Dimension Characterizations of Comlexity Classes Xiaoyang Gu Jack H. Lutz Abstract We use derandomization to show that sequences of ositive sace-dimension in fact, even ositive k-dimension for suitable k have, for many uroses, the full ower of random oracles. For examle, we show that, if S is any binary sequence whose 3-dimension is ositive, then BPP P S and, moreover, every BPP romise roblem is P S -searable. We rove analogous results at higher levels of the olynomial-time hierarchy. The dimension-almost-class of a comlexity class C, denoted by dimalmost-c, is the class consisting of all roblems A such that A C S for all but a Hausdorff dimension 0 set of oracles S. Our results yield several characterizations of comlexity classes, such as BPP = dimalmost-p and AM = dimalmost-np, that refine reviously known results on almost-classes. They also yield results, such as Promise-BPP = almost-p-se = dimalmost-p-se, in which even the almost-class aears to be a new characterization. 1 Introduction Assessing the comutational ower of randomness is one of the most fundamental challenges facing comutational comlexity theory. Concrete questions involving the best algorithms for rimality testing, factoring, etc., are instances of this challenge, as are structural questions concerning BPP, AM, and other randomized comlexity classes. One aroach to studying the ower of a randomized comlexity class C is to address the following question: If C 0 is the nonrandomized version of C, then how weak an assumtion can we lace on an oracle S and still be assured that C C0 S? For examle, how weak an assumtion can we lace on an oracle S and still be assured that BPP P S? For this articular question, it was a result of folklore that BPP P S holds for every oracle S that is algorithmically random in the sense of Martin-Löf [22]; it was shown by Lutz [18] that BPP P S holds for every oracle S that is sace-random; and it was shown by Allender and Strauss [3] that BPP P S holds for every oracle S that is -random, or even random relative to a articular sublinear-time comlexity class. In this aer, we extend this line of inquiry by considering oracles S that have ositive dimension (a comlexity-theoretic analog of classical Hausdorff dimension [11, 8]) with resect to various resource bounds. Secifically, we rove that every oracle S that has ositive 3-dimension (hence every oracle S that has ositive sace-dimension) satisfies BPP P S. Our main theorem is a generalization of this fact that alies to randomized romise classes at various levels of the olynomial-time hierarchy. (Promise roblems were introduced by Grollman and Selman [10]. The randomized romise class Promise-BPP was introduced by Buhrman Deartment of Comuter Science, Iowa State University, Ames, IA 50011, USA. Email: xiaoyang@cs.iastate.edu Deartment of Comuter Science, Iowa State University, Ames, IA 50011, USA. Email: lutz@cs.iastate.edu Research suorted in art by National Science Foundation Grant 0344187. 1
and Fortnow [6] and shown by Fortnow [9] to characterize a strength level of derandomization hyotheses. The randomized romise class Promise-AM was introduced by Moser [25].) For every integer k 0, our main theorem says that, for every oracle S with ositive k+3-dimension, every BP Σ P k romise roblem is ΣP,S k -searable. In articular, if S has ositive 3-dimension, then every BPP romise roblem is P S -searable, and, if S has ositive 4-dimension, then every AM romise roblem is NP S -searable. We use our results to investigate classes of the form dimalmost-c = { A dim H ( { B A / C B } ) = 0 } for various comlexity classes C. It is clear that dimalmost-c is contained in the extensively investigated class almost-c = { A Pr[A / C B ] = 0 }, where the robability is comuted according to the uniform distribution (Lebesgue measure) on the set of all oracles B. We show that dimalmost-σ P k -Se = almost-σp k -Se = Promise-BP ΣP k holds for every integer k 0, where Σ P k -Se is the set of all ΣP k -searable airs of languages. This imlies that dimalmost-p = BPP, refining the roof by Bennett and Gill [5] that almost-p = BPP. Also, for all k 1, dimalmost-σ P k = BP ΣP k, refining the roof by Nisan and Wigderson [26] that almost-σ P k = BP ΣP k. The 1997 derandomization method of Imagliazzo and Wigderson [16] is central to our arguments. 2 Resource-Bounded Dimension and Relativized Circuit Comlexity This section reviews and develos those asects of resource-bounded dimension and its relationshi to relativized circuit-size comlexity that are needed in this aer. It is convenient to use entroy rates as an intermediate ste in this develoment. 2.1 Resource-Bounded Dimension Resource-bounded dimension is an extension of classical Hausdorff dimension that imoses dimension structure on various comlexity classes. There are now several equivalent ways to formulate resource-bounded dimension. Here we sketch the elements of the original formulation that are useful in this aer. We work in the Cantor-sace C of all infinite binary sequences. Definition. ([19]). Let s [0, ). 1. An s-gale is a function d : {0, 1} [0, ) satisfying d(w) = 2 s [d(w0) + d(w1)] for all w {0, 1}. 2
2. An s-gale succeeds on a sequence S C if lim su d(s[0..n 1]) =, where S[0..n 1] denotes the n-bit refix of S. n 3. The success set of an s-gale d is S [d] = {S C d succeeds on S }. The following gale characterization of Hausdorff dimension is the key to resource-bounded dimension. In this aer we will use this characterization in lace of the original definition of Hausdorff dimension [11, 8], which we refrain from reeating here. Theorem 2.1. (Lutz [19]). The Hausdorff dimension of a set X C is dim H (X) = inf {s there is an s-gale d such that X S [d]}. To extend Hausdorff dimension to comlexity classes, we define a resource bound to be one of the following classes of functions. all = {f f : {0, 1} {0, 1} } = { f all f is comutable in n O(1) time } k = ΣP k 1 for k 2 sace = { f all f is comutable in n O(1) sace } Each of these resource bounds is associated with a result class R( ) defined as follows. R(all) = C R() = E = TIME(2 linear ) R( k ) = E k = EΣP k 1 R(sace) = ESPACE = SPACE(2 linear ) A real-valued function f : {0, 1} [0, ) is -comutable if there is a function ˆf : {0, 1} N Q such that ˆf (where the inut (w, r) {0, 1} N is suitably encoded with r in unary) and, for all w {0, 1} and r N, ˆf(w, r) f(w) 2 r. We now define resource-bounded dimension by imosing resource bounds on the gale characterization in Theorem 2.1. Definition. ([19]). Let be a resource bound, and let X C. (We identify each S X with the language whose characteristic sequence is S.) 1. The -dimension of X is (X) = inf {s there is a -comutable s-gale d such that X S [d]}. 2. The dimension of X in R( ) is dim(x R( )) = (X R( )). As shown in [19], these definitions endow the above-mentioned comlexity classes R( ) with dimension structure. In general, and dim(r( ) R( )) = 1. Also, 0 dim(x R( )) (X) 1, = (X) (X), e.g., dim sace (X) 3 (X). It is clear that dim all(x) = dim(x C) = dim H (X). 3
Our main results involve -dimensions of individual sequences S, by which we mean (S) = ({S}). We use the easily verified fact that, if is any of the countable resource bounds above, then dim H ({S (S) = 0}) = 0. (2.1) For more discussion, motivation, examles, and results, see [19, 14, 20, 12, 23]. 2.2 Entroy Rates We use a recent result of Hitchcock and Vinodchandran [15] relating entroy rates to dimension. Entroy rates were studied by Chomsky and Miller [7], Kuich [17], Staiger [27, 28], Hitchcock [12], and others. Definition. The entroy rate of a language A {0, 1} is where A =n = A {0, 1} n. H A = lim su n log A =n, n Definition. Let C be a class of languages, and let X C. The C-entroy rate of X is where H C (X) = inf { H A A C and X A i.o.}, A i.o. = {S C ( n)s[0..n 1] A}. The following result is a routine relativization of Theorem 5.5 of [15]. Theorem 2.2. (Hitchcock and Vinodchandran [15]). For all X C and k Z +, k+2 (X) H Σ P k (X). 2.3 Relativized Circuit-Size Comlexity Definition.1 [29]. For f : {0, 1} n {0, 1} and A {0, 1}, size A (f) is the minimum size of (i.e., number of wires in) an n-inut oracle circuit γ such that γ A comutes f. 2. For x {0, 1} and A {0, 1}, size A (x) = size A (f x ), where f x : {0, 1} log x {0, 1} is defined by { x[i] if 0 i < x f x (w i ) = 0 if i x, w 0,...,w 2 log x 1 lexicograhically enumerate {0, 1} log x, and x[i] is the ith bit of x. Lemma 2.3. For all A, S C, H NP A({S}) lim inf n size A (S[0..n 1])log n. n 4
Proof. Assume that α > β > lim inf n size A (S[0..n 1])log n. n It suffices to show that H NP A({S}) α. Let B be the set of all strings x such that size A (x) < β x log x. By standard circuit-counting arguments (e.g., see [21]), there is a constant c N such that, for all sufficiently large n, if we choose m N with 2 m 1 n < 2 m and write γ = 2 m n, so that β n log n = β γ2 m log(γ2 m ) βγ 2m m 1, then so ) βγ 2 B =n c (4eβγ m 2m m 1, m 1 ) log B =n log c + βγ (4eβγ 2m m 1 log 2m m 1 [ m = log c + βγ2 m log 4eβγ log(m 1) + m 1 m 1 αn, ] whence H B = lim su n log B =n n α. By our choice of β, S B i.o.. Since B NP A, it follows that H NP A({S}) α. Notation. For k N and x {0, 1}, we write size ΣP k (x) = size K k (x), where K k is the canonical Σ P k -comlete language [4]. By Theorem 2.2 and Lemma 2.3, we have the following. Theorem 2.4. For all S C and k N, size ΣP k (S[0..n 1]) (S) lim inf k+3 n n 3 Positive-Dimension Derandomization In order to state our main theorem, we review the notion of searability and give a formulation of Promise-BP-classes that is suitable for our uroses. Definition. Given a class C of languages, an ordered air A = (A +, A ) of (disjoint) languages is C-searable if there exists a language C C such that A + C and A C =. We write C-Se = { (A +, A ) (A +, A ) is C-searable }. 5
Definition. Fix a standard aring function, : {0, 1} {0, 1} {0, 1}. 1. A witness configuration is an ordered air B = (B, g) where B {0, 1} and g : N N. 2. Given a witness configuration B = (B, g), the B-critical event for a string x {0, 1} is the set { } B x = w {0, 1} g( x ) x, w B, interreted as an event in the samle sace {0, 1} g( x ) with the uniform robability measure. (That is, the robability of B x is Pr(B x ) = 2 g( x ) B x.) 3. Given a class C of languages, we define the class Promise-BP C to be the set of all ordered airs A = (A +, A ) of languages for which there is a witness configuration B = (B, q) with the following four roerties. (i) B C. (ii) q is a olynomial. (iii) For all x A +, Pr(B x ) 2 3. (iv) For all x A, Pr(B x ) 1 3. Note that Promise-BP is an oerator that mas a class C of languages to a class Promise-BP C of disjoint airs of languages. In articular, Promise-BP P = Promise-BPP is the class of BPP romise roblems investigated by Buhrman and Fortnow [6] and Moser [24], and Promise-BP NP = Promise-AM is the class of Arthur-Merlin romise roblems investigated by Moser [25]. The following result is the main theorem of this aer. Theorem 3.1. For every S C and k N, (S) > 0 = Promise-BP ΣP k+3 k ΣP,S k -Se. Before roving Theorem 3.1, we derive some of its consequences. First, the cases k = 0 and k = 1 are of articular interest: Corollary 3.2. For every S C, and 3 (S) > 0 = Promise-BPP PS -Se 4 (S) > 0 = Promise-AM NPS -Se. We next note that our results for romise roblems imly the corresonding results for decision roblems. (Note, however, that the results of Fortnow [9] suggest that the results on romise roblems are in some sense stronger.) 6
Corollary 3.3. For every S C and k N, In articular, and k+3 (S) > 0 = BP ΣP k ΣP,S k. 3 (S) > 0 = BPP PS (3.1) 4 (S) > 0 = AM NPS. (3.2) Intuitively, (3.1) says that even an oracle S with 3-dimension 0.001 which need not be random relative to any reasonable distribution contains enough randomness to carry out a deterministic simulation of BPP. To ut the matter differently, to rove that P = BPP, we need only show how to disense with such an oracle S. As in section 1, for each relativizable comlexity class C (of languages or airs of languages), define the dimension-almost-class dimalmost-c = { A dim H ( { S A / C S } ) = 0 }, noting that this is contained in the reviously studied almost-class almost-c = { A Pr[A C S ] = 1 }, where the robability is comuted according to the uniform distribution (Lebesgue measure) on the set of all oracles S. Theorem 3.4. For every k N, dimalmost-σ P k -Se = almost-σp k -Se = Promise-BP ΣP k. Nisan and Wigderson s unconditional seudorandom generator for constant deth circuit is used in the roof for Theorem 3.4. We state it here. Theorem 3.5 (Nisan and Wigderson [26]). Let d Z +. There exists a function G NW : {0, 1} {0, 1} defined by a collection {G n : {0, 1} ln {0, 1} n } such that l n = O((log n) 2d+6 ), G n is comutable by a logsace uniform family of circuits of olynomial size and deth d+4, and for any circuit family {C n : {0, 1} n {0, 1}} of olynomial size and deth d, Pr[C n (x) = 1] Pr[C n (G n (y))] 1/n. Proof of Theorem 3.4. We only rove for k > 0, since when k = 0, the roof is easier. Since every set of Hausdorff dimension less than 1 has Lebesgue measure 0, it is clear that dimalmost-σ P k -Se almost-σ P k -Se. To see that almost-σ P k -Se Promise-BP ΣP k, we use Nisan and Wigderson s roof. Let A = (A +, A ) almost-σ P k -Se. Then by the Lebesgue density theorem, there exists ΣP k oracle machine N with time bound n m such that Pr R [N R searates A] 3/4. Note that when inut x is fixed and x = n, the comutation of N (x) may be reresented as a deth k +2 circuit of size 2 (k+1)nm with 2 (k+1)nm oracle queries as inut. This is a linear size (with resect to oracle inut length) deth k + 2 circuit. We can use Theorem 3.5 to derandomize the exonential number of queries on random oracle to n (2k+10)m random oracle queries. Let G NW be the Nisan-Wigderson seudorandom generator. Let l n = n (2k+10)m. Let N be the following oracle Turing machine. 7
inut x n = x inut s {0, 1} ln let R = G NW (s) simulates N R(x) outut the outut of the simulation For all x A +, Pr R [N R (x) = 1] 3/4. By the seudorandomness of G NW, then, Similarly, for all x A, Let Pr s {0,1} ln[n GNW (s) (x) = 1] 2/3. (3.3) Pr s {0,1} ln[n GNW (s) (x) = 1] 1/3. (3.4) B = { x, s N( x, s ) = 1}. It is clear that B Σ P k. Also by (3.3), for all x A+, and, by (3.4), for all x A, Pr(B x ) 2/3, Pr(B x ) 1/3. Then (B, n (2k+10)m ) is a witness configuration for A, hence A Promise-BP Σ P k. To see that Promise-BP Σ P k dimalmost-σp k -Se, let A Promise-BP ΣP k. Let { } X = S A / Σ PS k -Se. By Theorem 3.1, every element of X has k+3 -dimension 0. By (2.1), this imlies that dim H(X) = 0, whence A dimalmost-σ P k -Se. Corollary 3.6. For every k N, dimalmost-σ P k = BP ΣP k. In articular, and dimalmost-p = BPP (3.5) dimalmost-np = AM. (3.6) We now turn to the roof of Theorem 3.1. We use the following well-known derandomization theorem. Theorem 3.7 (Imagliazzo and Wigderson [16]). For each ɛ > 0, there exists constants c > c > 0 such that, for every A {0, 1} and integer n > 1, the following holds. If f : {0, 1} clog n {0, 1} is a Boolean function that cannot be comuted by an oracle circuit of size at most n cɛ relative to A, 8
then the generator G IW97 f : {0, 1} c log n {0, 1} n has the roerty that, for every oracle circuit γ with size at most n, Pr r Un [γ A (r) = 1] Pr x U c log n [γ A (G IW97 f (x)) = 1] < 1 n, where U m denotes {0, 1} m with the uniform robability measure. Proof of Theorem 3.1. We rove the theorem for k > 0, since when k = 0, the roof is easier. Assume that (S) = α > 0. It suffices to show that for every A Promise-BP ΣP k+3 k, A Σ P,S k -Se. Note that Promise-BP Σ P k does not have oracle access to S. So we actually rove A NP ΣP k 1,S -Se. By Theorem 2.4, we have size ΣP k (S[0..n 1]) > αn 2log n for all but finitely many n. Let A = (A +, A ) Promise-BP Σ k. There exists B ΣP k and olynomial q such that (B, q) is a witness configuration for A. Therefore, there exists olynomial-time oracle Turing machine M and olynomial such that, for all x A +, and, for all x A, Pr r [( w {0, 1} ( x ) )M ΣP k 1 (x, r, w) = 1] 2/3 Pr r [( w {0, 1} ( x ) )M ΣP k 1 (x, r, w) = 1] 1/3. Let n d be the uer bound of the running time of M on x of length n with r and w of corresonding lengths. Let ɛ = α/2 and let c, c be fixed in Theorem 3.7, and let f : {0, 1} cd log n {0, 1} be (the Boolean function whose truth table is) given by the first 2 cd log n bits of S. By Theorem 3.7, G IW97 f derandomizes linear size circuits with Σ P k oracle and linear size nondeterministic circuits with Σ P k 1 oracle. Let NΣP k 1,S be the following nondeterministic Turing machine with oracles Σ P k 1 and S. inut x n = x guess w 1, w 2,...,w 2 c d log n {0, 1} (n) query the first 2 cd log n bits of S Let f : {0, 1} cd log n {0, 1} be given by the first 2 cdlog n bits of S for each string s i {0, 1} c dlog n do Let r i = G IW97 f (s i ) end for Let r = 0 for each r i if M ΣP k 1 (x, ri, w i ) = 1 then r = r + 1 end for r if 1/2 then outut 1 2 c d log n else outut 0. 9
By Theorem 3.7, for all x A +, there exists a witness w 1, w 2,...,w 2 c d log n such that N ΣP k 1,S (x)= 1, and, for all x A, such a witness does not exist. Therefore, the above NP ΣP k 1 machine searates A with oracle S and hence A Σ P,S k -Se. It should be noted that derandomization lays a significantly larger role in the roof of Corollary 3.6 than in the roofs of the analogous results for almost-classes. For examle, the roof by Bennett and Gill [5] that almost-p = BPP uses the easily roven fact that the set X = { S P S BPP S } has Lebesgue measure 0. Hitchcock [13] has recently roven that this set has Hausdorff dimension 1, so the Bennett-Gill argument does not extend to a roof of (3.5). Instead, our roof of (3.5) relies, via (3.1), on Theorem 3.7 to rove that the set Y = { S BPP P S } has Hausdorff dimension 0. Similarly, the roof by Nisan and Wigderson [26] that almost-np AM uses derandomization, but their roof that AM almost-np is elementary. In contrast, both directions of the roof of (3.6) use derandomization: The inclusion dimalmost-np AM relies on the fact that almost-np AM (hence on derandomization), and our roof that AM dimalmost-np relies, via (3.2), on Theorem 3.7. 4 Conclusion We conclude with a brief remark on relativization. Theorem 3.7 relativizes to arbitrary oracles, as to all our arguments here. For examle, imlication (3.1), 3 (S) > 0 = BPP PS, holds relative to every oracle. Note, however, that, if we consider this imlication relative to an oracle S of ositive 3 -dimension, then the relativized 3-dimension of this S will be 0, so we cannot use the relativized imlication to conclude that P S = BPP S. Indeed, by Hitchcock s just-mentioned result and (2.1), there must exist languages S of ositive 3-dimension for which P S BPP S. Acknowledgment. We are grateful to Eric Allender, whose discussions of the ideas in [1, 2] were a significant motivation of our work. We thank an anonymous referee for ointing out a flaw in our earlier roof. We thank John Hitchcock for useful discussions and suggestions. We also thank Manindra Agrawal for a useful discussion. References [1] E. Allender. When worlds collide: Derandomization, lower bounds, and kolmogorov comlexity. In Proceedings of the 21st annual Conference on Foundations of Software Technology and Theoretical Comuter Science, volume 2245 of Lecture Notes in Comuter Science, ages 1 15. Sringer-Verlag, 2001. [2] E. Allender, H. Buhrman, M. Koucký, D. van Melkebeek, and D. Ronneburger. Power from random strings. In Proceedings of the 43rd Annual IEEE Symosium on Foundations of Comuter Science, ages 669 678, 2002. SIAM Journal on Comuting. To aear. [3] E. Allender and M. Strauss. Measure on small comlexity classes with alications for BPP. In Proceedings of the 35th Symosium on Foundations of Comuter Science, ages 807 818, 1994. 10
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