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In Search of an Easy Witness: Exponential Time Vs. Probabilistic

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In Search of an Easy Witness: Exponential Time Vs. Probabilistic In Search of an Easy Witness Exp onential Time vs Probabilistic Polynomial Time y Valentine Kabanets Russell Impagliazzo Department of Computer Science Department of Computer Science University of California San Diego University of California San Diego La Jolla CA La Jolla CA kabanetscsucsdedu russellcsucsdedu z Avi Wigderson Department of Computer Science Hebrew University Jerusalem Israel and Institute for Advanced Study Princeton NJ aviiasedu May Abstract n Restricting the search space f g to the set of truth tables of easy Bo olean functions e establish on log n variables as well as using some known hardnessrandomness tradeos w a numb er of results relating the complexityof exp onentialtime and probabilistic p olynomial time complexity classes In particular we show that NEXP Ppoly NEXP MA this can be interpreted as saying that no derandomization of MA and hence of promiseBPP is p ossible unless NEXP contains a hard Bo olean function Wealsoprove several downward closure results for ZPP RP BPP and MA eg weshow EXP BPP EE BPE where EE is the doubleexp onential time class and BPE is the exp onentialtime analogue of BPP Intro duction One of the most imp ortant question in complexity theory is whether probabilistic algorithms are more p owerful than their deterministic counterparts A concrete formulation is the op en question of whether BPP P Despite growing evidence that BPP can be derandomized ie simulated deterministically without a signicant increase in the running time so far it has not b een ruled out that NEXP BPP Research supp orted byNSFAward CCR and USAIsrael BSF Grant y This researchwas done while the author was at the Department of Computer Science UniversityofToronto and at the Scho ol of Mathematics Institute for Advanced Study Princeton At IAS the researchwas supp orted by NSF grant DMS z This research was supp orted by grant numb er of the Israel Science Foundation founded by the Israel Academy for Sciences and Humanities and partially supp orted by NSF grants CCR and CCR Anumb er of conditional derandomization results are known which are based on the assumption that EXP contains hard Bo olean functions ie those of high circuit complexity NW BFNW O n ACR IW STV For instance it is shown in IW that BPP P if DTIME contains n a language that requires Bo olean circuits of size Results of this form usually called hardness randomness tradeos are proved by showing that the truth table of a hard Bo olean function can b e used to construct a pseudorandom generator which is then used to derandomize BPP or some other probabilistic complexity class It is well known that such pseudorandom generators exist if and only if there exist hard Bo olean functions in EXP However it is not known whether the existence of hard Bo olean functions in EXP is actually necessary for derandomizing BPP That is it is not known if BPP SUBEXP EXP Ppoly Obtaining such an implication would yield a normal form for derandomization b ecause hardnessvsrandomness results actually conclude that BPP can b e derandomized in a very sp ecic way Think of a probabilistic algorithm after xing its input x as dening a Bo olean function f r on the random bits r Since the algorithm is fast we know f is easy ie has low x x circuit complexity For an algorithm accepting a language L in BPP f is either almost always x if x L or almost always otherwise To decide which it suces to approximate the fraction of r s with f r to within a constant additive error To do this the derandomization rst x computes all p ossible sequences that are outputs of a generator G say r r and tries f r t x i for each i If G is a pseudorandom generator the nal output is the ma jority of the bits f r x i Other constructions such as the hittingset derandomization from ACR are more complicated but have the same general form In particular We never use the acceptance probability guarantees for the algorithm on other inputs Thus w e can derandomize algorithms even when acceptance separations arent guaranteed for all inputs ie we can derandomize promise BPP For KRC Intuitively this means that randomized heuristics that only p erform well on some inputs can also be simulated by a deterministic algorithm that p erforms well on the same inputs as the randomized algorithm The derandomization pro cedure only uses f as an oracle Although its correctness relies on x the existence of a small circuit computing f the circuit itself is only used in a blackbox x fashion Derandomization along the lines ab ove is equivalent to proving circuit lower b ounds which seems dicult One might hop e to achieve derandomization unconditionally by relaxing the ab ove restrictions In particular one could hop e that it is easier to approximate the acceptance probability using the circuit itself than it is treating it as a black box In fact recent results of a circuit indicate that in general having access to the circuit computing a function is stronger than having the function as an oracle BGI Bar However we show that this hop e is illfounded for nondeterministic algorithms solving the approximation problem for circuit acceptance oracle access is just as powerful as access to the circuit In particular any even nondeterministic derandomization of promise BPP yields a circuit lower b ound for NEXP and hence a blackb ox circuit approximation algorithm running in non deterministic sub exp onential time Thus unconditional results in derandomization require either er b ound for NEXP making a distinction b etween BPP and promise BPP or proving a circuit low More precisely we show that NEXP Ppoly MA NEXP and hence no derandomization of MA is p ossible unless there are hard functions in NEXP Since derandomizing promise BPP also allows one to derandomize MA the conclusion is that no full derandomization result is p ossible without assuming or proving circuit lower b ounds for NEXP Another piece of evidence that it will be dicult to show EXP BPP or NEXP MA comes from the downward closure results for these classes It is a basic fact in computational complexity that the equalities of complexity classes translate upwards For example if NP P then NEXP EXP by a simple padding argument Thus a separation at a higher level implies a separation at a lower level which suggests that higherlevel separations are probably harder to prove We show that separating EXP from BPP is as hard as separating their higher time complexity analogues More precisely we show that EXP BPP i EE BPE where EE is the O n O n class of languages accepted in deterministic time and BPE is the time analogue of BPP Weprove similar downward closures for ZPP RPandMA Main Techniques One of the main ideas that we use to derive our results can be informally describ ed as the easy witness metho d invented by Kabanets Kab It consists in searching that have for a desired ob ject eg a witness in a NEXP search problem among those ob jects concise descriptions eg truth tables of Bo olean functions of low circuit complexity Since there are few binary strings with small descriptions such a search is more ecient than the exhaustive search On the other hand if our search fails then we obtain a certain hardness test an ecient algorithm that accepts only those binary strings which do not have small descriptions With such a hardness test we can guess a truth table of a hard Bo olean function and then use it as a source of pseudorandomness via known hardnessrandomness tradeos Recall that the problem SuccinctSAT is to decide whether a prop ositional formula is satisable when given a Bo olean circuit which enco des the formula eg the truth table of the Bo olean function computed by the circuit is an enco ding of the prop ositional formula it is easy to see that SuccinctSATisNEXP complete Thus the idea of reducing the search space for NEXP problems to easy witnesses is suggested by the following natural question Is it true that every satisable prop ositional formula that is describ ed by a small Bo olean circuit must have at least one satisfying assignment that can also b e describ ed by a small Bo olean circuit We will show that this is indeed the case if NEXP Pp oly This idea was applied in Kab to RP search problems in order to obtain certain uniform setting derandomization of RP In this pap er we consider NEXP search problems which allows us to prove our results in the standard setting Remainder of the pap er In Section we present the necessary background In Section we describ e our main technical to ols In particular as an application of the easy witness metho d we show that nontrivial derandomization of AM can be achieved under the uniform complexity assumption that NEXP EXP cf Theorem where the class AM is a probabilistic version of NP see the next section for the denitions In Section we proveseveral results on complexityof NEXP In particular Section contains the pro of of the equivalence NEXP Ppoly NEXP MA In Section we show that every polyn NEXP search problem can be solved in deterministic time if NEXP AM we also prove that if NEXP Ppoly then every language in NEXP has memb ership witnesses of p olynomial circuit complexity from Section for the Section contains several interesting implications of our main result circuit approximation problem and natural pro ofs In Section we establish our downward closure results for ZPP RP BPP and MA We also prove gap theorems for ZPE BPE and MA in particular our gap theorem for ZPE states that Such closure results were also obtained byFortnow and Miltersen
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