Separating Complexity Classes Using Autoreducibility

Home , EXPSPACE, NL (complexity)

Separating Complexity Classes using Autoreducibility

z y

Dieter van Melkeb eek Lance Fortnow Harry Buhrman

The University of Chicago The University of Chicago CWI

Leen Torenvliet

University of Amsterdam

Abstract

A set is autoreducible if it can b e reduced to itself by a Turing machine that do es not

ask its own input to the oracle We use autoreducibility to separate the p olynomialtime hi

erarchy from p olynomial space by showing that all Turingcomplete sets for certain levels of

the exp onentialtime hierarchy are autoreducible but there exists some Turingcomplete set for

doubly exp onential space that is not

Although we already knew how to separate these classes using diagonalization our pro ofs

separate classes solely by showing they have dierent structural prop erties thus applying Posts

Program to complexity theory We feel such techniques may prove unknown separations in the

future In particular if we could settle the question as to whether all Turingcomplete sets

for doubly exp onential time are autoreducible we would separate either p olynomial time from

p olynomial space and nondeterministic logarithmic space from nondeterministic p olynomial

time or else the p olynomialtime hierarchy from exp onential time

We also lo ok at the autoreducibility of complete sets under nonadaptive b ounded query

probabilistic and nonuniform reductions We show how settling some of these autoreducibility

questions will also lead to new complexity class separations

Key words complexity classes completeness autoreducibility coherence

AMS classication Q Q D

Part of this research was done while visiting the Univ Politecnica de Catalunya in Barcelona Partially supp orted

by the Dutch foundation for scientic research NWO through NFI Pro ject ALADDIN under contract NF

and a TALENT stip end Email buhrmancwinl URL httpwwwcwinlcwipeopleHarryBuhrmanhtml

Supp orted in part by NSF grant CCR Research partly done while visiting CWI Email

fortnowcsuchicagoedu URL httpwwwcsuchicagoedufortnow

Partially supp orted by the Europ ean Union through Marie Curie Research Training Grant ERBGT

by the US National Science Foundation through grant CCR and by the Fields Institute Re

search partly done while visiting CWI and the University of Amsterdam Email dietercsuchicagoedu URL

httpwwwcsuchicagoedudieter

Partially supp orted by HCM grant ERBPL Email leenwinsuvanl URL

httpturingwinsuvanlleen

Intro duction

While complexity theorists have made great strides in understanding the structure of complexity

classes they have not yet found the prop er to ols to do nontrivial separation of complexity classes

such as P and NP They have develop ed sophisticated diagonalization combinatorial and algebraic

techniques but none of these ideas have yet proven very useful in the separation task

Back in the early days of computability theory Post wanted to show that the set of noncom

putable computably enumerable sets strictly contains the Turingcomplete computably enumerable

sets In what we now call Posts Program see Post tried to show these classes dier

by nding a prop erty that holds for all sets in the rst class but not for some set in the second

We would like to resurrect Posts Program for separating classes in complexity theory In

particular we will show how some classes dier by showing that their complete sets have dierent

structure While we do not separate any classes not already separable by known diagonalization

techniques we feel that renements to our techniques may yield some new separation results

In this pap er we will concentrate on the prop erty known as autoreducibility A set A is

autoreducible if we can decide whether an input x b elongs to A in p olynomialtime by making

queries ab out memb ership of strings dierent from x to A

Trakhtenbrot rst lo oked at autoreducibility in b oth the computability theory and space

b ounded mo dels Ladner showed that there exist Turingcomplete computably enumerable sets

that are not autoreducible Amb osSpies rst transferred the notion of autoreducibility to the

p olynomialtime setting More recently Yao and Beigel and Feigenbaum have studied a

probabilistic variant of autoreducibility known as coherence

In this pap er we ask for what complexity classes do all the complete sets have the autore

ducibility prop erty In particular we show

EXP EXP

All Turingcomplete sets for are autoreducible for any constant k where denotes

k k

the sets that are exp onentialtime Turingreducible to

There exists a Turingcomplete set for doubly exp onential space that is not autoreducible

EXP

Since the union of all sets coincides with exp onentialtime hierarchy we obtain a separation

of the exp onentialtime hierarchy from doubly exp onential space and thus of the p olynomialtime

hierarchy from exp onential space Although these results also follow from the space hierarchy

theorems which we have known for a long time our pro of do es not directly use diagonalization

rather separates the classes by showing that they have dierent structural prop erties

Issues of relativization do not apply to this work b ecause of oracle access see A p olynomial

time autoreduction can not view as much of the oracle as an exp onential or doubly exp onential

computation To illustrate this p oint we show that there exists an oracle relative to which some

complete set for exp onential time is not autoreducible

Note that if we can settle whether the Turingcomplete sets for doubly exp onential time are

all autoreducible one way or the other we will have a ma jor separation result If there ex

ists a Turingcomplete set for doubly exp onential time that is not autoreducible then we get

that the exp onentialtime hierarchy is strictly contained in doubly exp onential time thus that the

p olynomialtime hierarchy is strictly contained in exp onential time If all of the Turingcomplete

sets for doubly exp onential time are autoreducible we get that doubly exp onential time is strictly

contained in doubly exp onential space and thus p olynomial time strictly in p olynomial space We

will also show that this assumption implies a separation of nondeterministic logarithmic space from

nondeterministic p olynomial time Similar implications hold for space b ounded classes see Section

Autoreducibility questions ab out doubly exp onential time and exp onential space thus remain

an exciting line of research

We also study the nonadaptive variant of the problem Our main results scale down one

exp onential as follows

P P

All truthtablecomplete sets are truthtableautoreducible for any constant k where

k k

denotes that sets p olynomialtime Turingreducible to

There exists a truthtablecomplete set for exp onential space that is not truthtableautoreducible

Again nding out whether all truthtablecomplete sets for intermediate classes namely p olynomial

space and exp onential time are truthtableautoreducible would have ma jor implications

In contrast to the ab ove results we exhibit the limitations of our approach For the restricted

reducibility where we are only allowed to ask two nonadaptive queries all complete sets for EXP

EXPSPACE EEXP EEXPSPACE etc are autoreducible

We also argue that uniformity is crucial for our technique of separating complexity classes b e

cause our nonautoreducibility results fail in the nonuniform setting Razb orov and Rudich show

that if strong pseudorandom generators exist natural pro ofs cannot separate certain nonuniform

complexity classes Since this pap er relies on uniformity in an essential way their result do es not

apply

Regarding the probabilistic variant of autoreducibility mentioned ab ove we can strengthen

our results and construct a Turingcomplete set for doubly exp onential space that is not even

probabilistically autoreducible We leave the analogue of this theorem in the nonadaptive setting

op en Do es there exist a truthtable complete set for exp onential space that is not probabilistically

truthtable autoreducible We do show that every truthtable complete set for exp onential time

is probabilisticall y truthtable autoreducible So a p ositive answer to the op en question would

establish that exp onential time is strictly contained in exp onential space A negative answer on the

other hand would imply a separation of nondeterministic logarithmic space from nondeterministic

p olynomial time

Here is the outline of the pap er First we intro duce our notation and state some preliminaries

in Section Next in Section we establish our negative autoreducibility results for the adaptive

as well as the nonadaptive case Then we prove the p ositive results in Section where we also

briey lo ok at the randomized and nonuniform settings Section discusses the separations that

follow from our results and would follow from improvements on them Finally we conclude in

Section and mention some p ossible directions for further research

Errata in conference version

An previous version of this pap er erroneously claimed pro ofs showing all Turing complete sets

for EXPSPACE are autoreducible and all truthtable complete sets for PSPACE are nonadaptively

autoreducible Combined with the additional results in this version we would have a separation of

NL and NP see Section

However the pro ofs in the earlier version failed to account for the growth of the running time

when recursively computing previous players moves We use the pro of technique in Section

though unfortunately we get considerably weaker theorems The original results claimed in the

previous version remain imp ortant op en questions as resolving them either way will yield new

separation results

Notation and Preliminaries

Most of our complexity theoretic notation is standard We refer the reader to the textb o oks by

Balcazar Daz and Gabarro and by Papadimitriou

We use the binary alphab et f g We denote the dierence of a set A with a set B ie

the subset of elements of A that do not b elong to B by A n B

For any integer k > a formula is a Bo olean expression of the form

n n n

y y y z y y Q y

k k k

where is a Bo olean formula Q denotes if i is o dd and otherwise and the n s are p ositive

i i

integers We say that has k alternations A formula is just like except starts with

a quantier It also has k alternations A QBF formula is a formula without free

k k

variables z

denotes the k th level of the p olynomialtime hierarchy We For any integer k >

P P

dene these levels recursively by NP The levels of the p olynomialtime P and

P EXP

The and exp onentialtime hierarchy are dened as P resp ectively EXP

k k

p olynomialtime hierarchy PH equals the union of all sets and the exp onentialtime hierarchy

EXP

EXPH similarly the union of all sets

A reduction of a set A to a set B is a p olynomialtime oracle Turing machine M such that

B P

M A We say that A reduces to B and write A 6 B T for Turing The reduction M

is nonadaptive if the oracle queries M makes on any input are indep endent of the oracle ie the

queries do not dep end up on the answers to previous queries In that case we write A 6 B tt

for truthtable Reductions of functions to sets are dened similarly If the numb er of queries on

P P

an input of length n is b ounded by q n we write A 6 B resp ectively A 6 B if it is

q nT q ntt

B b for b ounded We denote the set of queries b ounded by some constant we write A 6

btt

x in case of nonadaptive reductions we omit the oracle of M on input x with oracle B by Q

B in the notation If the reduction asks only one query and answers the answer to that query we

write A 6 B m for manyone

P P P

For any reducibili ty 6 and any complexity class C a set C is 6 hard for C if we can 6 reduce

r r r

every set A C to C If in addition C C we call C 6 complete for C For any integer k >

P P

the set TQBF of all true QBF formulae is 6 complete for For k this reduces to the

k k

fact that the set SAT of satisable Bo olean formulae is 6 complete for NP

Now we get to the key concept of this pap er

Denition A set A is autoreducible if there is a reduction M of A to itself that never queries

x We cal l such M an autoreduc its own input ie for any input x and any oracle B x Q

tion of A

We will also discuss randomized and nonuniform variants A set is probabilistical ly autoreducible if it

has a probabilistic autoreduction with b ounded twosided error Yao rst studied this concept

under the name coherence A set is nonuniformly autoreducible if it has an autoreduction that uses

p olynomial advice For all these notions we can consider b oth the adaptive and the nonadaptive

case For randomized autoreducibility nonadaptiveness means that the queries only dep end on the

input and the random seed

Nonautoreducibility Results

In this section we show that large complexity classes have complete sets that are not autoreducible

Theorem There is a 6 complete set for EEXPSPACE that is not autoreducible

Most natural classes containing EEXPSPACE eg EEEXPTIME and EEEXPSPACE also have

this prop erty

We can even construct the complete set in Theorem to defeat every probabilistic autoreduc

tion

complete set for EEXPSPACE that is not probabilistical ly autore Theorem There is a 6

ducible

In the nonadaptive setting we obtain

complete set for EXPSPACE that is not nonadaptively autore Theorem There is a 6

ducible

Unlike in case of Theorem our construction do es not seem to yield a truthtable complete set

that is not probabilistically nonadaptively autoreducible In fact as we shall show in Section

such a result would separate EXP from EXPSPACE See also Section

We will detail in Section that our nonautoreducibility results do not hold in the nonuniform

setting

Adaptive Autoreductions

Supp ose we want to construct a 6 complete set A for a complexity class C that is not autoreducible

If C has a 6 complete set K we could try to enco de K in A and at the same time diagonalize

against all autoreductions A straightforward implementation would b e to enco de K y as Ah y i

and stagewise diagonalize against all 6 reductions M by picking for each M an input x not of

the form h y i that is not queried during previous stages and setting Ax M x However

the computation of M x might require deciding K y on inputs y of order of size the running

time of M on input x Since we have to do this for all p otential autoreductions M we can only

b ound the time complexity of A by a function in tn where tn denotes the running time of

some deterministic Turing machine accepting K That do es not suce to keep A inside C

In order to solve this problem we consider two p ossible co ding regions at every stage the left

region L containing strings of the form h y i and the right region R containing strings of the

form h y i Now supp ose we want to diagonalize against 6 reduction M on input x not of the

form h y i or h y i

Observation Either it is the case that for any setting of L there is a setting of R such that

A A

M x accepts or for any setting of R there is a setting of L such that M x rejects

In the former case we will set Ax and enco de K in L at that stage otherwise we will set

Ax and enco de K in R In any case

K y AhAx y i

so deciding K is still easy when given A Moreover the diagonalization ie determining Ax no

longer requires computing K y for large inputs y We just have to decide which of the ab ove cases

holds and that is indep endent of the part of K we want to enco de Provided C is p owerful enough

we can do it in C The price we have to pay is that in order to force M x to Ax we have

to set the bits in the nonco ding region so as to satisfy the underlying prop erty of Observation

In addition to determining the co ding region this still requires the knowledge of K y for p ossibly

large inputs y We will use a slightly stronger version of Observation to circumvent the need to

compute K y for to o large inputs y by grouping the quantiers corresp onding to inputs of ab out

the same length in Observation and rearranging them

This is what happ ens in the next lemma which we prove in a more general form b ecause we

will need the generalization later on in Section

Lemma Fix a set K and suppose we can decide it simultaneously in time tn and space

sn Let N be a constructible monotone unbounded function and suppose there is

a deterministic Turing machine accepting TQBF that takes time t n and space sn on QBF

formulae of size with at most log n alternations Then there is a set A such that

A is not autoreducible

K 6 A

n n

We can decide A simultaneously in time O tn t n and space O sn sn

Pro of of Lemma

Fix a function satisfying the hyp otheses of the Lemma Let M M b e a standard enumeration

of autoreductions clo cked such that M runs in time n on inputs of length n Our construction

starts out with A b eing the empty set and then adds strings to A in subsequent stages i

dened by the following sequence

(

n n

Fix and integer i > and let m n For any integer j such that 6 j 6 log m let I

i j

denote the set of all strings with lengths in the interval m minm m Note that

log m

forms a partition of the set I of strings with lengths in m m n n fI g

i i j

During the ith stage of the construction we will enco de the restriction K j of K to I into

fhb y i j b and y I g and use the string for diagonalizing against M applying the next

strengthening of Observation to do so

Claim For any set A at least one of the fol lowing holds

r r

y y I y y I y y I y y I

A m

r M accepts

y y I y y I

log m log m

r r

y y I y y I y y I y y I

A m

r M rejects

y y I y y I

log m log m

where A denotes A fh y i j y I and g fh y i j y I and r g

y y

Here we use Q z as a shorthand for Q z Q z Q z where Y fy y y g and

y y Y y y y

jY j

all variables are quantied over f g Without loss of generality we assume that the range of the

pairing function h i is disjoint from

Pro of of Claim

Fix A If do es not hold then its negation holds ie

r r

y y I y y I y y I y y I

A m

M rejects r

y y I y y I

log m log m

pairwise for every 6 j 6 log m in yields and r Switching the quantiers

y y I y y I

j j

the weaker statement

Claim

Figure describ es the ith stage in the construction of the set A Note that the lexicographically

rst values in this algorithm always exist so the construction works ne We now argue that the

resulting set A satises the prop erties of Lemma

Anf g

m m

holds The construction guarantees that by the end of stage i A M

m m

Since M on input cannot query b ecause M is an autoreduction nor any of the

i i

strings added during subsequent stages b ecause M do es not even have the time to write

m A m

down any of these strings A M holds for the nal set A So M is not an

autoreduction of A Since this is true of any autoreduction M the set A is not autoreducible

During stage i we enco de K j in the left region i we do not put into A otherwise

we enco de K j in the right region So for any y I K y AhA y i Therefore

A K 6

First note that A only contains strings of the form and hb y i for b and y

The computation intensive steps during the ith stage of the construction of A are deciding

QBF formulae of size O like and deciding K on inputs from I All stages

log m

m m

up to but not including i can b e done within time O tm t m To decide we

additionally have to check the validity of To decide hb y i with b and y I we also

have to do this and we either have to evaluate K y or run the part of stage i corresp onding

to values of j in Figure up to and including k The latter we can p erform within time

O t n where n jy j A crucial p oint here is that n upp er b ounds the length tn

of the elements of I

All together this yields the time b ound claimed for A The analysis of the space complexity

is analogous

if formula holds

then for j log m

K y

y I y y I

j j

the lexicographically rst value satisfying r

y y I

r r

y y I y y I y y I y y I

j j j j

A m

M accepts r

y y I y y I

log m log m

where A A fh y i j y I and g fh y i j y I and r g

y y

endfor

A A fh y i j y I and g fh y i j y I and r g

y y

else f formula holds g

for j log m

r K y

y y I y I

j j

the lexicographically rst value satisfying

y y I

r r

y y I y y I y y I y y I

j j j j

A m

M accepts r

y y I y y I

log m log m

where A A fh y i j y I and g fh y i j y I and r g

y y

endfor

A A f g fh y i j y I and g fh y i j y I and r g

y y

endif

Figure Stage i of the construction of the set A in Lemma

Lemma

Using the upp er b ound for sn the smallest standard complexity class to which Lemma

applies seems to b e EEXPSPACE This results in Theorem

Pro of of Theorem

In Lemma set K a 6 complete set for EEXPSPACE and n n

In section we will see that 6 in the statement of Theorem is optimal Theorem

shows that Theorem fails for 6

EXPSPACE

We note that the pro of of Theorem carries through for 6 reductions with p olyno

mially b ounded query lengths This implies the strengthening given by Theorem

Nonadaptive Autoreductions

In case of a nonadaptive autoreduction M running in time n when diagonalizing against it on

an input of length n there are only n co ding strings that can interfere as opp osed to

for autoreductions This allows to reduce the complexity of the set constructed in Lemma as

follows

Lemma Fix a set K and suppose we can decide it simultaneously in time tn and space

sn Let N be a constructible monotone unbounded function and suppose there is

a deterministic Turing machine accepting TQBF that takes time t n and space sn on QBF

formulae of size n with at most log n alternations Then there is a set A such that

A is not nonadaptively autoreducible

A K 6

n n

We can decide A simultaneously in time O tn t n and space O sn sn

Pro of of Lemma

The construction of the set A is the same as in Lemma see Figure apart from the following

dierences

M M now is a standard enumeration of nonadaptive autoreductions clo cked such that

M runs in time n on inputs of length n

During stage i > of the construction I denotes the set of all strings in Q with lengths

m m

in m m n n and I for 6 j 6 m denotes the set of strings in Q

i i j M

j j

with lengths in m minm m

At the end of stage i we additionally union A with fhb y i j b y with m 6 jy j

m y I and K y g

Adjusting time and space b ounds appropriately the pro of that A satises the prop erties claimed

carries over For the third one note that has b ecome a QBF formula of length O n

log n

Lemma

As a consequence we can lower the space complexity in the equivalent of Theorem from

doubly exp onential to singly exp onential yielding Theorem In section we will show we

cannot reduce the numb er of queries from to in Theorem

If we restrict the numb er of queries the nonadaptive autoreduction is allowed to make to some

xed p olynomial the pro of technique of Theorem also applies to EXP In particular we obtain

P P

Theorem There is a 6 complete set for EXP that is not 6 autoreducible

btt

Autoreducibili ty Results

For small complexity classes all complete sets turn out to b e autoreducible Beigel and Feigenbaum

established this prop erty of all levels of the p olynomialtime hierarchy as well as of PSPACE

the largest class for which it was known to hold b efore our work In this section we will prove it

for the levels of the exp onentialtime hierarchy

As to nonadaptive reductions the question was even op en for all levels of the p olynomialtime

hierarchy We will show here that the 6 complete sets for the levels of the p olynomialtime

hierarchy are nonadaptively autoreducible For any complexity class containing EXP we will prove

P P

autoreducible complete sets are 6 that the 6

tt tt

Finally we will also consider nonuniform and randomized autoreductions

Throughout this section we will assume without loss of generality an enco ding of a compu

tation of a given oracle Turing machine M on a given input x with the following prop erties will

b e a marked concatenation of successive instantaneous descriptions of M starting with the initial

instantaneous description of M on input x such that

Given a p ointer to a bit in we can nd out whether that bit represents the answer to an

oracle query by probing a constant numb er of bits of

If it is the answer to an oracle query the corresp onding query is a substring of the prex of

up to that p oint and we can easily compute a p ointer to the b eginning of that substring

without probing any further

If it is not the answer to an oracle query we can p erform a lo cal consistency check for that

bit which only dep ends on a constant numb er of previous bit p ositions of and the input x

We call such an enco ding a valid computation of M on input x i the lo cal consistency test for all

the bit p ositions that do not corresp ond to oracle answers are passed and the other bits equal the

oracles answer to the corresp onding query Any other string we will call a computation

Adaptive Autoreductions

complete set for EXP is autoreducible and then generalize to all We will rst show that every 6

levels of the exp onentialtime hierarchy

Theorem Every 6 complete set for EXP is autoreducible

Here is the pro of idea For any of the standard deterministic complexity classes C we can decide

each bit of the computation on a given input x within C So if A is a 6 complete set for C we can

6 reduce this decision problem for A to A Now consider the two p ossibly invalid computations

Afxg Anfxg

pjxj pjxj

hx i hx i R if R

Afxg

pjxj

then accept i R hx i

Afxg

else i R x

Anfxg

accept i lo cal consistency test for R hx i

b eing the computation of M on input x

fails on the ith bit

endif

Figure Autoreduction for the set A of Theorem on input x

we obtain by applying for every bit p osition the ab ove reduction answering all queries except for

x according to A and assuming x A for one computation and x A for the other

Note that the computation corresp onding to the right assumption ab out Ax is certainly

correct So if b oth computations yield the same answer which we can eciently check using A

without querying x that answer is correct If not the other computation contains a mistake We

cannot check b oth computations entirely to see which one is right but given a p ointer to the rst

incorrect bit of the wrong computation we can eciently verify that it is mistaken by checking

only a constant numb er of bits of that computation The p ointer is again computable within C

In case C EXP using a 6 reduction to A and assuming x A or x A as ab ove we

can determine the p ointer with oracle A but without querying x in p olynomial time since the

p ointers length is p olynomially b ounded

We now ll out the details

Pro of of Theorem

complete set A for EXP Say A is accepted by a Turing machine M such that the Fix a 6

computation of M on an input of size n has length for some xed p olynomial p Without loss

of generality the last bit of the computation gives the nal answer

Let hx ii denote the ith bit of the computation of M on input x We can compute in

EXP so there is an oracle Turing machine R 6 reducing to A

Anfxg Afxg

pjxj

Let x b e the rst i 6 i 6 such that R hx ii R hx ii provided it

reduction R from to A exists Again we can compute in EXP so there is a 6

Consider the algorithm in Figure for deciding A on input x The algorithm is a p olynomial

time oracle Turing machine with oracle A that do es not query its own input x We now argue that

it correctly decides A on input x We distinguish b etween two cases

Anfxg Afxg

pjxj pjxj

case R hx i R hx i

Afxg Anfxg

hx i coincides with the hx i or R Since at least one of the computations R

actual computation of M on input x and the last bit of the computation equals the nal

decision correctness follows

Anfxg Afxg

pjxj pjxj

case R hx i R hx i

Anfxg Anfxg

pjxj

If x A R hx i so R hx i contains a mistake Variable i gets the

correct value of the index of the rst incorrect bit in this computation so the lo cal consistency

test fails and we accept x

Anfxg

If x A R hx i is a valid computation so no lo cal consistency test fails and we reject

Theorem

The lo cal checkability prop erty of computations used in the pro of of Theorem do es not

relativize b ecause the oracle computation steps dep end on the entire query ie on a numb er of

bits that is only limited by the resource b ounds of the base machine in this case exp onentially

many We next show that Theorem itself also do es not relativize

Theorem Relative to some oracle EXP has a 6 complete set that is not autoreducible

Pro of

Note that EXP has the following prop erty

Prop erty There is an oracle Turing machine N running in EXP such that for any oracle B

B P

the set accepted by N is 6 complete for EXP

n B

Without loss of generality we assume that N runs in time Let K denote the set accepted by

complete for EXP and is not We will construct an oracle B and a set A such that A is 6

6 autoreducible

The construction of A is the same as in Lemma see Figure with n log n and

K K except for that the reductions M now also have access to the oracle B

We will enco de in B information ab out the construction of A that reduces the complexity of A

completeness of A for EXP relative to B but do it high enough so as not to destroy the 6

nor the diagonalizations against 6 autoreductions

We construct B in stages along with A We start with B empty Using the notation of Lemma

at the b eginning of stage i we add to B i prop erty do es not hold and at the end of

substage j we union B with

fh y i j y I and r y g if holds at stage i

fh y i j y I and y g otherwise

Note that this do es not aect the value of K y for jy j m nor the computations of M on

log m m

inputs of size at most m for suciently large i such that m It follows from the analysis

P P

in the pro of of Lemma that the set A is 6 hard for EXP and not 6 autoreducible

T T

Regarding the complexity of deciding A relative to B note that the enco ding in the oracle B

allows us to eliminate the need for evaluating QBF formulae of size Instead we just

log n

query B on easily constructed inputs of size O Therefore we can drop the terms corresp onding

to the QBF formulae of size in the complexity of A Consequently A EXP

log n

Theorem

Theorem applies to any complexity class containing EXP that has Prop erty eg

EXPSPACE EEXP EEXPSPACE etc

Sometimes the structure of the oracle allows to get around the lack of lo cal checkability of

oracle queries This is the case for oracles from the p olynomialtime hierarchy and leads to the

following extension of Theorem

P EXP

Theorem For any integer k > every 6 complete set for is autoreducible

T k

The pro of idea is as follows Let A b e a 6 complete set accepted by the deterministic oracle

Turing machine M with oracle TQBF First note that there is a p olynomialtime Turing machine

N such that a query q b elongs to the oracle TQBF i

y y Q y N q y y y accepts

k k k

where the y s are of size p olynomial in jq j

We consider the two purp orted computations of M on input x constructed in the pro of of

Theorem One of them b elongs to a party assuming x A the other one to a party assuming

x A The computation corresp onding to the right assumption is correct the other one might not

b e

Now supp ose the computations dier and we are given a p ointer to the rst bit p osition

where they disagree which turns out to b e the answer to an oracle query q Then we can have

the two parties play the k round game underlying The party claiming q TQBF plays the

existentially quantied y s the other one the universally quantied y s The players strategies will

i i

reducing consist of computing the game history so far determining their optimal next move 6

this computation to A and nally pro ducing the result of this reduction under their resp ective

assumption ab out Ax This will guarantee that the party with the correct assumption plays

optimally Since this is also the one claiming the correct answer to the oracle query q he will win

the game ie N q y y y will equal his answer bit

The only thing the autoreduction for A has to do is determine the value of N q y y y

in p olynomial time using A as an oracle but without querying x It can do that along the lines

of the base case algorithm given in Figure If during this pro cess the lo cal consistency test for

N s computation requires the knowledge of bits from the y s we compute these via the reduction

dening the strategy of the corresp onding player The bits from q we need we can retrieve from

the M computations since b oth computations are correct up to the p oint where they nished

generating q Once we know N q y y y we can easily decide the correct assumption ab out

The construction hinges on the hyp othesis that we can 6 reduce determining the players

moves to A Computing these moves can b ecome quite complex though b ecause we have to

recursively reconstruct the game history so far The numb er of rounds k b eing constant seems

crucial for keeping the complexity under control The conference version of this pap er erroneously

claimed the pro of works for EXPSPACE which can b e thought of as alternating exp onential time

with an exp onential numb er of alternations Establishing Theorem for EXPSPACE would

actually separate NL from NP as we will see in Section

Pro of of Theorem

EXP P

EXP complete set for accepted by the exp onentialtime oracle Turing Let A b e a 6

k T

machine M with oracle TQBF Without loss of generality there is a p olynomial p and a p olynomial

time Turing machine N such that on inputs of size n M makes exactly oracle queries all of

the form

pn pn pn

y y Q y N q y y y accepts

k k k

p n p n

where q has length Moreover the computations of N in each have length and

their last bit represents the answer the same holds for the computations of M on inputs of length

EXP

We rst dene a bunch of functions computable in For each of them say we x an

oracle Turing machine R that 6 reduces to A and which the nal autoreduction for A will

EXP

use The pro ofs that we can compute these functions in are straightforward

TQBF

Let hx ii denote the ith bit of the computation of M on input x and x the rst i

Anfxg Afxg

if any such that R hx ii R hx ii The roles of and are the same as in the pro of

of Theorem We will use R to gure out whether b oth p ossible answers for the oracle query

x A lead to the same nal answer and if not use R to nd a p ointer i to the rst incorrect

bit in any of the simulated computation getting the negative oracle answer x A If i turns out

not to p oint to an oracle query we can pro ceed as in the pro of of Theorem Otherwise we will

make use of the following functions and asso ciated reductions to A

Anfxg

Let 6 6 k The function takes an input x such that the ith bit of R hx i where

Afxg

i R x is the answer to an oracle query say For 6 m 6 let

(

Afxg Afxg

x if m R hx ii mo d R

Anfxg

x otherwise R

pjxj

We dene x as the lexicographically least y such that

Q y Q y Q y N q y y y accepts mo d

k k k

pjxj

if this value do es not exist we set x

In case i p oints to the answer to an oracle query the function and the reduction R

incorp orate the moves during the th round of the game underlying The function denes an

optimal move during that round provided it exists The reduction R together with the players

assumption ab out memb ership of x to A determines the actual move which may dier from the

one given by in case the players assumption is incorrect The condition on the righthand side

of ensures that each opp onent plays his rounds

Finally we dene the functions and which have a similar job as the functions resp ectively

TQBF

but for the computation of N q y y y instead of the computation of M x More

precisely hx r i equals the r th bit of the computation of N q y y y where the y s

k m

Afxg Anfxg

are dened by and the bit with index i R x in the computation R hx i is the

Anfxg

hx r i answer to the oracle query We dene x to b e the rst r if any for which R

Afxg Afxg Anfxg

R hx r i provided the bit with index i R x in the computation R hx i is the

answer to an oracle query

Now we have these functions and corresp onding reductions we can describ e an autoreduction for

A On input x it works as describ ed in Figure We next argue that the algorithm correctly decides

A on input x Checking the other prop erties required of an autoreduction for A is straightforward

Afxg Anfxg

p jxj p jxj

hx i and i p oints to the hx i R We only consider the cases where R

Anfxg

hx i We refer to the analysis in the pro of of Theorem answer to an oracle query in R

for the remaining cases

Anfxg Afxg

p jxj p jxj

case R hx i R hx i

Anfxg

If x A variable i p oints to the rst incorrect bit of R hx i which turns out to b e the

Anfxg

answer to an oracle query say Since the query is correctly describ ed in R hx i

Afxg Anfxg

p jxj p jxj

hx i hx i R if R

Afxg

p jxj

then accept i R hx i

Afxg

else i R x

Anfxg

if the ith bit of R hx i is not the answer to an oracle query

Anfxg

then accept i lo cal consistency test for R hx i

b eing the computation of M on input x

fails on the ith bit

Anfxg Afxg

p jxj p jxj

else if R hx i R hx i

Anfxg Anfxg

p jxj

then accept i R hx ii R hx i

Afxg

x else r R

Anfxg

accept i lo cal consistency test for R hx i

b eing the computation of N q y y y

fails on the r th bit

Anfxg

where q denotes the query describ ed in R hx i

to which the ith bit in this computation is the answer

and

(

Afxg Afxg

x if m R hx ii mo d R

Anfxg

x otherwise R

endif

Figure Autoreduction for the set A of Theorem on input x

and the setting of the y s forces the outcome of the game underlying with input q to

Afxg Afxg

p jxj

hx i gives the correct answer Since hx ii R the correct oracle answer R

Anfxg

hx ii is incorrect R

Anfxg Afxg p jxj Anfxg p jxj

R hx ii R hx i R hx i

and we accept x

Anfxg Anfxg

p jxj

hx i give the correct answer to the oracle hx ii and R If x A b oth R

Anfxg

hx i So we reject x query i p oints to in the computation R

Anfxg Afxg

p jxj p jxj

case R hx i R hx i

Anfxg Anfxg

p jxj

If x A R hx i is incorrect so R hx i has an error as a computation

of N q y y y Variable r gets assigned the index of the rst incorrect bit in this

computation so the lo cal consistency check fails and we accept x

Anfxg

If x A R hx i is a valid computation of N q y y y so every lo cal consistency

test is passed and we reject x

Theorem

Nonadaptive Autoreductions

So far we constructed autoreductions for 6 complete sets A On input x we lo oked at the

two candidate computations obtained by reducing to A answering all oracle queries except for x

according to A and answering query x p ositively for one candidate and negatively for the other If

the candidates disagreed we tried to nd out the right one which there always was We managed

to get the idea to work for quite p owerful sets A eg EXPcomplete sets by exploiting the lo cal

checkability of computations That allowed us to gure out the wrong computation without going

through the entire computation ourselves With help from A we rst computed a p ointer to the

rst mistake in the wrong computation and then veried it lo cally

We cannot use this adaptive approach for constructing nonadaptive autoreductions It seems

like guring out the wrong computation in a nonadaptive way requires the autoreduction to p erform

the computation of the base machine itself so the base machine has to run in p olynomial time

Then checking the computation essentially b oils down to verifying the oracle answers Using the

game characterization of the p olynomialtime hierarchy along the same lines as in Theorem we

can do this for oracles from the p olynomialtime hierarchy

P P

is nonadaptively autore Theorem For any integer k > every 6 complete set for

ducible

Parallel to the adaptive case an earlier version of this pap er stated Theorem for unb ounded

k ie for PSPACE However we only get the pro of to work for constant k In Section we will

see that proving Theorem for PSPACE would separate NL from NP

The only additional diculty in the pro of is that in the nonadaptive setting we do not know

which player has to p erform the even rounds and which one the o dd rounds in the k round game

underlying a query like But we can just have them play b oth scenarios and afterwards gure

out the relevant run

Pro of of Theorem

P P

accepted by the p olynomialtime oracle Turing machine P Let A b e a 6 complete set for

M with oracle TQBF Without loss of generality there is a p olynomial p and a p olynomialtime

Turing machine N such that on inputs of size n M makes exactly pn oracle queries q all of the

form

pn pn pn

y y Q y N q y y y accepts

k k k

TQBF

on input x Note where q has length p n Let q x i denote the ith oracle query of M

that q FP

P P

so there is a 6 reduction R Let Q fhx ii j q x i TQBF g The set Q b elongs to

tt k

from Q to A

Anfxg Afxg

agree on hx j i for every 6 j 6 pjxj we are home and R If for a given input x R

Q Q

Afxg

hx j i as the answer to the j th oracle query We can simulate the base machine M using R

Otherwise we will make use of the following functions computable in and

corresp onding oracle Turing machines R R R dening 6 reductions to A Let 6 6 k

The function is dened for inputs x such that there is a smallest 6 i 6 pjxj for which

Afxg Anfxg

hx ii For 6 m 6 let hx ii R R

Q Q

(

Afxg Afxg

hx ii mo d x if m R R

Anfxg

x otherwise R

pjxj

We dene x as the lexicographically least y such that

Q y Q y Q y N q x i y y y accepts mo d

k k k

pjxj

we set x if such string do es not exist

The intuitive meaning of the functions and the reductions R is similar to in the pro of

of Theorem They capture the moves during the th round of the game underlying for

q q x i The function encapsulates an optimal move during round if it exists and the

reduction R under the players assumption regarding memb ership of x to A pro duces the actual

move in that round The condition on the righthand side of guarantees the correct alternation

of rounds

Consider the algorithm in Figure Note that the only queries to A the algorithm in Figure

Afxg Anfxg

hx j i for every 6 j 6 pjxj hx j i R if R

Q Q

Afxg

then accept i M accepts x when the j th oracle query is answered R hx j i

Afxg Anfxg

hx j i hx j i R else i rst j such that R

Q Q

Afxg

hx ii accept i N q y y y R

where q denotes the ith query of M on input x

Afxg

hx j i when the answer to the j th oracle query is given by R

and

(

Afxg Afxg

hx ii mo d x if m R R

Anfxg

x otherwise R

endif

Figure Nonadaptive autoreduction for the set A of Theorem on input x

needs to make are the queries of R dierent from x on inputs hx j i for 6 j 6 pjxj and the

queries of R dierent from x on input x for 6 m 6 k Since R and the R s are nonadap

m m

tive it follows that Figure describ es a 6 reduction to A that do es not query its own input A

similar but simplied argument as in the pro of of Theorem shows that it accepts A So A is

nonadaptively autoreducible Theorem

Next we consider more restricted reductions Using a dierent technique we show

P P

Theorem For any complexity class C every 6 complete set for C is 6 autoreducible

tt tt

provided C is closed under exponentialtime reductions that only ask one query which is smal ler in

length

In particular Theorem applies to C EXP EXPSPACE and EEXPSPACE In view of

Theorems and this implies that Theorems and are optimal

The pro of exploits the ability of EXP to simulate all p olynomialtime reductions to construct

an auxiliary set D within C such that any 6 reductions of D to some xed complete set A has

a prop erty that induces an autoreduction on A

Pro of of Theorem

P i

Let M M b e a standard enumeration of 6 reductions such that M runs in time n on

inputs of size n Let A b e a 6 complete set for C

Consider the set D that only contains strings of the form h xi for i N and x and is

decided by the algorithm of Figure on such an input Except for deciding Ax the algorithm

case truthtable of M on input h xi with the truthvalue of query x set to Ax

constant

A i

accept i M rejects h xi

of the form y A

accept i x A

otherwise

accept i x A

endcase

Figure Algorithm for the set D of Theorem on input h xi

runs in exp onential time Therefore under the given conditions on C D C so there is a 6

reduction M from D to A

The construction of D diagonalizes against every 6 reduction M of D to A whose truth

table on input h xi would b ecome constant once we lled in the memb ership bit for x Therefore

for every input x one of the following cases holds

The reduced truthtable is of the form y A with y x

Then y A M accepts h xi x A

The reduced truthtable is of the form y A with y x

Then y A M accepts h xi x A

The truthtable dep ends on the memb ership to A of strings dierent from x

A j

Then M do es not query x on input h xi and accepts i x A

The ab ove analysis shows that the algorithm of Figure describ es a 6 reduction of A

Theorem

Probabilistic and Nonuniform Autoreductions

The previous results in this section trivially imply that the 6 complete sets for the levels of the

exp onentialtime hierarchy are probabilistically autoreducible and the 6 complete sets for the

if jQ h xi n fxgj

A j

then accept i M accepts h xi

else f jQ h xi n fxgj g

y unique element of Q h xi n fxg

accept i y A

endif

Figure Autoreduction constructed in the pro of of Theorem

levels of p olynomialtime hierarchy are probabilistically nonadaptively autoreducible Randomness

allows us the prove more in the nonadaptive case

First we can establish Theorem for EXP

complete set for EXP is probabilis Theorem Let f be a constructible function Every 6

f ntt

P P

tical ly 6 autoreducible In particular every 6 complete set for EXP is probabilistical ly

O f ntt

nonadaptively autoreducible

Pro of of Theorem

Let A b e a 6 complete set for EXP We will apply the PCP Theorem for EXP to A

f ntt

Lemma There is a constant k such that for any set A EXP there is a polynomialtime

Turing machine V and a polynomial p such that for any input x

If x A then there exists a proof oracle such that

Pr V x r accepts

jr jpjxj

If x A then for any proof oracle

Pr V x r accepts 6

jr jpjxj

Moreover V never makes more than k proof oracle queries and there is a proof oracle EXP

independent of x such that holds for in case x A

Translating Lemma into our terminology we obtain

Lemma There is a constant k such that for any set A EXP there is a probabilistic 6

k tt

reduction N and a set B EXP such that for any input x

If x A then N x always accepts

If x A then for any oracle C N x accepts with probability at most

P A

Let R b e a 6 reduction of B to A and consider the probabilistic reduction M that on

f ntt

Afxg A P

input x runs N on input x with oracle R M is a probabilistic 6 reduction to A

k f ntt

that never queries its own input The following shows it denes a reduction from A

Afxg A A B

If x A R R B so M x N x always accepts

Afxg A C

If x A then for C R M x N x accepts with probability at most

Theorem

Note that Theorem makes it plausible why we did not manage to scale down Theorem

by one exp onent to EXPSPACE in the nonadaptive setting as we were able to do for our other

results in Section when going from the adaptive to the nonadaptive case This would separate

EXP from EXPSPACE

We suggest the extension of Theorem to the levels of the exp onentialtime hierarchy as

an interesting problem for further research

Second Theorem also holds for NP

Theorem Al l 6 complete sets for NP are probabilistical ly nonadaptively autoreducible

Pro of of Theorem

P P

Fix a 6 complete set A for NP Let R denote a length nondecreasing 6 reduction of A to

tt m

SAT

Dene the set

i m

W fh i j is a Bo olean formula with say m variables and a a and a g

Since W NP there is a 6 reduction R from W to A

We will use the following probabilistic algorithm by Valiant and Vazirani

Lemma There exists a polynomialtime probabilistic Turing machine N that on input

a Boolean formula with n variables outputs another quantier free Boolean formula N

such that

If is satisable then with probability at least has a unique satisfying assignment

If is not satisable then is never satisable

Now consider the following algorithm for A On input x run N on input R x yielding a

Bo olean formula with say m variables and it accepts i

Afxg Afxg Afxg Afxg

m i

h i h i R h i R h i R R

W W W W

evaluates to true Note that this algorithm describ es a probabilistic 6 reduction to A that never

queries its own input Moreover

the ValiantVazirani algorithm N pro duces a If x A then with probability at least

jxj

Bo olean formula with a unique satisfying assignment a In that case the assignment we

Afxg Afxg Afxg Afxg

m i

h i equalsa and h i R h i R h i R use R

W W W W

we accept x

If x A any Bo olean formula which N pro duces has no satisfying assignment so we always

reject x

Executing n indep endent runs of this algorithm and accepting i any of them accepts yields a

probabilistic nonadaptive autoreduction for A Theorem

So for probabilistic autoreductions we get similar results as for deterministic ones Low end

complexity classes turn out to have the prop erty that their complete sets are autoreducible whereas

high end complexity classes do not As we will see in more detail in the next section this structural

dierence yields separations

If we allow nonuniformity the situation changes dramatically Since probabilistic autoreducibil

ity implies nonuniform autoreducibility all our p ositive results for small complexity classes carry

over to the nonuniform setting But as we will see next the negative results do not b ecause also

the complete sets for large complexity classes b ecome autoreducible b oth in the adaptive and in the

nonadaptive case So uniformity is crucial for separating complexity classes using autoreducibility

and the Razb orovRudich result do es not apply

Feigenbaum and Fortnow dene the following concept of Probustness of which we also

consider the nonadaptive variant

A A A

Denition A set A is Probust if P FP A is nonadaptively Probust if P

Nonadaptive Probustness implies Probustness For the usual deterministic and nonde

terministic complexity classes containing PSPACE all 6 complete sets are Probust For the

deterministic classes containing PSPACE it is also true that the 6 complete sets are nonadap

tively Probust

The following connection with nonuniform autoreducibility holds

Theorem Al l Probust sets are nonuniformly autoreducible Al l nonadaptively Probust

sets are nonuniformly nonadaptively autoreducible

Pro of

Feigenbaum and Fortnow show that every Probust language is randomselfreducible Beigel

and Feigenbaum prove that every randomselfreducible set is nonuniformly autoreducible or

weakly coherent as they call it Their pro ofs carry over to the nonadaptive setting

It follows that the 6 complete sets for the usual deterministic complexity classes containing

PSPACE are all nonuniformly nonadaptively autoreducible The same holds for adaptive reduc

tions in which case the prop erty is also true of nondeterministic complexity classes containing

PSPACE In particular we get the following

complete sets for NEXP EXPSPACE EEXP NEEXP EEXPSPACE Corollary Al l 6

are nonuniformly autoreducible Al l 6 complete sets for PSPACE EXP EXPSPACE are

nonuniformly nonadaptively autoreducible

Separation Results

In this section we will see how we can use the structural prop erty of all complete sets b eing

autoreducible to separate complexity classes Based on the results of Sections and we only

get separations that were already known EXPH EEXPSPACE by Theorems and

EXP EEXPSPACE by Theorems and and PH EXPSPACE by Theorems and

and also by scaling down EXPH EEXPSPACE However settling the question for certain

other classes would yield impressive new separations

We summarize the implications in Figure

Theorem In Figure a positive answer to a question from the rst column implies the

separation in the second column and a negative answer the separation in the third column

question yes no

Are all 6 NL NP PH PSPACE complete sets for EXPSPACE autoreducible

NL NP

Are all 6 complete sets for EEXP autoreducible PH EXP

P PSPACE

P P

Are all 6 NL NP PH PSPACE complete sets for PSPACE 6 autoreducible

tt tt

NL NP

P P

Are all 6 complete sets for EXP 6 autoreducible PH EXP

tt tt

P PSPACE

Are all 6 complete sets for EXPSPACE

probabilisticall y 6 NL NP P PSPACE autoreducible

Figure Separation results using autoreducibility

Most of the entries in Figure follow directly from the results of the previous sections In order to

nish the table we use the next lemma

Lemma If NP NL we can decide the validity of QBFformulae of size t and with alter

O c

nations on a deterministic Turing machine M in time t and on a nondeterministic Turing

machine M in space O c log t for some constant c

Pro of of Lemma

Since coNP NP by Co oks Theorem we can transform in p olynomial time a formula with

free variables into an equivalent formula with the same free variables and vice versa Since

NP P we can decide the validity of formulae in p olynomialtime Say b oth the transformation

algorithm T and the satisability algorithm S run in time n for some constant c

Let b e a QBFformula of size t with alternations Consider the following algorithm for

deciding Rep eatedly apply the transformation T to the largest sux that constitutes a or

formula until the whole formula b ecomes and then run S on it

This algorithm correctly decides the truth of Since the numb er of alternations decreases by

one during every iteration it makes at most calls to T each time at most raising the length of

O c

the formula to the p ower c It follows that the algorithm runs in time t

Moreover since P NL a padding argument shows that DTIME NSPACElog for any

time constructible function Therefore the result holds Lemma

This allows us to improve Theorems and as follows under the hyp othesis NP NL

Theorem If NP NL there is a 6 complete set for EXPSPACE that is not probabilisti

cal ly autoreducible The same holds for EEXP instead of EXPSPACE

Pro of

Combine Lemma with the probabilistic extension of Lemma used in the pro of of Theorem

We have studied the question whether all complete sets are autoreducible for various complexity

Theorem If NP NL there is a 6 complete set for PSPACE that is not nonadaptively

autoreducible The same holds for EXP instead of PSPACE

Pro of

Combine Lemma with Lemma

Now we have all ingredients for establishing Figure

Pro of of Theorem

The NL NP implications in the yescolumn of Figure immediately follow from Theorems

and by contrap osition

By Theorem a p ositive answer to the nd question in Figure would yield EEXP

EEXPSPACE and by Theorem a p ositive answer to the th question would imply EXP

EXPSPACE By padding b oth translate down to P PSPACE

Similarly by Theorem a negative answer to the nd question would imply EXPH EEXP

which pads down to PH EXP A negative answer to the th question would yield PH EXP

directly by Theorem By the same token a negative answer to the st question results in

EXPH EXPSPACE and PH PSPACE and a negative answer to the rd question in PH

PSPACE By Theorem a negative answer to the last question implies EXP EXPSPACE and

P PSPACE

We note that we can tighten all of the separations in Figure a bit b ecause we can apply

Lemmata and to smaller classes than in Theorems resp ectively One improvement

along these lines that might warrant attention is that we can replace NL NP in Figure by

coNP NP NSPACElog n This is b ecause that condition suces for Theorems and

since we can strengthen Lemma as follows

Lemma If coNP NP NSPACElog n we can decide the validity of QBFformulae of

O c

size t and with alternations on a deterministic Turing machine M in time t and on a

nondeterministic Turing machine M in space O d log t for some constants c and d

Conclusion

classes and various reducibili ties We obtained a p ositive answer for lower complexity classes in

Section and a negative one for higher complexity classes in Section This way we separated

these lower complexity classes from these higher ones by highlighting a structural dierence The

resulting separations were not new but we argued in Section that settling the very same question

for intermediate complexity classes would provide ma jor new separations

We b elieve that renements to our techniques may lead to them and would like to end with a

few words ab out some thoughts in that direction

One do es not have to lo ok at complete sets only Let C C Supp ose we know that all

complete sets for C are autoreducible Then it suces to construct eg along the lines of Lemma

a hard set for C that is not autoreducible in order to separate C from C

As we mentioned at the end of Section we can improve Theorem a bit by applying Lemma

to smaller spaceb ounded classes than EEXPSPACE We can not hop e to gain much though

since the co ding in the pro of of Lemma seems to b e DSPACE complete b ecause of the

QBF formulae of size involved for inputs of size n The same holds for Theorem

log n

and Lemma

Generalizations of autoreducibility may allow us to push things further For example one could

lo ok at k nautoreducibility where k n bits of the set remain unknown to the querying machine

Theorem go es through for k n O log n Perhaps one can exploit this leeway in the co ding of

Lemma and narrow the gap b etween the p ositive and negative results As discussed in Section

that would yield interesting separations

Finally one may want to lo ok at other prop erties than autoreducibility to realize Posts Program

in complexity theory Perhaps another concept from computability theory or a more articial

prop erty can b e used to separate complexity classes

Acknowledgments

We would like to thank Manindra Agrawal and Ashish Naik for very helpful discussions We are

also grateful to Carsten Lund and Muli Safra for their help regarding the PCP Theorem

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