Alternation in Interaction

x z y

R Sundaram D Spielman A Russell C Lund M Kiwi

MIT MIT MIT ATT MIT U Chile

Novemb er

Abstract

We study competingprover oneround interactive proof systems We show that oneround proof sys

tems in which the rst prover is trying to convince a verier to accept and the second prover is trying

to make the verier reject recognize languages in NEXPTIME and with restrictions on communica

tion and randomness languages in NP We extended the restricted model to an alternating sequence

P

of k competing provers which we show characterizes Alternating oracle proof systems are also

k 1

examined

Dept of Applied Mathematics Massachusetts Institute of Technology Cambridge MA mkiwimathmitedu On

leave of absence from Dept de Ingeniera Matematica U de Chile Supp orted by an ATT Bell Lab oratories PhD Scholarship

Part of this work was done while the author was at Bell Lab oratories

y

ATT Bell Lab oratories Ro om C Mountain Avenue P O Box Murray Hill NJ USA

lundresearchattcom

z

Dept of Applied Mathematics Massachusetts Institute of Technology Cambridge MA acrtheorylcsmitedu

Partially supp orted by an NSF Graduate Fellowship and NSF AFOSR FJ and DARPA NJ

x

Dept of Applied Mathematics Massachusetts Institute of Technology Cambridge MA spielmanmathmitedu

Partially supp orted by the Fannie and John Hertz Foundation Air Force Contract FJ and NSF grant

CCR Part of this work was done while the author was at Bell Lab oratories

Lab oratory for Computer Science Massachusetts Institute of Technology

Cambridge MA koodstheorylcsmitedu Research supp orted by DARPA contract NJ and NSF

CCR

Intro duction

Voter V is undecided ab out an imp ortant issue A Republican wants to convince her to vote one way and

a Demo crat the other What happ ens if the Republican has stolen the Demo crats brieng b o ok and thus

knows the Demo crats strategy We show that if the voter can conduct private conversations with the

Republican and the Demo crat then she can b e convinced of how to vote on NEXPTIME issues and with

suitable restrictions on communication and randomness of issues in NP

The framework of co op erating provers has received much attention Babai Fortnow and Lund BFL

showed that NEXPTIME is the set of languages that have twoco op eratingprover multiround pro of systems

This characterization was strengthened when LS FL showed that NEXPTIME languages have two

co op eratingprover oneround pro of systems Recently Feige and Kilian FK have proved that NP is

characterized by twoco op eratingprover oneround pro of systems in which the verier has access to only

O log n random bits and the provers resp onses are constant size We show that twocompetingprover

pro of systems have similar p ower

Sto ckmeyer St used games b etween comp eting players to characterize languages in the p olynomial

time hierarchy Other uses of comp eting players to study complexity classes include Reif PR Feige

Shamir and Tennenholtz FST prop osed an interactive pro of system in which the notion of comp etition

is present Recently Condon Feigenbaum Lund and Shor CFLSa CFLSb characterized PSPACE

by systems in which a verier with O log n random bits can read only a constant numb er of bits of a

P

p olynomialround debate b etween two players We show that is the class of languages that can b e

k

recognized by a verier with similar access to a k round debate We call this a system of k comp eting

oracles

We are naturally led to consider the scenario of k competing provers In a comp etingprover pro of

system two teams of comp eting provers P and P I f k g interact with a probabilistic

i iI i

iI

p olynomialtime verier V The rst team of provers tries to convince V that is in for some presp ecied

word and language L The second team has the opp osite ob jective but V do es not know which team to

trust Before the proto col b egins the provers x their strategies in the order sp ecied by their subindices

These strategies are deterministic To mo del the situation of k comp eting provers we prop ose and study

the class k APP of languages that have k alternatingprover oneround pro of systems

Denition A language L is said to have a k APP system with parameters n q n Q Q

k

Q f g and error if there is a probabilistic p olynomialtime nonadaptive verier V that

i acc r ej

interacts with two teams of comp eting provers such that

if L then Q P Q P such that

k k

Prob V P P r accepts

r k acc

if L then Q P Q P such that

k k

Prob V P P r accepts

r k r ej

where the verier is allowed one round of communication with each prover and where the probabilities are

taken over the random coin tosses of V Furthermore V uses O r n coin ips and the provers resp onses

are of size O q n

As in FRS no prover has access to the communication generated by or directed to any other prover We

adopt the following conventions when the parameters r n and q n are omitted they are assumed to b e

log n and resp ectively when the quantiers do not app ear they are assumed to alternate b eginning with

when we say the system has error We dene the class k APP to b e the set of languages

acc r ej

that have a k APP system with error

Pro of systems related to the one given in Denition but where the provers do not have access to each

others strategies have b een studied in FST FKS

We dene k alternatingoracle pro of systems analogously

Denition A language L is said to have a k AOP system with parameters r n q n Q Q

k

Q f g and error if there is a probabilistic p olynomialtime nonadaptive verier V that

i acc r ej

queries two teams of comp eting oracles such that

if L then Q O Q O such that

k k

Prob V O O r accepts

r k acc

if L then Q O Q O such that

k k

Prob V O O r accepts

r k r ej

where the probabilities are taken over the random coin tosses of V Furthermore V uses O r n coin ips

and queries O q n bits

The fundamental dierence b etween an alternatingprover pro of system and an alternatingoracle pro of

system is that provers are asked only one question whereas oracles may b e asked many questions but their

answers may not dep end on the order in which those questions are asked This requirement was lab eled

in FRS as the oracle requirement It follows that a k AOP system can b e viewed as a k APP system in

which we have imp osed the oracle requirement on the provers and are allowed to ask many questions of each

The results of LS FL FK show that in many dierent scenarios twoco op eratingprover oneround

pro of systems are equivalent in p ower to oracle pro of systems In this work we establish conditions under

which alternatingprover pro of systems are equivalent in p ower to alternatingoracle pro of systems

In Section we intro duce some of the techniques that we will use to prove our main theorem Along the

way we show that the class of languages that have twoalternatingprover pro of systems do es not dep end on

the error parameter that this class is equivalent to NP and that twoalternatingprover pro of systems lose

p ower if the provers are allowed to cho ose randomized strategies In Section we show that k alternating

P

We use Section to summarize work of Feige and Kilian FK that oracle systems characterize

k

we will need to prove our main theorem In Section we show that the class of languages that have a

k alternatingprover pro of systems do es not dep end on the error parameter and conclude that

P

k APP Theorem

k

TwoAlternating Prover Pro of Systems for NP

Supp ose a language L can b e recognized by a twoco op eratingprover oneround pro of system with verier

b b b b b

V and provers P and P quantied by in which V on random string r asks prover P question

b

q r asks prover P question q r and uses the answers to the questions to decide whether or not to

accept We will transform this system into a twoalternatingprover pro of system with verier V and provers

b b

P P quantied by In this latter system prover P claims that there exist provers P and P that

b

would convince V to accept Prover P is trying to show that P is wrong The verier V will simulate the

b b

verier V from the original system to generate the questions q r and q r that V would have asked the

b b

co op erating provers To justify his claim P will tell the verier what P or P would have said in answer to

any question To test P s claim V will pick one of the two questions q r and q r at random and ask

P to resp ond with what the corresp onding prover would have said in answer to the question Unfortunately

this do es not enable V to get the answer to b oth questions To solve this problem we recall that P knows

what P would answer to any question Thus the verier V will send b oth questions to P and request that

P resp ond by saying how P would have answered the questions

If L then P is honest and P has to lie ab out what P would say in order to get the verier to

reject If P lies ab out what P said he will b e caught with probability at least b ecause P do es not

know which question V sent to P Thus the verier will accept with probability at least

On the other hand if L then P will honestly answer V by telling V what P would have answered

b

to b oth questions In this case V will accept only if the provers that P represent would have caused V to

accept Thus we obtain a twoalternatingprover pro of system with error

We will now show how we can balance these error probabilities by a parallelization of the ab ove proto col

with an unusual acceptance criteria In our parallelized proto col we have

The Queries

Or n

b

V generates m pairs of questions as V would ie V selects r r f g and generates

m

q r q r q r q r

q r q r

m m

m

V then picks one question from each pair ie cho oses j j f g and sends to P the

m

j j j

1 2 m

questions q r q r q r and the tuple j j

m m

V sends the questions

q r q r q r q r

q r q r

m m

to P

The Resp onses

P is supp osed to resp ond with

j j

2 1

b b

q r q r P P

j j

2 1

j

m

b

q r P

m j

m

b b

which is how the provers P and P would have answered the questions

i i

1 m

P is supp osed to resp ond with what P would have said to q r q r and i i

m m

m

say a a for every one of the p ossible assignments to we allow P to answer this way b ecause

m

a cheating prover P may vary its strategy according to the value of that it receives

Acceptance Criteria

V accepts if P did not correctly represent what P said on the tuple that P was asked

V accepts if there exists a round k and index vector such that and switch and

k k

b

if necessary and such that V on random string r would accept if given answers a and a

k

k

k

Lemma A twocooperatingprover oneround proof system with error can be simulated by a two

m m

alternatingprover oneround proof system with error m an mfold increase in the veriers

randomness and an exponential in m increase in the length of the answers returned by the provers

Pro of Use the pro of system describ ed ab ove If L then P is going to honestly answer the question

he is asked in each round We will see that this forces P to lie in resp onse to all but the set of indices If

there is any mtuple of questions indexed by for which P honestly represents what P would say

then there must b e a k such that Thus if P honestly represented P s answers on tuple the

k k

verier would accept If L then the probability that V accepts is at most for each k and such that

m

for a total error of m 2

k k

We will combine this Lemma with a Theorem of Feige and Kilian FK which proves a weak version

of the parallel rep etition conjecture FL For a formal statement of the parallel rep etition conjecture

see FL

Theorem FK For any constant A language L is in NP i L has a APP system with

parameters and error

Corollary For any constant A language L is in NP i L has a APP system with error

Pro of The reverse direction is trivial the other direction is a direct consequence of Theorem and

Lemma 2

Corollary A language L is in NEXPTIME i L has a APP system with parameters pol y n pol y n

and error pol y n expn

Pro of Again the reverse implication is trivial To prove the forward direction observe that in FL it

is shown that a language in NEXPTIME has a APP system with parameters pol y n pol y n and

error expn Hence the corollary follows again from Lemma 2

Remark In Section we wil l want to use the fol lowing stronger version of Lemma assume that the

b b b

original twocooperatingprover oneround proof system with verier V and honest provers P and P had

b

twosided error We wil l say that V accepts in round k if there exists an index vector such that

acc r ej

b

and switch and if necessary and such that V on random string r would accept if

k k k

given answers a and a After P and P have answered V s questions we wil l al low player P to arbitrarily

k

k

b

choose some fraction of the m rounds and we wil l say that V accepts in a round if either V accepted in that

round or if that round was in the fraction that P chose Imagine that P is able to alter the computation

of V on those rounds We wil l change the veriers acceptance criteria to

V accepts if P did not correctly represent what P said on the tuple that P was asked

V accepts if it accepts for a fraction of the rounds we wil l set so that ln

acc

because we wil l use a Cherno bound to bound the probability of error

m m m

acc acc

m This twoalternatingprover proof system has error e

r ej

In twocooperatingprover pro of systems the p ower of the pro of system is unchanged if the provers are

allowed to cho ose randomized strategies We end this section by observing that the situation for comp eting

prover pro of systems diers

Say language L has a twoalternating randomizedprover proof system denoted ARP if it can b e

recognized by a APP system where the provers are allowed to have randomized strategies that is b efore

the proto col b egins the provers now cho ose randomized strategies instead of only deterministic strategies

in the order sp ecied by their subindices Equivalently a ARP system is a APP system where the provers

have access to private coins

Lemma For any constant a language L is in P i L has a ARP system with error

Pro of Sketch The forward direction is trivial To prove the converse observe rst that if i f g

i

is the probability distribution over deterministic strategies that prover P cho oses then the probability

i

p that the verier V accepts input n j j dep ends on through the probabilities q a

i i i i

P q a where q ranges over the set Q of p ossible questions to prover P and a over Prob

i i i i i i i P

i i

the set A of p ossible resp onses of prover P

i i

Let

q a

i i i i q Q a A

i i i i

R b e the set of random strings that the verier may generate on input

b e the probability that the verier generates random string r

r

q r b e the question that the verier sends to prover P on random string r

i i

b e equal to if on input random string r and provers resp onses a and a the verier V

ra a

0 1

accepts and otherwise

c

1 c n

resp ectively for some p ositive constant c poly n and expn refer to O n and O

Applying the technique of FL it follows that

C sub ject to p max

0

X

q q a

a

0

where C

X

min q r a q r a V

r ra a

0 1

1

ra a

0 1

sub ject to

X

q q a

a

1

where r q q a and a range over R Q Q A and A resp ectively By strong duality see Sc

C can b e expressed as the optimum of a linear program in max form Thus to compute p it is enough

Olog n

to solve a linear program of pol y n jA j jA j jRj size Since in our case jA j jA j O jRj

and linear programming is p olynomialtime solvable the lemma follows 2

An analogous result for EXPTIME for the only if part was indep endently obtained in FKS

P

AOP Systems for

k

Feige and Kilians pro of of Theorem uses an amplication of the main result of AS ALMSS which

states that NP PCPlog n In our terminology this says that NP languages have AOP pro of systems

In order to extend our results b eyond NP we will need analogous to ols that we can apply to languages in

P P

We will b egin by showing that languages in have k AOP pro of systems

k k

We consider from now on only the case in which k is o dd Our results have analogous statements for

even k

The next theorem is implicit in CFLSa

P

Theorem k o dd For any constant Every language L in has a k AOP system with error

k

In the pro of of this theorem we will make use of some facts ab out Justesen co des For a string x let

E x denote the Justesen enco ding of x MS The following facts are standard

The length of E x is linear for our purp oses p olynomial would suce in the length of x

There is a p olynomial time algorithm that on input x returns E x

There is a constant and a p olynomial time algorithm C such that if y and E x dier in at most

J J

an fraction of their bits then C y x in which case we say that x and y are closer than

J J J

Otherwise C y outputs FAILURE in which case we say that y is farther than from any

J J

co deword

P

Pro of of Theorem Let L b e a language in That is there exists a p olynomialtime Turing machine

k

V such that L if and only if X X X V X X accepts CKS As in CFLSa

k k

we will view the acceptance condition as a game b etween an player and a player who take turns writing

down p olynomiallength strings X with the player writing on the o dd rounds

i

In our k AOP the player who writes in round i purp orts to write down a Justesen enco ding of X In

i

addition in the k th round the player is to write down enco dings of everything the player said and

a PCP log n pro of that V X X accepts If each player actually wrote down co dewords then

k

using the techniques from ALMSS the verier would read a constant numb er of random bits from each

oracle and a constant numb er of bits to check the PCP log n pro of and accept if V X X would

k

have accepted or reject with high probability if V would have rejected

In reality one of the players will b e trying to cheat and will have little incentive to write co dewords Let

Y denote the oracle that is written in the ith round If a player writes an oracle Y that is within of

i i J

a co deword then the other player will pro ceed as if Y was that co deword On the other hand if a player

i

writes an oracle Y that is farther than from a co deword then with high probability the verier will detect

i J

that players p erdy

In the last round the player writes down strings Z which he claims are enco dings of Y for each even

i i

i In addition the player will for each bit of each oracle Y provide a PCP log n pro of that when

i

deco ded Z agrees with Y on that bit For each even i the verier will test these PCP log n pro ofs to

i i

check that C Z agrees with Y on some constant numb er of randomly chosen bits If any of these tests

J i i

fails then the verier will know that the player has cheated and will reject accordingly If the Z s pass

i

all the tests then the verier will b e condent that C Z is close to Y for all even i The verier then

J i i

accepts if the last players pro of indicates either that

C Y FAILURE for some even i or

J i

V X C Y C Y X accepts note that the PCP log n pro of refers to the X s for

J J k k i

o dd i through their enco ding as Y and to the Y s for i even through their enco ding as Z

i i i

To see why this proto col works assume that L In this case the player will always write co dewords

Moreover regardless of what oracle Y the player writes in turn i the exist player will write Z E Y

i i i

Thus the Z s will always pass the consistency tests If one of the players oracles is farther than from

i J

a co deword then the last player will include this fact in his pro of and the verier will reject On the other

hand if for each even i Y is close to some co deword X then the application of C to Y will result in Y

i i J i i

b eing treated as the enco ding of X in the players PCP log n pro of

i

If x L then the player will have to cheat in order to win If the player writes a message Y that is

i

not close to a unique co deword then this will b e detected with high probability when the verier checks the

validity of the PCP log n pro of supplied in the last round If one of the Z s misrepresents the oracle

i

Y E X of a player then either C Z and Y will have to dier in at least an fraction of their

i i J i i J

bits or the player will have to falsify the certication of the computation of C on C Z so that it do es

J J i

not deco de to X In either case the player will b e caught with high probability 2

i

P

b ecause an alternatingTuring machine with k alter We note that Theorem exactly characterizes

k

nations can guess the k oracles and then compute the acceptance probability of the k AOP system

Using Theorem and the standard technique of FRS for simulating an oracle by a pair of provers

we can transform a k alternatingoracle pro of system into a k alternatingprover pro of system

P

Corollary k o dd L i L has a k APP system with parameters and

k

error where N is a large constant depending on k

N

In order to prove Theorem we have to reduce the error in Corollary and show how to change the

parameters to In the next two sections we present some of the

necessary ideas that we need to achieve these goals

Previous Work

In Section we will prove an analogue of the theorem of Feige and Kilian FK which applies to k

comp eting provers The techniques used in FK provide a deep insight into how a few random variables

inuence the value of a multivariate function In order to prove our analogue of their theorem we will need

a b etter understanding of some of their results which we will summarize in this section

Consider a prover P to which we send m randomly suggested questions q r q r and to which P

m

answers according to a xed strategy f f f a function from mtuples the mtuple of questions

m

that P receives to mtuples the mtuple of answers that P resp onds with We would like P to use a global

strategy f in which each f is only a function of q r We say that such a prover b ehaves functional ly

i i

Below we state the consequences of a provers failure to b ehave in this way

Say that an f chal lenge I q r is live if a such that

i i

iI iI

Prob f q r q r a

i m i

iI iI

r

i ~

i62I

where is a parameter to b e xed later If the ab ove inequality holds say that a is a live answer set for

i

iI

the challenge I q r Intuitively if we know the questions Q q r that P receives on rounds

i i

iI iI

I and that A a is not a live answer set for the challenge I Q we should not b e willing to b et that

i

iI

P will answer A on rounds I If P acts functionally in each round then every challenge I Q is live since

the questions that P is sent on a round completely determine his answer on that round

1

ln M ln M 3

n max For any choice of parameters such that

M M

where M m n the following lemmas are implicit in the work of Feige and Kilian

Lemma If A a is not a live answer set for the chal lenge I q r then with probability

i i

iI iI

at least n over the choices of fi i g and r where I fi i g and I fi i g

j n i n j

iI nI

it holds that

h i

f q r q r a for al l i I n Prob

i m i

r

i i62I

Intuitively the preceding Lemma says that if A is not a live answer set for the challenge I q r

i

iI

and in addition to knowing the questions that P receives in rounds I we know the questions sent to P in

rounds I I we usually should still not b e willing to b et that P will answer A in rounds I Observe that

knowledge of all the questions that P receives completely determines the answers in rounds I

Lemma There is a good j for f j n such that if more than a fraction of the chal lenges

I q r over the choices of I fi i g and r are live then for a fraction of

i j i

iI iI

the live chal lenges I Q q r for every i I and for each live answer set A a for the

i i

iI iI

chal lenge I Q it holds that there is a function F such that

f iQA

fi I n I f q r q r F q r g

i 1 m i ~ ~

f iQ A

Pr

jI n I j

where the probability is taken over the choices of r and I n I fi i g

i j n

iI

In other words to every prover P we can asso ciate a j such that if a nonnegligible fraction of the

challenges I Q are live where jI j j and if P answers the challenge I Q with live answer set A then

P acts almost functionally in a signicant fraction of the rounds I n I for most of the live challenges I Q

P

APP systems for

k

P

where In Corollary we characterized in terms of alternatingprover pro of systems with error

k

N

N is a large constant and where the quantiers asso ciated to the provers except the last two alternated

P

The goal of this Section is to again provide alternatingprover pro of systems for languages which in

k

addition achieve

Low error rates and

The quantiers asso ciated to al l the provers alternate

2

We omit f whenever it is clear from context

To achieve these goals we follow an approach similar to the one taken in Section First we prove that

languages that have a k APP system with parameters and error can

N

b e recognized by a k APP system with parameters and error for any

constant Notice that we achieve this without increasing the numb er of alternations of the

quantiers asso ciated to the provers This step can b e viewed as an analogue of Theorem which applies to

comp etingprover pro of systems We then show how to convert the pro of systems obtained in the rst step

into k APP systems in which all the quantiers asso ciated to the provers alternate This latter step is

p erformed without signicantly increasing the error probability

We describ e b elow a proto col which achieves the rst goal of this section Consider a language that has a

b b b

with verier V and provers P P k APP with parameters and error

k

N

l

b b

Let q r denote the question that the verier V on random string r asks the prover P Consider now

l

a verier V interacting with provers P P quantied by resp ectively The

k

underlying idea of the proto col we are ab out to describ e is the following the verier V tries to parallelize

b

V s proto col In order to force cheating provers to b ehave functionally a Consistency Test is implemented

This test requires the use of two one for each comp eting team of provers additional provers P and

k

P If a team of provers fails the consistency test then their claim will b e rejected Honest provers

k

will b ehave functionally in each round and thus they will always b e able to PASS the Consistency Test

Nevertheless it may b e that cheating provers have a signicant probability of passing the Consistency Test

but honest provers cannot determine the strategy which the cheating provers use to answer each round from

the knowledge of their strategy alone The proto col implemented by the verier V will allow honest provers

to make a set of educated guesses regarding the strategies that the cheating provers might b e using to

answer each of the questions p osed to them Thus either cheating provers will fail the Consistency Test

or the verier V will end with high probability with a set of rounds most of which can b e treated as

b

indep endent executions of V s proto col

In what follows we describ e the questions that V makes the format in which the provers are supp osed

to answer and the acceptance criteria

The Queries

V cho oses I f mg jI j n at random and selects for every i I random string r as the old

i

b

verier V would have

j

b

V selects j f k g i f mg n I random string r as the old verier V would have

i

For i f mg j f k g let

r if i I

i

j

j

i

r otherwise

i

S

I jI j n at random together with a random ordering of I for V cho oses I I n

j l l l l

j l

l k we will later set

k

S

l

l

V sends to prover P l f k g q i f mg n I when a question is sent to

l j

i

j l

a prover the verier indicates the round i f mg to which the question corresp onds

j

the set of indices I and the V sends to prover P l f k g all questions q r

j l i iI

j

orderings for all j f l g

j

l

the set of indices I and the orderings V sends to provers P and P the questions q r

l k k i iI

l

for all l f k g

l

The Resp onses

3

In fact a k APP with parameters v and similar error rates suces but proving this would

unnecessarily complicate our exp osition

The format of each provers answer is a tree

For a graph tree T we will only consider ro oted trees we say that a no de v V T is at level i if its

distance to the ro ot of T is i We say that an edge e E T is at level i if it is ro oted at a no de of

level i

Denition Let J f r g b e a set of indices We say that a tree T is a J tree if no des at

level j J have only one child

We will consider the no des and edges of our trees as b eing lab eled Internal no des will b e lab eled by

sets of indices Edges ro oted at a no de lab eled J will b e lab eled by a tuple of strings a We

j j J

denote the lab el of a no de v resp edge e by L v resp L e

T T

Denition Let v b e a no de of level l of tree T and v v the sequence of no des along the

l

path from the ro ot of T to v Dene the history of v as follows

T

H L v L v v L v L v v

T T T l T l

v

P resp onds with a tree T

l l

T for l f k g is a tree of depth l lab eled as follows

l

An internal no de v at level j is lab eled by I

j

Every internal no de v at level j has an edge ro oted at v for every p ossible set of answers of P to

j

j

and lab eled by this tuple of answers questions q r

i iI

j

A leaf is lab eled CONCEDE DEFEAT GARBAGE or by the answers to the questions

S

l

q r i f mg n I

i j

j l

T resp T is a K tree of depth k where K f k k g resp K f k g

k k

lab eled as follows

No des and edges at levels j K are lab eled as in the trees describ ed ab ove

j

The only edge of level j K ro oted at v is lab eled by a tuple of answers a

iI

j

i

Assume now that T T are in the prop er format Dene the sequence of leaves v V T

k l l

l f k g as follows v is the only no de of T if v v have b een determined and

l

S

j

a i f mg n I is the lab el of v then v is the leaf of tree T with history

s j l l

i

sj

l

I a I a

l iI iI

l 0

i i

Dene for j f k g the following path

j

I a P T T I a

j iI j iI

j 0

i i

Note that there is only one leaf in T and T that have the same history The history of b oth these

k k

leaves determines a common path of T and T which we denote by

k k

j

I a P T T I a

iI j iI j

j i 0 i

Acceptance Criteria

The verier V checks that

i The resp onses of the provers have the prop er format If a prover do es not comply with the format

of the resp onses its claim is immediately rejected honest provers will always answer in the prop er

format

ii The following Consistency Test PASSES

V compares P P T T and P P T T Two dierent situations may arise

k k k

a P P in this case we say that the consistency test PASSES

j j

V then rejects a b P P in this case let l b e the smallest j such that a

iI iI

j j

i i

P s claim

l

S

b

V then accepts i for more than a fraction of the rounds i I n I the old verier V

j

j k

cN

k

b b

on random string r would accept if given answers a a from provers P P resp ectively

i k

i i

where c is a large constant whose value we will set later

Lemma k o dd For any constant Every language L recognized by a k APP

where N is a large constant has a k APP with with parameters and error

N

parameters and error

Pro of Sketch Use the proto col describ ed ab ove First we show how honest provers resp ond Assume P

l

is honest l k then P will generate a tree T of the prop er format We only have to show how the leaves

l l

on T are lab eled Let u u b e a path in T from the ro ot u to the leaf u In lab eling u P considers

l l l l l l

the strategies f f that P P use in lab eling the leaves abusing notation u u of

l l l

T T

l l

resp ectively For s f l g let j b e the smallest go o d j H T T with history H

s l

u u

l1 0

as dened by Lemma for f Let I I b e the set of the rst j indices of I as determined by Let

s s s s s s

s

s

Q q r b e the set of questions on rounds I and A a b e the set of answers in rounds

s i s s

i

iI iI

s s

I induced by the lab el of the leaf u

s s

Dene the following events

Event E s f l g the answer sets A are live for the f challenge I Q

s s s s

s

g q r Event E s f l g more than a fraction of the f challenges I fi i

i s s j

s

iI

s

are live g and r over the choices of I fi i

i s j

s

iI

s

For an appropriate choice of parameters we can distinguish two cases

S

Events E and E o ccur i f mg n I s f l g P determines if p ossible the

s l

sl

u

l

of as in Lemma and the optimal strategy for answering round i say P functions F

i

f iQ A

s s s

an l th prover of a k APP of the form given in Corollary where the rst l provers are given

he lab els u with If P is unable to determine some F by F F

l l

f iQ A f iQ A f iQ A

s s s 0 0 0 l1 l1 l1

CONCEDE DEFEAT that is P recognizes that cheating provers have outsmarted him Otherwise

l

u

l

P answers round i of u according to P

l l

i

Otherwise P lab els u with GARBAGE that is P expresses its condence that the cheating provers

l l l

will not PASS the consistency test

It follows that honest provers never fail the consistency test since in each leaf of their resp ective resp onse

tree honest provers answer each round functionally

Consider now the leaf v of T and the leaves v v of T T with history P T

k k

P T T x ab ove u v for i f k g l k and dene events E and E as

k i i

b efore

If E do es not o ccur then from Lemma cheating provers will PASS the consistency test with prob

ability at most k n assuming k n

If E do es not o ccur then from the preceding paragraph we conclude that cheating provers will PASS

the consistency test with probability at most

k

the honest provers are If E and E o ccur then Lemma implies that with probability at most

k

outsmarted by the cheating provers otherwise with probability at least at least k

S

k

n of the rounds i I n F I are answered according to F

j

f iQ A j k f iQ A

k k k 0 0 0

p

k

Let p small enough so and n k n Now cho ose

k

p

k k

n

and p cN k hence k n

0

n cN

large enough so n N log and n small enough so and k n M large

ln M ln M

Rememb er that all the ab ove mentioned parameters enough so if M m n max

M M

are constants

If L with probability at least p the old verier will accept for a fraction of at least

S

I and thus V will accept of the rounds i I n

j

j k

cN

0

n N

If L then the probability of V accepting is at most p where the b ound on the

log e

probability of acceptance follows by cho osing c to b e a constant large enough so that h

cN N cN

N where h is the binary entropy function

Finally note that the order of the provers might as well have b een P P P P P P

k k k k k

2

Note that the proto col used to prove Lemma is more generous with the cheating provers than actually

required We can mo dify the proto col by requiring that prover P resp ond with a tree T which is also an

l l

S tree where s S if s l and prover P is in P s team The lab els of the edges of T at level s

l l s l l

s S can now b e required to b e consistent with the lab els that P assigns to the leaves of T Combining

l s s

this mo died proto col the proto col that Feige and Kilian FK use in their pro of of Theorem and the

proto col describ ed in Remark we can prove

Lemma For any constant Every language L recognized by a k APP system with

parameters and error for any constant can be recognized by a k APP system

with parameters and error

Pro of Omitted 2

We can now prove our main theorem

P

if and only if it has a k APP Theorem For any constant A language L is in

k

system with error

P

Turing machine can guess the Pro of If L has a k APP system with error then a

k

strategy of the rst k provers compute for each one of the p olynomially many questions that the last prover

can receive the optimal constantsize resp onse and then calculate the acceptance probability of the verier

The other direction follows from Theorem Lemmas and 2

We observe that Theorem is b est p ossible in the sense that unless the p olynomialtime hierarchy

P

collapses all languages do not have k APP systems with onesided error

k

Remark For any constant If language L has a k APP system with error resp

P

then L is in

k

Pro of Omitted but not dicult 2

We hop e that the theorems presented in this pap er can b e used to prove nonapproximability results for

nonarticial ly constructed problems

Acknowledgments

This pap er owes its existence to Joan Feigenbaum we give her a sp ecial thanks We would like to thank Uri

Feige and Jo e Kilian for explaining their early results to us and for providing us with an early draft of their

pap er We would like to thank Mario Szegedy for suggesting that we generalize Corollary and for atypical

contributions We would also like to thank Michel Go emans Sha Goldwasser and Mike Sipser for helpful

discussions

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