab ove case using the complementary

L classes for example for the intersection if we have two sets L and L in K then

and L are in coK and L L coK L L L L K The union case is

analogous

ii Q is C

Complement

Let L b e a set in CK There is a set A in K a function f in FP and a p olynomial

p such that for any x

x L C y jy j pjxj hx y i A

f x

C y jy j pjxj hx y i A

p(jxj)

f x

a Q is

The set A in the ab ove formula is in K There is a set B in K such that

L C y jy j pjxj z hx y z i B x

p(jxj)

f x

C hy z i hx y z i B and z z z hx y z i B

p(jxj)

f x

B D C hy z i hx y z i

p(jxj)

f x

b eing D a set in K B K by fact iii if K K or by fact ii if

K CK By induction hyp othesis B D K and then L CK We

have shown coCK CK It follows coCK CK

b Q is

coC C since coCK CcoK CcoK coCcoK CK

c Q is C

In this case set A in is in the class CK which by hyp othesis is closed under

complement

Symmetric dierence

Let L L b e two sets in CK By lemma we can change the threshold of the rst

quantier to b e more than half of the p ossible strings There are two sets B B in

K and a p olynomial p such that for any x

x L C y jy j pjxj hx y i B

p(jxj)1

i i

Following the same idea as in Russos pro of that PP is closed under symmetric dier

ence Ru let x and let a and a b e the two integers not necesarily p ositive

such that

pn

jjfy hx y i B gjj a

i i i i

Let

t jjfhy y i hx y i B and hx y i B or hx y i B and hx y i B jj

pn pn pn pn pn

t a a a a a a

If x L L then either a and a or a and a In b oth cases

pn

t

If x L L then either a and a or a and a and in b oth

pn

cases t Therefore

x L L C hy y i hx y i B and hx y i B

2p(jxj)1

or hx y i B and hx y i B

C hy y i hx y i hx y i B IN IN B

2p(jxj)1

a Q is

The ab ove sets B are then in the class K There are two sets D D in K

i

such that

x L L C hy y i z hx y z i D or hx y z i D

2p(jxj)1

and z z hx y z i D or hx y z i D

C hy y z i hx y z i D or hx y z i D

2p(jxj)1

and z z hx y z i D or hx y z i D

and z z z hx y z i D and hx y z i D

It follows that L L CcoK As we have seen this class is closed under

complements and L L coCcoK CK

b Q is C

By lemma the sets B IN and IN B in are in CK and by induction

hyp othesis the symmetric dierence of these sets is in CK and L L CCK

iii Q is

a Q is

K is the class K and K K by induction hyp othesis coK K and

therefore coK coK K K

b Q is C

By induction hyp othesis K coK Therefore coK K K

tu

Later we will need a stronger version of part ii of the theorem for any class K in

CH K and K are closed under unb ounded cartesian pro duct The pro of of this fact is

analogous to the ab ove one

Corollary For any class K in CH

=

i CK C K

=

ii CCK CC K

= = =

iii C CK C C K

=

iv C K CK

The next result shows that it is not likely that the classes starting with quantier C

are closed under unb ounded cartesian pro duct As mentioned b efore for a language class

K Olog n

K P denotes the class of languages accepted by a p olynomial time deterministic

machine that queries an oracle in K at most a logarithmic numb er of times

Theorem For any class K in CH if CK is closed under unb ounded cartesian

CK Olog n

pro duct then CK P

Pro of The inclusion from left to right is straightforward for the other one let K b e a

CK Olog n

class in CH and L P via a p olynomial time deterministic Turing machine M

quering at the most c log n times an oracle A CK For a given input x we can enco de

the oracle answers from A on the computation of M by a string of c log jxj bits y

x L y jy j c log jxj

y

M x f x y A A f x y A A f x y A A

jy j

y

where M x means that M accepts x following the oracle answers enco ded in y and

f x y hw ai b eing w the ith query that M on input x and following y makes to the

i

oracle and a the ith bit of y We can write the ab ove expression as

y

x L y jy j c log jxjM x hf x y f x y f x y i A A

jy j

denote hf x y f x y f x y i by hx y and A A by B By the closure of

jy j

CK under complements unb ounded cartesian pro duct hyp othesis and intersection with

p olynomial time predicates the set B b elongs to CK and

y

x L y jy j c log jxj M x hx y B

Since the y in the quantier has logarithmic length we can avoid it by writing explicitel y

all the strings of this length

B x L hhx y hx y hx y c i

jxj

Being y the ith string of length c log jxj in lexicographical order Using again

i

that CK is closed under complements and the hyp othesis it follows that L CK and

CK Olog n

P CK tu

Another consequence of theorem is that deterministic p olynomial time oracle

Turing machines that can make just one question to an oracle in a CH class whose char

acterization starts with the counting quantier can only recognize those languages in the

class

CK

Corollary For any class K in CH P CK

Pro of The inclusion from left to right is straightforward for the converse let A b e a

CK p

A mcomplete set in CK Wa It is clear that for any language L in P L A

m

By the closure of CK under complements and mreducibility L CK

tu

Characterizing the counting hierarchy with oracles

In this section we show that the counting hierarchy coincides with the hierarchy

obtained by iterating nondeterministic and probabilistic machines with oracles We unify

b oth concepts by giving an oracle characterization of the hierarchy similar to the oracle

characterization of the p olynomial time hierarchy the dierence is that here instead of

using only nondeterministic Turing machines we also use probabilistic machines This

p

characterization extends a result from Wa where it is shown that for any class in

k

p

p

k

C PH PP

k

Theorem For any class K in CH

K

i PP CK

K K CK

ii NP NP K and NP CK

Statement ii is divided in two cases dep ending of the quantier characterization of the

class in the oracle

Pro of We prove i we will see later that the pro of of ii is completely analogous

Straightforward

Let K b e a class in CH K Q K b eing Q the rst quantier characterizing the

class and K in CH

K

Let L b e a set in PP There is a probabilistic Turing machine M a p olynomial p

b ounding the computation time of M and a set A in K such that L LM A For every

x

A

x L C y M x accepts

p(jxj)1

y

a Q is

Let B b e the set

B f hx y q z q z q q i

k k k m

m pjxj and M with input x following computation path y asks all questions

q in the list not neccessarily in the same order and answering them yes if

i

i k and no if i k M accepts and

for i k q A and z is the smallest string witnessing this fact and

i i

A g for i k m q

i

A

x We claim that B coK Since for every x and for every y such that M

y

accepts there is exactly one string v such that hx y v i B it is clear that

x L C w hx w i B

p(jxj)1

It is only left to show that B coK can b e checked in p olynomial time

Since A is in coK and this class is closed under unb ounded cartesian pro duct

is a predicate in coK Condition can b e written

for i k hq z i D and z z z hq z i D

i i i i

b eing D a set in K By theorem the predicate b etween is in coK

Condition is therefore an unb ounded cartesian pro duct of predicates in co

K and by theorem it is a predicate in coK and B coK

K

CcoK but by theorem CcoK CK We have shown PP

CK

b Q is C

A is also The set A is in the class CK By theorem and lemma the set A

in CK and there is a function f PF and a set D K such that for every u

A C v hu v i D u A

f u

Let B b e the set

B f hx y q a z q a z i

m m m

m pjxj and M with input x following computation path y asks all questions

q in the list not neccessarily in the same order and answering them yes if

i

a and no if a M accepts and

i i

A and hq a z i D and there are exactly f q strings for i k q a A

i i i i i i

z greater than or equal to z such that hq a z i D g

i i i

i i

A

=

We claim that B C K Again since for every x and for every y such that M x

y

accepts there is exactly one string v such that hx y v i B

x L C w hx w i B

p(jxj)1

=

and it is left to show that B C K is a predicate that can b e checked in

p olynomial time and condition can b e written in the following way

=

z z z and hq a z i D for i m hq a z i D and C

i i i i i i i

f q

i i

i

and hq a z i D

i i

i

=

K is therefore an unb ounded cartesian pro duct of predicates in C and by theorem

CK

= = =

CC K But by it is a predicate in C K It follows that B C K and PP

=

corollary CC K CCK CK

pjxj

ii The pro of is completely analogous to the one ab ove b eing instead of

the threshold of the machine tu

Observe that from the ab ove pro of it can also b e derived that every language recog

nized by nondeterministic or probabilistic oracle machines with an oracle in CH can b e

decided by a machine of the same typ e quering the oracle just once

Separations

In this section we try to show that the containments b etween the classes in the studied

language hierarchy are strict Absolute separations are very hard to accomplish since they

would immediately imply PPSPACE solving a long time standing op en problem A more

mo dest approach is to try to nd relativized separations These relativized separations are

still imp ortant for dierent reasons One of them is that these separations together with

the relativizati on in which PSPACE is used as oracle forcing all the classes in CH to

collapse together show that there are contradictory relativizati ons for this classes giving

stronger evidence that the absolute separation problem is very hard Quoting Hartmanis

the pro of that a problem can b e relativized in two contradictory ways serves to day in

theoretical computer science almost the same role as proving a problem NP hard in the

study of algorithms If a problem is NP hard we are very unlikely to solve it in reasonable

time suciently big instances of this problem Similarly the contradictory relativizat ion

of a suciently rich problem is go o d indication that it can not b e solved with our

current mathematical techniques Har Another reason for the imp ortance of the

relativized separations of the classes in CH is that they provide absolute separations for

the corresp onding classes in the logarithmic time counting hierarchy Toa

We intro duce a new technique based on counting the numb er of accepting computa

tions of the machines over which we diagonalize Lemma is the main result in which

our constructions are based we try to motivate it with the example of the separation of

=

NP from C Apart from this mentioned separation we also separate P from PP and NP

from P P is the class parity of languages recognized by nondeterministic p olynomial

time machines with an even numb er of computation paths for words in the language and

an o dd numb er of accepting paths for words that are not in the language Using the

known inclusions b etween these classes we are able to prove other results separating the

lower levels of CH As a consequence we obtain absolute separations for the lower levels of

the logarithmic time counting hierarchy

Although there are several references in the literature of relativizati ons separating

counting classes from classes in the p olynomial time hierarchy An Ya Has

to our knowledge these are the rst relativizat ions separating counting classes from other

A A

counting classes In BeGi it was claimed that P PP for a random oracle A but

the pro of of this result was incorrect

=

We start separating NP from C

Let M M b e an enumeration of all the probabilistic Turing machines and p p

an enumeration of the p olynomials Wlog we can supp ose that for every k M has

k

computation time b ounded by p

k

A A

=

Theorem There is an oracle A such that NP C

n

Pro of For every set A dene L f w jw j n and w Ag clearly for every set

A

A A

=

A L NP We construct in stages a set A such that L C

A A

Stage A n

Stage s Let n b e the smallest integer such that

s

n n

s s

n maxfp n i sg

s i s

n

s

p n

s s

n n

s s

A A B b eing B such that B LM A B

s s s s

Here for a string x and an oracle B x LM B means that machine M has

s s

exactly th accepting computation paths for this input b eing th the threshold of the machine

for input x

S

A It is clear following the same ideas as in BaGiSo that if we Let A

s

s

A

=

prove the existence of set B in then the set L is not in C In the following we

A

show that the set B in always exists

B

Notation Q denotes the numb er of accepting paths from M with oracle

s

a a b b

1 k 1 l

n

s

A B and input in which all the words a a are queried and none of the

s k

w w

1 3

denotes the numb er of words b b is queried to the oracle For example Q

l

w w w w

1 2 3 4

n

s

accepting paths from M with oracle A fw w g and input in which the words

s s

w and w are queried and none of the words w w are queried to the oracle Notice

that we have omitted the fs from the set notation from the sup erscript of the Q Also

observe that if a word is not queried it do esnt make any dierence if we drop it into or

w w

1 3

is equivalent take it away from the oracle for example the ab ove expression Q

w w w w

1 2 3 4

w

1

to Q

w w w w

1 2 3 4

The following lemma is needed for proving the result We omit the pro of since it is

straightforward It just says that we can decomp ose the set of accepting computations

quering w w into two those that query also w and those that do not

k k

n

s

the following equality Lemma For any set B and any k words w w

k

holds

B B B

Q Q Q

w w w w w

w w w

1 k k+1 1 k

1 k k+1

B B

Q Sometimes we will use the ab ove equality in the form Q

w w w

w w w

1 k k+1

1 k k+1

B

which should not create any confussion Q

w w

1 k

In order to op erate in a concise form the dened Qexpressions representing the

numb er of accepting paths with dierent oracles will b e group ed into a combinatorial

formula The motivation for this is presented with more detail in Toa

n

s

with B D Notation For any B D

kD k

X X

i B A B

Q J

D D

i

AD

kAki

This formalism is needed for our pro ofs since otherwise using only the Qs we would

have to carry very long symb ol strings

We intro duce now the main lemma in this section that would enable us to prove the

existence of the oracles separating the classes

n n

s s

Lemma For any sequence of words w w w in and any set B

k k

with B fw w w g

k k

B B B fw g

k+1

J J J

w w w w w w w

1 k k+1 1 k 1 k

Pro of For the pro of rst we decomp ose the J s into Qs following the denition and

then manipulate the Qs either decomp osing them into two by lemma or deleting from

the oracle some words that are not queried

Let D fw w g

k

k

X X

B fw g

k+1

i B fw gA

k+1

Q J

w w

D

1 k

AD i

kAki

k

X X

B fw gA

k+1

i B fw gA

k+1

Q Q

w w w

w w w 1 k k+1

1 k k+1

i

AD

kAki

k

X X

B fw gA B A i

k+1

Q Q

w w w

w w w

1 k k+1

1 k k+1

i

AD

kAki

k

X X

i B fw gA B A B A

k+1

Q Q Q

w w w w w w

1 k+1 1 k 1 k+1

i

AD

kAki

k k

X X X X

i B A B fw gA i B A

k+1

Q Q Q

w w w w w w

1 k+1 1 k+1 1 k

i i

AD AD

kAki kAki

k

X X X X

i B A B A B A B

Q Q Q J

w w w w w w D

1 k+1 1 k+1 1 k+1

i

AD AD

AD fw g

k+1

kAki kAki

kAki

k

X X

i B B A B

Q Q J

w w w w D

1 k+1 1 k+1

i

AD fw g

k+1

kAki

k

X X

i B B A B

Q Q J

w w w w D

1 k+1 1 k+1

i

AD fw g

k+1

kAki

B B

J J

D

D fw g

k+1

Mayb e the step taken to obtain the expression in needs some clarication Observe

that in the expression b efore the two last sums only dier in the size of k A k Since

i

these sums are part of another sum and are multiplied by the terms cancel the sum

B

and telescop es remaining only Q

w w

1 k+1

X

k B A

Q

w w

1 k+1

AD

kAkk

but this last term is since jjD jj k tu

n

s

Lemma For s if for every set R it is true that

n

s

LM A R R

s s

n

s

then for any nonempty sequence of words w w in and any oracle B with B

k

n B

s

B and fw w g B it holds that J Moreover J J

k

w w w w w

1 k 1 k 1

Q th

B B

Pro of By hyp othesis and using the denition of J for every set B J Q th

By lemma

B B B fw g

k+1

J J J

w w w w w w

1 k+1 1 k 1 k

We prove the rst claim by induction on k

B B B fw g

1

For k J J J th th

w

1

B fw g

k

B B

For k J J J where by induction hyp othesis b oth terms

w w

1 k1

w w w w

1 k 1 k1

are

The second claim is proved also by induction on k

For k

w

w w w

1

1 1 1

Q Q Q th Q Q Q Q Q Q J

w w w

w

w 1 1 1

1

1

w

k

but by the rst part of the result J J For k J

w w w w w w

1 k1 1 k1 1 k

w

k

tu J Thus J J By induction hyp othesis J J

w w w w w w w w

1 1 k 1 1 k1 1 k1

Now we are ready to prove the existence of the set B in

n n

s s

such that B Lemma For every s there is a set B

LM A B

s s

n

s

Supp ose that the mentioned Pro of Let th b e the threshold of the machine for input

B n

s

B Q th We are in the hyp othesis set B do es not exist then for every set B

of lemma

n

s

is b ounded by p the n Since the running time of M on input Let p p

s s n

s

machine can make at the most p queries to the oracle on every computation path and

is a sum of computation paths in which Recall that J therefore J

w w w w

1 p+1 1 p+1

all words w w are queried

p

Q th It follows that Q th J On the other hand by lemma J

w w w

1 1 p+1

n

s

which contradicts the hyp othesis since LM A tu

s s

A A

=

Corollary There is an oracle A such that C C

Pro of Straightforward from the ab ove separation considering that the pro of of NP C

relativizes tu

A

=

Corollary There is an oracle A such that C is not closed under complements

=

Pro of Follows from theorem and the fact that coNP is included in C and the pro of

relativizes tu

We present now separations dealing with the class P parity This class was dened

in PaZa Recently some results have app eared separating this class from the p olyno

mial time hierarchy FuSaSi Ya Has We show relativizati ons separating PP

and P As a consequence these results will imply separations in the lower levels of PH

Denition PfL there is a nondeterministic p olynomial time machine

recognizing L with an even numb er of accepting computation paths for input strings in L

Lg and an o dd numb er of accepting computation paths for input strings in

We present now an oracle separating NP from P

A A

Theorem There is an oracle A such that NP P

Let M M b e an enumeration of all the nondeterministic Turing machines and

p p an enumeration of the p olynomials Wlog we can supp ose that for every k

M has computation time b ounded by p

k k

n

For every set A dene L f w jw j n and w Ag Clearly for every set

A

A A

A L NP We construct in stages a set A such that L P

A A

Stage A n

Stage s Let n b e the smallest integer such that

s

n n

s s

n maxfp n i sg

s i s

n

s

p n

s s

n n

s s

LM A B such that B A A B b eing B

s s s s

S

Let A A Observe that since we are trying to diagonalize away from parity

s

s

n n

s s

the expression LM A B means that machine M on input and oracle

s s s

A B has an o dd numb er of computation paths It should b e clear that if we manage

s

A

to prove the existence of set B in then the set L cannot b e in P In the following

A

we show that the set B in always exists we will make use of lemma

n

s

Lemma For s if for every set R it is true that

n

s

LM A R R

s s

n

s

then for any nonempty sequence of words w w in and any oracle B with B

k

n B

s

is o dd is even Moreover J B and fw w g B it holds that J

k

w w w w

1 k 1 k

Pro of By the denition of acceptance of M and the hyp othesis for every set B B

s

B

J is even and by lemma

B B B fw g

k+1

J J J

w w w w w w

1 k+1 1 k 1 k

We prove the rst claim by induction on k

B B B fw g B B fw g B

1 1

For k J J J since J and J are even is J

w w

1 1

B fw g

k

B B

For k J J J and by induction hyp othesis b oth

w w

1 k1

w w w w

1 k 1 k1

members of the right hand side of the equation are even

The second claim is proved also by induction on k

w w

1 1

J J J is o dd by hyp othesis and J is even by the rst part For k J

w

1

of the result

w

k

For k J J J but by the rst part of the result

w w w w w w

1 k 1 k1 1 k1

w

k

J is even By induction hyp othesis J is o dd and it follows that J

w w w w w w

1 k1 1 k1 1 k

is also o dd tu

Now we are ready to prove the existence of set B in

n n

s s

Lemma For every s there is a set B such that B

LM A B

s s

Pro of Supp ose that the mentioned set B do es not exist then we are in the hyp othesis

of lemma

n

s

is b ounded by p the n Since the running time of M on input Let p p

s s n

s

machine can make at most p queries to the oracle on each computation path and therefore

J

w w

1 p+1

is o dd This is a contradiction and it On the other hand by lemma J

w w

1 p+1

follows that the mentioned set B always exists tu

A A

Corollary There is an oracle A such that PP P

Pro of Straightforward considering that the pro of of NPPP relativizes tu

We present now the last separation this time separating P from PP This result will

bring as a consequence the separation of dierent classes in the counting hierarchy

A A

Theorem There is an oracle A such that P PP

Let M M b e an enumeration of all the probabilistic Turing machines and p p

an enumeration of the p olynomials Wlog we can supp ose that for every k M has

k

computation time b ounded by p

k

n n

For every set A dene L f jjA jj is eveng Clearly for every set A L

A A

A A

P We construct in stages a set A such that L C

A

Stage A n

Stage s Let n b e the smallest integer such that

s

n n

s s

n maxfp n i sg

s i s

n

s

p n

s s

n n

s s

A A B b eing B such that LM A B jjB jj is o dd

s s s s

S

A In the following we show that the set B in always exists which Let A

s

s

A

implies that L PP

A

n

s

Lemma For s if for every set R it is true that

n

s

LM A R jjRjj is even

s s

n n

s s

then for any nonempty sequence of words w w in and any oracle B with B

k

B

and fw w g B it holds that J More precisely if jjB jj is even then

k

w w

1 k

B B

and if jjB jj is o dd then J J

w w w w

1 k 1 k

Pro of By the denition of acceptance of M and the hyp othesis for every set B jjB jj is

s

B

even J th and by lemma

B fw g B B

k+1

J J J

w w w w w w

1 k 1 k 1 k+1

By induction on k Supp ose jjB jj is even the o dd case is analogous

B B fw g B B fw g B B

1 1

J J since J th and J th we obtain J For k J

w w

1 1

B fw g

k

B B B

J J For k J and by induction hyp othesis J

w w

1 k1

w w w w w w

1 k1 1 k 1 k1

B fw g

k

B

and J tu It follows that J

w w

1 k1

w w

1 k

Now we are ready to prove the existence of set B in

n

s

such that Lemma For every s there is a set B

n

s

LM A B jjB jj is o dd

s s

n

s

Pro of Let th b e the threshold of the machine for input Supp ose that the mentioned

set B do es not exist then we are in the hyp othesis of lemma

n

s

is b ounded by p the n Since the running time of M on input Let p p

s s n

s

machine can make at most p queries to the oracle on each computation path and therefore

J

w w

1 p+1

On the other hand by lemma J This is a contradiction and it

w w

1 p+1

follows that the mentioned set B always exists tu

A A A A

Corollary There is an oracle A such that C C and C C

In Toa a counting hierarchy of classes op erating in logarithmic time has b een

dened It is not hard to prove that the ab ove relativized separations imply absolute

separations for the lower levels of the logarithmic time counting hierarchy

Conclusions and further research areas

Our work has b een motivated by the study of a hierarchy connected with the idea

of counting the p olynomial time counting hierarchy We have studied the closure of

the classes in CH under b o olean op erations and unb ounded cartesian pro duct showing

that for these prop erties this hierarchy b ehaves in a dierent way as PH Using these

results we have given an oracle characterization of CH parallel to the one existing for the

p olynomial time hierarchy closing an op en problem and unifying concepts From the oracle

characterization of CH follows also that probabilistic oracle machines can b e simmulated

by machines of the same typ e querying a new oracle once at the most on every computation

path

Finally in section we have separated some of the studied classes Three relativiza

=

tions have b een given separating NP from C NP from P and P from PP These

=

relativizati ons provoke other separations such as C from C C and C from C etc as

well as some other classes related with the closure under b o olean op erations of the counting

classes The relativized separations for the classes in CH imply absolute separations for

the corresp onding logarithmic time classes and thus we have separated the lower levels

of the logarithmic time counting hierarchy

Although we have obtained many new results solving some op en problems there

are still several questions connected with the counting hierarchies that remain op en In

the following we give a list of some of these questions We will not include in this list

obvious op en problems of typ e P PP or NP PP which we b elieve are still far from

b eing solved we will concentrate more in problems that apparently can b e solved with the

existing techniques of structural complexity theory or at least they do not seem so far as

the ones mentioned in the rst place

If the class NP is closed under complements then the p olynomial hierarchy collapses

do es the counting hierarchy collapse if PP is closed under intersection Observe that in

PPlog n

theorem we have shown a collapse of the class P to PP in case PP is closed

under unb ounded cartesian pro duct a certain kind of unb ounded intersection

PP

Recently To da To d has obtained a remarkable result showing that PHP This

implies that if C is included in PH then the p olynomial time hierarchy collapses Is this

=

fact also true if C PH

The obvious op en problem related with the last section is to know if there is an

oracle separating every classes in the counting hierarchy in analogy with the result for PH

Ya Has Ko This question is closely related to the existence of exp onential

lower b ounds for constant depth circuits made of threshold gates and seems to need new

techniques Nevertheless there are interesting relativizati on questions that still remain

op en and might b e easier for example is there an oracle for which PP is not closed under

intersection

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