We show that BPP has either SUBEXP-dimension zero (randomness is easy) or
BPP=EXP (randomness is intractable).Comment: to be published in information processing letter

Moser, Philippe

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We show that BPP has either SUBEXP-dimension zero (randomness is easy) or BPP =

EXP (randomness is intractable)

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A zero-one SUBEXP-dimension law for BPP

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arXiv:1101.4848v1  [cs.CC]  25 Jan 2011A zero-one SUBEXP-dimension law for BPPPhilippe Moser ∗AbstractWe show that BPP has either SUBEXP-dimension zero (randomness is easy) or BPP =EXP (randomness is intractable).1 IntroductionLutz’s resource bounded-measure [6] allows to quantify the size of standard complexity classessuch as BPP,NP, . . . where any class can be either small, large or in between, correspondingto having p-measure 0, 1 or being non-measurable (p stands for polynomial-time computablemartingales). It was recently refined by the introduction of effective dimension [7] with (amongothers) the notion of p-dimension. One of the motivations is to quantify the structure of setsthat have resource-bounded measure zero, in the same way as classical Hausdorff dimensioncan be used to quantify many sets of Lebesgue measure zero. Examples include classes oflanguages decidable by (nonuniform) circuits that all have measure zero, but whose dimensiondiffers depending on a parameter of the circuit size (see [3]).Among many applications, resource-bounded measure was used in derandomization, whenvan Melkebeek showed that the standard probabilistic complexity classes ZPP and BPP haveeither p-measure zero or one [11]. Impagliazzo and Moser later proved that RP also satisfiesthe zero-one law [4]. These results ruled out the non-measurable case, thus yielding furtherevidence that if randomness is not intractable (i.e. C 6= EXP, where C ∈ {ZPP,RP,BPP})then randomness is easy (i.e. C has p-measure zero).In this paper we improve van Melkebeek’s zero-one law for BPP [11], by strengtheningthe statement “randomness is easy” in two aspects; first by reducing the time-bounds ofthe martingales from polynomial to subpolynomial and second by replacing measure withdimension.Informally, a class has p-measure zero, if there is a polynomial time computable predictor(a martingale) that can predict infinitely often any language L in the class without makingtoo many mistakes. Quantifying how often a martingale makes correct predictions abouta language L, can be measured by a real number s ∈ [0, 1], where s = 0 means that themartingale predicts L correctly on a large fraction of all strings, whereas s = 1 means thepredictions are correct on a small fraction. The p-dimension of a p-measure zero class C, isthe infimum over all s ∈ [0, 1] for which there is a polynomial time martingale that predictsall languages in C correctly on at least a fraction s of all strings.Lutz’s formulation of resource bounded-dimension [7] (similarly to resource-bounded mea-sure) is a general theory where the time bound of the martingales is a parameter that can∗Department of Computer Science, National University of Ireland, Maynooth Co. Kildare, Ireland. Email:pmoser(at)cs.nuim.ie1be set depending on what classes one tries to measure in. For example choosing polynomialtime bounds yields the notion of p-dimension which is a dimension notion on the class E.Although it is easy to extend the theory by considering time bounds greater than polyno-mial (i.e. measuring in classes containing E), it is not easily extended to smaller time bounds(i.e. subpolynomial bounds that allow to measure in classes like P or SUBEXP for example).It is only recently that a dimension notion on subclasses of E was obtained in [9]. Previousmeasure notions on subclasses of E were introduced prior to [9] (see [1, 10] for measures on P(and above), and [8] for a measure on PSPACE), but they cannot be used to define dimensionbecause the associated martingales cannot place enough bets (see [9] for more details).In this paper we use the dimension on SUBEXP from [9], corresponding to subpolynomialtime computable martingales (instead of polynomial ones for p-dimension on E), to show thateither BPP has dimension zero or BPP = EXP. Our proof uses van Melkebeek’s argument [11]in the dimension on SUBEXP setting introduced in [9]. Because subpolynomial martingalesare weaker than polynomial ones, our result immediately holds for Lutz’s p-dimension.This result seems to be particular to BPP, and it is not obvious whether it carries over toZPP or RP.2 PreliminariesFor ǫ > 0, denote by Eǫ the class Eǫ =⋃δ<ǫDTIME(2nδ ). SUBEXP is the class ∩ǫ>0Eǫ. Let usfix some notations for strings and languages. Let s0, s1, . . . be the standard enumeration ofthe strings in {0, 1}∗ in lexicographical order, where s0 = λ denotes the empty string. Denoteby sn0 , sn1 , · · · , sn2n−1 all strings of size n ordered lexicographically, i.e. snt = s2n+t−1 for any0 ≤ t ≤ 2n − 1. A sequence is an element of {0, 1}∞. If w is a string or a sequence and0 ≤ i < |w| then w[i] and w[si] denotes the ith bit of w. Similarly w[i . . . j] and w[si . . . sj]denote the ith through jth bits. We identify language L with its characteristic function χL,where χL is the sequence such that χL[i] = 1 iff si ∈ L. L ↾ sn stands for L[s0 · · · sn]. If w1is a string and w2 is a string or a sequence extending w1, we write w1 ⊑ w2.The Shannon entropy is the following continuous functionH : [0, 1] → [0, 1]H(α) = α log1α+ (1− α) log11− αwhere H(0) = H(1) := 0.2.1 SUBEXP-dimensionLutz’s resource-bounded measure [6] is obtained by imposing appropriate resource boundson a game-theoretical characterization of classical Lebesgue measure, in which a gamblerplaces bets on the successive membership bits of a hidden language A. The game proceedsin infinitely many rounds where at the end of round n, it is revealed to the gambler whethersn ∈ A or not. The game starts with capital 1. Then, in round n, depending on the first n−1outcomes w = χA[0 . . . n− 1], the gambler bets a certain fraction a ∈ [0, 1] (resp. 1−a) of hiscurrent capital, that the nth word sn ∈ A (resp. sn 6∈ A). The game is fair, i.e. the amountplaced on the correct event is doubled, the one placed on the wrong guess is lost. The playerwins on a language A if he manages to make his capital arbitrarily large during the game.The wins of the player are measured by functions called rate-martingales.2Definition 2.1 [9] A rate-martingale is a function D : {0, 1}∗ → [0, 2] such that for everyw ∈ {0, 1}∗, D(w0) +D(w1) = 2.A rate-martingale outputs the factor by which the capital is increased after the bet.SUBEXP-dimension [9] is defined via rate-martingale families which share their wins.These rate-martingales are computed by Turing machines with random access to their in-put, i.e. machines that have oracle access to their input and can query any bit of it. Toenable such machines to compute the length of their input without reading it, they are givens|w| together with input w; this convention is denoted by Mw(s|w|).Definition 2.2 [9] Let α > 0. An Eα-family of rate-martingales ({Di}i, {Qi}i, ind), is afamily of rate-martingales {Di}i, where• Qi : N→ P({0, 1}∗) are disjoint Eα-printable query sets (i.e. there is a Turing machinethat on input (i, 1n) outputs all strings in Qi(n) in 2nα + ic steps with c ≥ 1), i.e.Qi(n) ∩ Qj(n) = ∅ (for every i 6= j), ∪i≤ 2nnQi(n) = {0, 1}≤n and Qi(m) ⊆ Qi(n) form < n,• ind : {0, 1}∗ → N is a DTIME(2nα)-computable function, such that Di(L ↾ x) is com-putable by a random access Turing machine M in time 2|x|αi.e. ML↾x(x, i) = Di(L ↾ x)where M queries its oracle only on strings in Qi(|x|), and ind(x) is the index such thatx ∈ Qind(x)(|x|).Each rate-martingale Di of the family only bets on strings inside its query set Qi (i.e. Di(x) =1 for x 6∈ Qi(|x|)). The function ind on input a string x, outputs which rate-martingale isto (possibly) bet on x. The idea is that the rate-martingales share their wins, and have theability to divide the bets along all members of the family. We are interested in the totalcapital such a family wins.Definition 2.3 [9] The win function of an Eα-family of rate-martingales ({Di}i, {Qi}i, ind)is the function W{Di}i : {0, 1}∗ → Q, where W{Di}i(L ↾ x) =∏i≤ 2|x||x|∏y≤xDi(L ↾ y).Because Di(L ↾ x) is the factor by which the capital is multiplied after the bet on x,the product in Definition 2.3 is exactly the total capital the whole family of rate-martingaleswould win, would they be able to share their wins after each bet.Lutz’s key idea to define resource-bounded dimension on E [7] is to tax the martingales’wins. The following definition formalizes this tax rate notion (the definition of s-successfor standard martingales is from [2], in [9] the definition was extended to families of ratemartingales).Definition 2.4 [9] Let s ∈ [0, 1], ({Di}i, {Qi}i, ind) be an Eα-family of rate-martingales, andlet L be a language. We say {Di}i s-succeeds on L, if lim supn→∞ 2(s−1)nW{Di}i(L ↾ sn−1) =∞.Similarly D s-succeeds on class C, if D s-succeeds on every language in C.The SUBEXP-dimension of a complexity class is the highest tax rate that can be leviedon the martingales’ wins without preventing them from succeeding on the class.Definition 2.5 [9] Let C be a class of languages. The Eα-dimension of C is defined asdimEα(C) = inf{s ∈ [0,∞) : there is an Eα-family of rate-martingales {Di}i that s-succeedson C}.3Eα-dimension satisfies the following union property.Theorem 2.1 [9] Let 0 < δ < α, {Cj}j be a family of classes, and let {tj}j with tj ∈ [0, 1]such that for every ǫ > 0 there exists an Eδ-family of rate-martingales {Di,j}i,j with identicalquery sets such that {Di,j}i (tj + ǫ)-succeeds on Cj. Then for C =⋃j Cj , dimEα(C) ≤supj{tj}.2.2 Derandomization of BPPWe need the following special case of a derandomization result for BPP under a uniformhypothesis from [5].Theorem 2.2 [5] If BPP 6= EXP then for every language A ∈ BPP and every δ > 0, thereexists a language B ∈ DTIME(2nδ), such that for infinitely many lengths mPr|x|=m[A(x) = B(x)] > 1−1m. (1)3 BPP satisfies the zero-one law for dimension in SUBEXPThe following result shows with respect to SUBEXP gamblers, randomness is either weak orintractable.Theorem 3.1 If BPP 6= EXP then for every α > 0, BPP has Eα-dimension 0.Proof. We need the following lemma.Lemma 3.1 Let C be closed under polynomial many-one reductions and such that for everyδ > 0 and for every language A ∈ C there exists a language B ∈ DTIME(2nδ), such that forinfinitely many lengths mPr|x|=m[A(x) = B(x)] > 1−1m. (2)Then for every α > 0, C has Eα-dimension 0.Let us prove the lemma. Let α > 0, ǫ > 0. Let us show that dimEα(C) < ǫ. Let δ < α′ < αand ǫ′ < ǫ (to be determined later).Let B ∈ DTIME(2⌊nδ⌋). Consider the following family of rate-martingales({DB,j}j∈N, {Qj}j∈N, ind) where the query sets are obtained by partitioning {0, 1}k into2k−⌊kδ⌋ query sets, i.e. letT kj := {skj2⌊kδ⌋, . . . , sk(j+1)2⌊kδ⌋−1} and Qj(n) = ∪nk=1Tkj (for all integers n)with the convention that Qj(n) is empty when j ≥ 2n−⌊nδ⌋. The corresponding index functionind is obvious.DB,j on Tnj bets according to the following list of “strategies” {pi}i∈Z+ , where pi onlybets on the strings s⌊ǫ′2i⌋−1, . . . , s2i−2 and bets that the corresponding membership bits aregiven by the sequence B[si⌊ǫ′2i⌋, . . . , si2i−1]. Thus on Tnj the only pi’s that bet are the ones forwhichzn := log(2n + j2⌊nδ⌋ + 2) ≤ i ≤ − log(ǫ′) + log(2n − 2 + (j + 1)2⌊nδ⌋) =: z′n.4DB,j upon starting betting on the first string in Tnj , divides its current capital c (c is unknownbut will cancel out in the final equation) into shares {c2−i}i∈Z+ where c2−i is devoted to pi.Hence after betting on the first string, the new capital c′ is given byc′ =∑i≥1c2−iD′i(L ↾ snj2⌊nδ⌋)= c[zn−1∑i=12−i +∑i≥z′n+12−i +z′n∑i=zn2−iD′i(L ↾ snj2⌊nδ⌋)]= c[1−z′n∑i=zn2−i +z′n∑i=zn2−iD′i(L ↾ snj2⌊nδ⌋)]whereD′i(L ↾ st) =2(1 − 2ǫ′) if ⌊ǫ′2i⌋ − 1 ≤ t ≤ 2i − 2 and L(st) = B(sit+1)4ǫ′ if ⌊ǫ′2i⌋ − 1 ≤ t ≤ 2i − 2 and L(st) 6= B(sit+1)1 otherwise (no bet).ThusDB,j(L ↾ snj2⌊nδ⌋) =c′c= 1−z′n∑i=zn2−i +z′n∑i=zn2−iD′i(L ↾ snj2⌊nδ⌋)and for any st ∈ TnjDB,j(L ↾ st) =1−∑z′ni=zn2−i +∑z′ni=zn2−i∏y∈Tnj ∩{s0,...,st}D′i(L ↾ y)1−∑z′ni=zn2−i +∑z′ni=zn2−i∏y∈Tnj ∩{s0,...,st−1}D′i(L ↾ y).Since both z′n < 2n (for n big enough), and D′i(L ↾ st) (with st ∈ Tnj ) requires com-puting B on strings of size at most 2n, DB,j(L ↾ st) can be computed in time less thanan2(2n)δ2nδ(for some constant a), which is smaller than 2nα′by an appropriate choice of δ.Thus ({DB,j}j∈N, {Qj}j∈N, ind) is a 2nα′-family of rate-martingales.Let us show that this family ǫ′-succeeds on C. Let A ∈ C and let D be the languagewhere for every length kD[sk1 . . . sk2k−1] = A[s0 . . . s2k−2]and let B ∈ DTIME(2⌊nδ⌋) be the language predicting D on infinitely many length i, i.e. thereexists an infinite set M such that for every i ∈M|{x : |x| = i and B(x) = D(x)}| ≥ (1−1i)2i.Let m ∈ M . By construction of {DB,j}j , the m-th strategy D′m of {DB,j} bets correctlyon at least 2m(1 − 2ǫ′) bits of A[s⌊ǫ′2m⌋−1 . . . s2m−2] (for m big enough), therefore because5Qj(m− 1) = ∅ whenever j > 2m−12−⌊m−1⌋δ− 1 =: vm we haveW{DB,j}j (A ↾ s2m−2) =vm∏j=1∏y≤s2m−2DB,j(A ↾ y)=vm∏j=1m−1∏k=1∏y∈T kjDB,j(A ↾ y)=vm∏j=1m−1∏k=1[1−z′k∑i=zk2−i +z′k∑i=zk2−i∏y∈T kjD′i(L ↾ y)]≥vm∏j=1m−1∏k=1[2−m∏y∈T kjD′m(L ↾ y)]=vm∏j=12−m(m−1)∏y∈Qj(m−1)D′m(L ↾ y)≥ 2−m(m−1)vm [2(1 − 2ǫ′)](1−2ǫ′)2m [4ǫ′](2ǫ′)2m= 2−m(m−1)vm2(1−H(2ǫ′))2m .This implies that {DB,j}j ǫ-succeeds on A, because for every m ∈MW{DB,j}j (A ↾ s2m−2)2(1−ǫ)(2m−3)≥ 2−m(m−1)vm+[ǫ−H(2ǫ′)]2m≥ 2[−m(m−1)2⌊(m−1)δ⌋+1+ǫ−H(2ǫ′)]2mwhich grows unbounded by choosing ǫ′ small enough such that − m(m−1)2⌊(m−1)δ⌋+1+ ǫ−H(2ǫ′) > 0(for m large enough).Let {Bi}i be an enumeration of DTIME(2nδ ), and let Di,j := DBi,j be the rate-martingaledefined as above. By construction, the query sets do not depend on language Bi, hence{Di,j}i,j is an Eα′-family of rate-martingales, such that for every ǫ > 0, {Di,j}j ǫ-succeedson Xi = {A ∈ C| ∃∞m : Pr|x|=m[A(x) = Bi(x)] > 1 −1m}. By Theorem 2.1, X = ∪iXihas Eα-dimension 0 (by choosing α′ < α accordingly). Since C ⊆ X, this ends the proof ofLemma 3.1. Since Theorem 2.2 implies the hypothesis of Lemma 3.1 with C = BPP, theproof is complete. ⊓⊔Final RemarkIt is not known whether a similar result holds for ZPP or RP. The martingales used in theproof of the zero-one law for RP from [4] do not bet often enough to yield a dimension result.It would be interesting to see whether any dimension type result can be obtained for thesetwo classes.6AcknowledgmentI thank John Hitchcock for commenting an earlier draft of this paper, and pointing out animprovement from dimension 1/2 to 0.References[1] Eric Allender and Martin Strauss. Measure on small complexity classes, with applicationsfor BPP. Proc. of the 35th Ann. IEEE Symposium on Foundations of Computer Science,pages 807–818, 1994.[2] Klaus Ambos-Spies, Wolfgang Merkle, Jan Reimann, and Frank Stephan. Hausdorffdimension in exponential time. In IEEE Conference on Computational Complexity, pages210–217, 2001.[3] John M. Hitchcock, Jack H. Lutz, and Elvira Mayordomo. The fractal geometry ofcomplexity classes. Complexity Theory Column (L.A. Hemaspaandra, ed.), SIGACTNews, 36:24–38, 2005.[4] Russell Impagliazzo and Philippe Moser. A zero-one law for RP and derandomization ofAM if NP is not small. Information and Computation, 207(7):787–792, 2009.[5] Russell Impagliazzo and Avi Wigderson. Randomness vs time: Derandomization undera uniform assumption. Journal of Computer and System Sciences, 63(4):672–688, 2001.[6] Jack H. Lutz. Almost everywhere high nonuniform complexity. Journal of Computerand System Science, 44:220–258, 1992.[7] Jack H. Lutz. Dimension in complexity classes. SIAM Journal on Computing, 32(5):1236–1259, 2003.[8] Elvira Mayordomo. Measuring in PSPACE. Proceedings of the 7th International Meetingof Young Computer Scientists (IMYCS’92). Gordon-Breach Topics in Computer Science6, 136:93–100, 1994.[9] Philippe Moser. Martingale families and dimension in P. Theoretical Computer Science,400(1-3):46–61, 2008.[10] M. Strauss. Measure on P: strength of the notion. Information and Computation,136:1:1–23, 1997.[11] Dieter van Melkebeek. The zero-one law holds for BPP. Theoretical Computer Science,244(1-2):283–288, 2000.7

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A zero-one SUBEXP-dimension law for BPP

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