BPP HAS SUBEXPONENTIAL TIME SIMULATIONS UNLESS EXPTIME HAS PUBLISHABLE PROOFS Laszlo Babai Lance Fortnow Noam Nisan and Avi Wigderson Abstract We show that BPP can b e simulated in sub exp onential time for innitely many input lengths unless exp onential time collapses to the second level of the p olynomialtime hierarchy has p olynomialsize circuits and has publishable pro ofs EXPTIME MA We also show that BPP is contained in sub exp onential time unless ex p onential time has publishable pro ofs for innitely many input lengths In addition we show BPP can b e simulated in sub exp onential time for innitely many input lengths unless there exist unary languages in MA P The pro ofs are based on the recent characterization of the p ower of multiprover interactive proto cols and on random selfreducibili ty via lowdegree p olynomials They exhibit an interplay b etween Bo olean circuit simulation interactive pro ofs and classical complexity classes An imp ortant feature of this pro of is that it do es not relativize One of the ingredients of our pro of is a lemma that states that if EXPTIME has p olynomial size circuits then EXPTIME MA This extends previous work by Alb ert Meyer Key words Complexity Classes Interactive Pro of Systems Sub ject classications Q Intro duction How much time is necessary to deterministically simulate a probabilistic machine We could simulate our machine for every p ossible choice of coin tosses Babai et al and then compute the probability of acceptance Simulating a probabilistic p olynomialtime Turing machine using this metho d would require exp onential time Do es there exist a faster way One solution involves pseudorandom numb er generators A pseudorandom numb er generator maps some set of random bits to a larger set of bits where the larger set of bits is computationally indistinguishable from random One could use a pseudorandom numb er generator to reduce the numb er of random bits needed for a probabilistic algorithm to accept a certain language and thus to reduce the numb er of computation paths to simulate Blum and Micali describ ed the rst secure pseudorandom generator based on the assumption of the hardness of the discrete log function These ideas are generalized by Yao who showed that one can convert any oneway p ermutation into a pseudorandom numb er generator to show that BPP has sub exp onential time simulations A series of results show that any oneway function is enough to create a pseudorandom generator that could b e used to show that BPP is in sub exp onential time Rather than making hardness assumptions on certain typ es of functions in this pap er we shall consider the eect of complexity theoretic assumptions Nisan and Wigderson show that if there exists a function computable in exp onential time that cannot b e approximated by a p olynomialsize circuit then there exists a pseudorandom numb er generator computable in sub exp onential time that lo oks random to p olynomialsize circuits for innitely many input lengths They use this pseudorandom numb er generator to show that if such a hard function exists then BPP has such a weak sub exp onential simulation We generalize this result by showing that a much weaker assumption will still give the same consequence We show that BPP has a weak sub exp onential n for innitely many values of n and every simulation ie a simulation in time unless exp onential time has p olynomial size circuits and is contained in the class MA of languages with probabilistically checkable pro ofs Since MA is P P known to lie within we deduce that sub exp onential simulations for 2 2 BPP exist unless EXPTIME lies within the second level of the p olynomialtime hierarchy Thus the main theorems of our pap er are the following Theorem BPP admits weak sub exp onential time simulations unless EXPTIME P MA P pol y 2 BPP is contained in sub exp onential time unless MA weakly simulates EXPTIME BPP and Sub exp onential Time We will use Theorem to also prove the following Theorem BPP admits weak sub exp onential time simulations unless there exist unary languages in MA P From the pro of of Theorem we will deduce the following corollary Corollary If EH E then P BPP Corollary is a rare instance of a collapse at the exp onentialtime level implying a collapse at the p olynomialtime level Background and Denitions Nisan and Wigderson showed a general theorem on how certain hard functions could b e used to create various pseudorandom numb er generators We will use a sp ecic corollary of their work k n and the notation E for We use the notation EXPTIME for DTIME k 0 P cn k DTIME We dene EH the exp onentialtime hierarchy as E c0 k 0 The class MA dened by Babai cf denotes the MerlinArthur class the class of languages accepted by an interactive pro of system consisting of a single message from the prover followed by probabilistic verication Formally we say that L MA if there exists a probabilistic p olynomial time machine M and a p olynomial q n such that For all x L there exists a y jy j q jxj and M x y accepts with probability at least twothirds and For all x L and for all y such that jy j q jxj M x y accepts with probability at most onethird Arguably this represents the class of languages with publishable proofs of mem b ership not requiring direct interaction b etween prover and verier the verier P P can ip coins at any later date Babai has shown that contains 2 2 MA Let f g Let f b e a function mapping to We say f is tn approximated by circuits of size sn if for all suciently large n there exists a circuit C of size sn on inputs of length n such that n PrC x f x n tn Babai et al n where x is chosen uniformly over We say that f is weakly tnapproximated by circuits of size sn if the ab ove statement holds for innitely many n Nisan and Wigderson use j approximated by circuits of size sn to mean weakly n approximated by circuits of size sn for any xed j in our terminology A function f cannot b e weakly approximated by p olynomialsize circuits j j if it cannot b e weakly n approximated by circuits of size n for any xed j A language cannot b e approximated if the characteristic function of that language cannot b e approximated A language class cannot b e approximated if some language in that class cannot b e approximated The class P pol y consists of all languages recognizable by a nonuniform family of p olynomialsize circuits We say that a machine M weakly computes the language L if for innitely many values of n n n L LM We say that L admits weak subexponential simulations if for every n time b ounded Turing machine which weakly computes L We there exists a say that a language class C admits weak sub exp onential simulations if each memb er of C do es A class D weakly simulates a class C if for every language in C there is a language in D that agrees with C for innitely many input lengths We will use and signicantly extend the following theorems due to Nisan and Wigderson Theorem If EXPTIME cannot b e approximated by p olynomialsize circuits then BPP admits weak sub exp onential simulations If EXPTIME cannot b e weakly approximated by p olynomialsize circuits n then BPP DTIME 0 To extend these theorems we will use some of the recent work by Babai Fortnow and Lund on multiple prover interactive pro of systems Interac tive pro of systems were intro duced by Babai and Goldwasser Micali and Racko as a probabilistic extension of NP The mo del consists of an in nitely p owerful but untrustworthy prover that tries to convince a probabilistic p olynomialtime verier that a string is in a certain language A recent series of results by Lund Fortnow Karlo and Nisan and Shamir show that this mo del accepts exactly those languages recognizable in p olynomial space BPP and Sub exp onential Time BenOr Goldwasser Kilian and Wigderson generalize this mo del to have many provers that cannot communicate with each other or see the communi cation of the others with the verier Babai Fortnow and Lund show that this mo del accepts exactly the languages recognizable in nondeterministic ex p onential time We will use the following theorem from regarding the p ower of EXPTIME provers Theorem Any language in EXPTIME has a multiprover interactive pro of system where the honest provers are limited to computing within de terministic exp onential time We note that this result implies that EXPTIME admits instance checking in the sense of Blum and Kannan cf Random SelfReducibil i ty in High Complexity Classes Following an idea of Beaver and Feigenbaum Lipton observed that the p ermanent function has the following random selfreducibility prop erty if p is a prime greater than n and an oracle tells the value of n n p ermanents 1 over Z correctly for a p ortion of the set of inputs then one can use this p 3n oracle to nd the correct value with exp onentially small probability of error on every input Blum Luby and Rubinfeld used similar ideas The idea of the pro of is that the value of p ermanent of A can b e computed using interp olation from the p ermanents of any n matrices A B where i B is a random matrix and Z i p A similar idea works for arbitrary p olynomials of low degree In particular by extending a Bo olean function f to a multilinear function g over Z we p f obtain a random selfreducible f hard and PSPACE easy function g We prove the following lemmas from for completeness s Lemma Let A f g Q b e a function Then A has a unique multi s e linear extension A Q Q e