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 L 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 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 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