complexity classes. Investigations of this structure y a numer of researchers have yielded many new insights over the past seven years. The recent sur

Transcription

1 Bias Invariance of Small Upper Spans 1 Jack H.Lutz Department of Computer Science Iowa State University Ames, Iowa U.S.A. Martin Strauss AT&T Las 180 Park Ave., P.O. Box 971 Florham Park, NJ U.S.A. Astract The resource-ounded measures of certain classes of languages are shown to e invariant under certain changes in the underlying proaility measure. Specically, for any real numer >0, any polynomial-time computale sequence ~ =( 0 1 :::)of iases i 2 [ 1 ; ], and any class C of languages that is closed upwards or downwards under positive, polynomial-time truth-tale resuctions with linear ounds on numer and length of queries, it is shown that the following two conditions are equivalent. (1) C has p-measure 0 relative to the proaility measure given y ~. (2) C has p-measure 0 relative to the uniform proaility measure. The analogous equivalences are estalished for measure in E and measure in E 2. (Breutzmann and Lutz [5] estalished this invariance for classes C that are closed downwards under slightly more powerful reductions, ut nothing was known aout invariance for classes that are closed upwards.) The proof introduces two new techniques, namely, the contraction of a martingale for one proaility measure to a martingale for an induced proaility measure, and a new, improved positive ias reduction of one ias sequence to another. Consequences for the BPP versus E prolem and small span theorems are derived. 1 Introduction Until recently, all research on the measure-theoretic structure of complexity classes has een restricted to the uniform proaility measure. This is the proaility measure that intuitively corresponds to a random experiment in which alanguage A f0 1g is chosen proailistically, using an independent toss of a fair coin to decide whether each string is in A. When eectivized y the methods of resource-ounded measure [15], induces measure-theoretic structure on E = DTIME(2 linear ), E 2 = DTIME(2 polynomial ), and other 1 This research was supported in part y National Science Foundation Grants CCR (with matching funds from Rockwell, Microware Systems Corporation, and Amoco Foundation) and CCR

2 complexity classes. Investigations of this structure y a numer of researchers have yielded many new insights over the past seven years. The recent surveys [3, 16, 6] descrie much of this work. There are several reasons for extending our investigation of resource-ounded measure to a wider variety of proaility measures. First, such variety is essential in cryptography, computational learning, algorithmic information theory, average-case complexity, and other potential application areas. Second, applications of the proailistic method [2] often require use of non-uniform proaility measures, and this is likely to hold for the resourceounded proailistic method [18, 16] as well. Third, resource-ounded measure ased on non-uniform proaility measures provides new methods for proving results aout resourceounded measure ased on the uniform proaility measure [5]. Motivated y such considerations, Breutzmann and Lutz [5] initiated the study of resource-ounded measure ased on an aritrary (Borel) proaility measure on the Cantor space C (the set of all languages). (Precise denitions of these and other terms appear in Appendix A.) Kautz [13] and Lutz [17] have furthered this study in dierent directions, and the present paper is another contriution. The principal focus of the paper [5] is the circumstances under which the -measure of a complexity class C is invariant when the proaility measure is replaced y some other proaility measure 0. For an aritrary class C of languages, such invariance can only occur if and 0 are fairly close to one another: Extending results of Kakutani [12], Vovk [24], and Breutzmann and Lutz [5], Kautz [13] has shown that the \square-summale equivalence" of and 0 is sucient to ensure p (C) = 0 () 0 p(c) = 0, ut very little more can e said when C is aritrary. Fortunately, complexity classes have more structure than aritrary classes. Most complexity classes of interest, including P, NP, conp, R, BPP, AM, P/Poly, PH, etc., are closed downwards under positive, polynomial-time truth-tale reductions ( P pos;tt-reductions), and their intersections with E are closed downward under P pos;tt-reductions with linear ounds on the length of queries ( P lin pos;tt-reductions). Breutzmann and Lutz [5] proved that every class C with these closure properties enjoys a sustantial amount ofinvariance in its measure. Specically, ifc is any such class and ~ and ~ 0 are strongly positive, P-sequences of iases, then the equivalences ~ p (C) =0 () ~ 0 p (C) =0 ~ (CjE) = 0 () ~ 0 (CjE) = 0 (1) ~ (CjE 2 )=0 () ~ 0 (CjE 2 )=0 2

3 hold, where ~ and ~ 0 are the proaility measures corresponding to the ias sequences ~ and ~ 0, respectively. Our primary concern in the present paper is to extend this ias invariance to classes that are closed upwards under some type P r of polynomial reductions. We have two reasons for interest in this question. First and foremost, many recent investigations in complexity theory focus on the resource-ounded measure of the upper P r -span P ;1 r (A) =fbja P r Bg of a language A. Such investigations include work on small span theorems [9, 14, 4, 11, 7] and work on the BPP versus E question [1, 7, 8]. In general, the upper P r -span of a language is closed upwards, ut not downwards, under P r -reductions. Our second reason for interest in upward closure conditions is that the aove-mentioned results of Breutzmann and Lutz [5] do not fully estalish the invariance of measures of complexity classes under the indicated changes of ias sequences. For example, if ~ is an aritrary strongly positive P-sequence of iases, the results of [5] show that ut they do not show that ~ (CjE) = 0 () (CjE) = 0 ~ (CjE) = 1 () (CjE) = 1: In general, the condition (CjE) = 1 is equivalentto(c c je) = 0, where C c is the complement of C. Since C is closed downwards under P r -reductions if and only if C c is closed upwards under P r -reductions, we are again led to consider upward closure conditions. Our main theorem, the Bias Invariance Theorem, states that, if C is any class of languages that is closed upwards or downwards under positive, polynomial-time, truth-tale reductions with linear ounds on numer and length of queries ( P lin pos;lin;tt -reductions), and if ~ and ~ 0 are strongly positive P-sequences of iases, then the equivalences (1) aove hold. The proof introduces two new techniques, namely, the contraction of a martingale for one proaility measure to a martingale for an induced proaility measure (dual to the martingale dilation technique introduced in [5]) and a new, improved positive ias reduction of one ias sequence to another. We also note three easy consequences of our Bias Invariance Theorem. First, in comination with work of Allender and Strauss [1] and Buhrman, van Melkeeek, Regan, Sivakumar, and Strauss [8], it implies that, if there is any strongly positive P-sequence of iases ~ such 3

4 that the complete P T -degree for E 2 does not have ~ -measure 1 in E 2, then E 6 BPP. Second, in comination with the work of Regan, Sivakumar, and Cai [19], it implies that, for any reasonale complexity class C, if there exists a strongly positive P-sequence of iases ~ such thatc has ~ -measure 1 in E, then E C( and similarly for E 2 ). Third, if P r is any polynomial reduciility such that A P lin pos;lin;tt B implies A P r B, and if ~ is a strongly positive P-sequence of iases, then the small span theorem for P r -reductions holds with respect to ~ if and only if it holds with respect to. Tantalizingly, thishypothesis places P r \just eyond" the small span theorem of Buhrman and van Melkeeek [7], which is the strongest small span theorem proven to date for exponential time. 2 Preliminaries We writef0 1g for the set of all (nite, inary) strings, and we write jxj for the length of a string x. The empty string,, is the unique string of length 0. The standard enumeration of f0 1g is the sequence s 0 = s 1 = 0 s 2 = 1 s 3 = 00 :::, ordered rst y length and then lexicographically. For x y 2f0 1g, we write x<y if x precedes y in this standard enumeration. For n 2 N, f0 1g n denotes the set of all strings of length n, and f0 1g n denotes the set of all strings of length at most n. If x is a string or an (innite, inary) sequence, and if 0 i j<jxj, thenx[i::j] is the string consisting of the i th through j th its of x. In particular, x[0::i ; 1] is the i-it prex of x. We write x[i] for x[i::i], the i th it of x. (Note that the leftmost it of x is x[0], the 0 th it of x.) If w is a string and x is a string or sequence, then we write w v x if w is a prex of x, i.e., if there is a string or sequence y such thatx = wy. The Boolean value of a condition is [[]] =if then 1 else 0. We work in the Cantor space C, consisting of all languages A f0 1g. We identify each language A with its characteristic sequence, which istheinnite inary sequence A dened y A [n] =[[s n 2 A] for each n 2 N. Relying on this identication, we also consider C to e the set of all innite inary sequences. The complement of a set X of languages is X c = C ; X. 4

5 For each string w 2f0 1g, the cylinder generated y w is the set C w = fa 2 C j w v A g : 3 Martingale Contraction Given a positive coin-toss proaility measure, an orderly truth-tale reduction (f g), and a (f g) -martingale d (where (f g) is the proaility measure induced y and (f g)), Breutzmann and Lutz [5] showed how to construct a -martingale (f g)d, called the (f g)- dilation of d, such that (f g)d succeeds on A whenever d succeeds on F (f g) (A). (See [5] or Appendix B for notation and terminology involving truth-tale reductions.) In this section we present a dual of this construction. Given and (f g) as aove and a -martingale d, weshowhow to construct a (f g) -supermartingale (f g) d, called the (f g)-contraction of d, such that (f g) d succeeds on A whenever d succeeds strongly on every element of F ;1 (fag). (f g) The notion of an (f g)-step, introduced in [5], will also e useful here. Denition. Let (f g) e an orderly tt -reduction. 1. An (f g)-step is a positive integer l such thatf (f g) (0 l;1 ) 6= F (f g) (0 l ). 2. For k 2 N, we let step(k) etheleast(f g)-step l such that l k. 3. For v w 2 f0 1g, we write v w to indicate that w v v and jvj = step(jwj + 1). (That is, v w means that v is a proper extension of w to the next step.) Our construction makes use of a special-purpose inverse of F (f g) that depends on oth (f g) and d. Denition. Let (f g) e an orderly tt -reduction, let e a positive proaility measure on C, and let d e a -martingale. Then the partial function is dened recursively as follows. F ;1 (f g) d : f0 1g ;!f0 1g 5

6 (i) F ;1 () =. (f g) d (ii) For w 2 f0 1g and 2 f0 1g, F ;1 (w) is the lexicographically rst string v (f g) d F ;1 (w) such that (f g) d F (f g)(v) =w and, for all v 0 F ;1 (w) such (f g) d thatf (f g)(v 0 )= w, we have d(v) d(v 0 ). (That is, v minimizes d(v) on the set of all v F ;1 (w) (f g) d satisfying F (f g) (v) =w.) Note that the function F ;1 is strictly monotone (i.e., (f g) d 6= w0 implies that F ;1 (f g) d 6= F ;1 (f g) d (w0 ), provided that these values exist), whence it extends naturally to a partial function F ;1 : C ;! C: (f g) d It is easily veried that F ;1 (f g) d inverts F (f g) in the sense that, for all x 2f0 1g [C, F ;1 (f g) d nds a preimage of F (f g) (x), i.e., F (f g) (F ;1 (f g) d (F (f g)(x))) = F (f g) (x): We nowdenethe(f g)-contraction of a -martingale d. Denition. Let (f g) e an orderly tt -reduction, let e a positive proaility measure on C, and let d e a -martingale. Then the (f g)-contraction of d is the function dened as follows. (f g) d : f0 1g ;!f0 1g (i) (f g) d() =d(): (ii) For w 2f0 1g and 2f0 1g, (f g) d(w) = ( d(f ;1 ;1 (w)) if d(f (w)) is dened (f g) d (f g) d 2 (f g) d(w) otherwise. Theorem 3.1 (Martingale Contraction Theorem). Assume that is a positive proaility measure on C, (f g) is an orderly tt -reduction, and d is a -martingale. Then (f g) d is a (f g) -supermartingale. Moreover, for every language A f0 1g,ifF ;1 (fag) (f g) S1 str[d], then A 2 S 1 [(f g) d]. 6

7 4 Bias Invariance In this section we present our main results. Denition. Let (f g) ea tt -reduction. 1. (f g) is positive (riey, a pos;tt -reduction) if, for all A B f0 1g, A B impliesf (f g) (A) F (f g) (B). 2. (f g) is polynomial-time computale (riey, a P tt-reduction) if the functions f and g are computale in polynomial time. 3. (f g) is polynomial-time computale with linear-ounded queries (riey, a P lin tt - reduction) if (f g) is a P tt-reduction and there is a constant c 2 N such that, for all x 2f0 1g, Q (f g) (x) f0 1g c(1+jxj). 4. (f g)ispolynomial-time computale with a linear numer of queries (riey, a P lin;tt - reduction) if (f g) is a P tt-reduction and there is a constant c 2 N such that,for all x 2f0 1g, jq (f g) (x)j c(1 + jxj). Of course, a P lin pos;tt -reduction is a tt-reduction with properties 1{3, and a P lin pos;lin;tt - reduction is a tt -reduction with properties 1{4. We now present thepositive Bias Reduction Theorem. This strengthens the identicallynamed result of Breutzmann and Lutz [5] y giving a P lin pos;lin;tt-reduction in place of a P lin pos;tt-reduction. This technical improvement, which is essential for our purposes here, requires a sustantially dierent construction. Details appear in Appendix D. Theorem 4.1 (Positive Bias Reduction Theorem). Let ~ and ~ 0 e strongly positive, P- exact sequences of iases, and let (f g) e the reduction dened in Appendix D. Then (f g) is an orderly P lin pos;lin;tt -reduction, and the proaility measure induced y ~ and (f g) is a coin-toss proaility measure ~ 00, where ~ 00 t ~ 0. The following result is our main theorem. 7

8 Theorem 4.2 (Bias Invariance Theorem). Assume that ~ and ~ 0 are strongly positive P- sequences of iases, and let C e a class of languages that is closed upwards or downwards under P lin pos;lin;tt-reductions. Then ~ p(c) =0() ~ 0 p (C) =0: The \downwards" part of Theorem 4.2 is a technical improvement of the Bias Equivalence Theorem of [5] from P lin pos;tt-reductions to P lin pos;lin;tt-reductions. The proof of this improvement is simply the proof in [5] with Theorem 4.1 used in place of its predecessor in [5]. The \upwards" part of Theorem 4.2 is entirely new. The proof of this result is similar to the proof of the Bias Equivalence Theorem in [5], ut now in addition to using our improved Positive Bias Reduction Theorem, we use the Martingale Contraction Theorem of section 3 in place of the Martingale Dilation Theorem of [5]. We also note that the linear ound on numer of queries in Theorem 4.1 is essential for the \upwards" direction. If P r is a polynomial reduciility, then a class C is closed upwards under P r -reductions if and only if C c is closed downwards under P r -reductions. We thus have the following immediate consequence of Theorem 4.2. Corollary 4.3. Assume that ~ and ~ 0 are strongly positive P-sequences of iases, and let C e a class of languages that is closed upwards or downwards under P lin -reductions. pos;lin;tt Then p(c) ~ =1() ~ 0 p (C) =1: We now mention some consequences of Theorem 4.2, eginning with a discussion of the measure of the complete P T-degree for exponential time, and its consequences for the BPP versus E prolem. For each class D of languages, we use the notations H T (D) = faja is P T-hard for Dg C T (D) = faja is P T-complete for Dg and similarly for other reduciilities. The following easy oservation shows that every consequence of (C T (E 2 )je 2 ) 6= 1 is also a consequence of (C T (E)jE) 6= 1. 8

9 Lemma 4.4. (C T (E)jE) 6= 1=) (C T (E 2 )je 2 ) 6= 1. Proof. Juedes and Lutz [10] have shown that, if X is a set of languages that is closed downwards under P m-reductions, then (XjE 2 )=0=) (XjE) = 0. Applying this result with X = H T (E) c = H T (E 2 ) c yields the lemma. Allender and Strauss [1] have proven that p (H T (BPP)) = 1. Buhrman, van Melkeeek, Regan, Sivakumar, and Strauss [8] have noted that this implies that (C T (E 2 )je 2 ) 6= 1=) E 6 BPP. Comining this argument with Corollary 4.3 yields the following extension. Corollary 4.5. If there exists a strongly positive P-sequence of iases ~ such that ~ (C T (E 2 )je 2 ) 6= 1,thenE6 BPP. Regan, Sivakumar, and Cai [19] have proven a \most is all" lemma, stating that if C is any class of languages that is either closed under nite unions and intersections or closed under symmetric dierence, then (CjE) = 1 =) E C. Comining this with Corollary 4.3 gives the following extended \most is all" result. Corollary 4.6. Let C e a class of languages that is closed upwards or downwards under P lin pos;lin;tt-reductions, and is also closed under either nite unions and intersections or symmetric dierence. If there is any strongly positive, P-sequence of iases ~ such that ~ (CjE) = 1, then E C. Of course, the analagous result holds for E 2. We conclude with a rief discussion of small span theorems. Given a polynomial reduciility P r, the lower P r -span of a language A is and the upper P r -span of A is We will use the following compact notation. P r (A) =fbjb P r Ag P ;1 r (A) =fbja P r Bg: 9

10 Denition. Let P r e a polynomial reduciility type, and let e a proaility measure on C. Then the small span theorem for P r -reductions in the class E over the proaility measure is the assertion SST ( P r E) stating that, for every A 2 E, (P r (A)jE) = 0 or p (P ;1 r (A)) = (P ;1 r (A)jE) = 0. When the proaility measure is, we omit it from the notation, writing SST( P r E) for SST ( P r E). Similar assertions for other classes, e.g., SST ( P r E 2), are dened in the now-ovious manner. Juedes and Lutz [9] proved the rst small span theorems, SST( P m E) and SST(P m E 2), and noted that extending either to P T would estalish E 6 BPP. Lindner [14] estalished SST( P 1;tt E) and SST(P 1;tt E 2), and Amos-Spies, Neis, and Terwijn [4] proved SST( P k;tt E) and SST(P k;tt E 2) for all xed k 2 N. Very recently, Buhrman and van Melkeeek [7] have taken a major step forward y proving SST( P g(n);tt E 2)forevery function g(n) satisfying g(n) =n o(1). We note that the Bias Invariance Theorem implies that small span theorems lying \just eyond" this latter result are somewhat roust with respect to changes of iases. Theorem 4.7. If P r is a polynomial reduciilitysuch that A P lin pos;lin;tt B implies A P r B, then for every strongly positive P-sequence of iases ~, and similarly for E 2. SST ~ ( P r E) () SST(P r E) Acknowledgment. The rst author thanks Steve Kautz for a very useful discussion. References [1] E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 807{818, Piscataway, NJ, IEEE Computer Society Press. [2] N. Alon and J. H. Spencer. The Proailistic Method. Wiley,

11 [3] K. Amos-Spies and E. Mayordomo. Resource-ounded measure and randomness. In A. Sori, editor, Complexity, Logic and Recursion Theory, Lecture Notes in Pure and Applied Mathematics, pages 1{47. Marcel Dekker, New York, N.Y., [4] K. Amos-Spies, H.-C. Neis, and S. A. Terwijn. Genericity and measure for exponential time. Theoretical Computer Science, 168:3{19, [5] J. M. Breutzmann and J. H. Lutz. Equivalence of measures of complexity classes. SIAM Journal on Computing. To appear. See also Proceedings of the 14th Symposium on Theoretical Aspects of Computer Science, Springer-Verlag, 1997, pp. 535{545. [6] H. Buhrman and L. Torenvliet. Complete sets and structure in surecursive classes. In Proceedings of Logic Colloquium '96, pages 45{78. Springer-Verlag, [7] H. Buhrman and D. van Melkeeek. Hard sets are hard to nd. In Proceedings of the 13th IEEE Conference on Computational Complexity, pages 170{181, New York, IEEE. [8] H. Buhrman, D. van Melkeeek, K. Regan, D. Sivakumar, and M. Strauss. A generalization of resource-ounded measure, with an application. In Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science, pages 161{171, Berlin, Springer-Verlag. [9] D. W. Juedes and J. H. Lutz. The complexity and distriution of hard prolems. SIAM Journal on Computing, 24(2):279{295, [10] D. W. Juedes and J. H. Lutz. Weak completeness in E and E 2. Theoretical Computer Science, 143:149{158, [11] D. W. Juedes and J. H. Lutz. Completeness and weak completeness under polynomialsize circuits. Information and Computation, 125:13{31, [12] S. Kakutani. On the equivalence of innite product measures. Annals of Mathematics, 49:214{224, [13] S. M. Kautz. Resource-ounded randomness and compressiility with repsect to nonuniform measures. In Proceedings of the International Workshop on Randomization and Approximation Techniques in Computer Science, pages 197{211. Springer-Verlag, [14] W. Lindner. On the polynomial time ounded measure of one-truth-tale degrees and p-selectivity, Diplomareit, Technische Universitat Berlin. [15] J. H. Lutz. Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences, 44:220{258,

12 [16] J. H. Lutz. The quantitative structure of exponential time. In L.A. Hemaspaandra and A.L. Selman, editors, Complexity Theory Retrospective II, pages 225{254. Springer- Verlag, [17] J. H. Lutz. Resource-ounded measure. In Proceedings of the 13th IEEE Conference on Computational Complexity, pages 236{248, New York, IEEE. [18] J. H. Lutz and E. Mayordomo. Measure, stochasticity, and the density of hard languages. SIAM Journal on Computing, 23:762{779, [19] K. W. Regan, D. Sivakumar, and J. Cai. Pseudorandom generators, measure theory, and natural proofs. In 36th IEEE Symposium on Foundations of Computer Science, pages 26{35. IEEE Computer Society Press, [20] C. P. Schnorr. Klassikation der Zufallsgesetze nach Komplexitat und Ordnung. Z. Wahrscheinlichkeitstheorie verw. Ge., 16:1{21, [21] C. P. Schnorr. A unied approach to the denition of random sequences. Mathematical Systems Theory, 5:246{258, [22] C. P. Schnorr. Zufalligkeit und Wahrscheinlichkeit. Lecture Notes in Mathematics, 218, [23] C. P. Schnorr. Process complexity and eective random tests. Journal of Computer and System Sciences, 7:376{388, [24] V. G. Vovk. On a randomness criterion. Soviet Mathematics Doklady, 35:656{660,

13 OPTIONAL TECHNICAL APPENDICES

14 Appendix A. Resource-Bounded -Measure In this appendix, we present the asic elements of resource-ounded measure ased on an aritrary proaility measure on C. The material in Appendices A and B is taken, with permission, from [5]. Denition. A proaility measure on C is a function : f0 1g ;! [0 1] such that() = 1, and for all w 2f0 1g, (w) =(w0) + (w1): Intuitively, (w) is the proaility that A 2 C w when we \choose a language A 2 C according to the proaility measure." We sometimes write (C w )for(w). Examples. 1. The uniform proaility measure is dened y for all w 2f0 1g. (w) =2 ;jwj 2. A sequence of iases is a sequence ~ =( :::), where each i 2 [0 1]. Given a sequence of iases ~, the ~ -coin-toss proaility measure (also called the ~ -product proaility measure) is the proaility measure ~ dened y for all w 2f0 1g. ~ (w) = jwj;1 Y i=0 ((1 ; i ) (1 ; w[i]) + i w[i]) Intuitively, ~ (w) is the proaility that w v A when the language A f0 1g is chosen proailistically according to the following random experiment. For each string s i in the standard enumeration s 0 s 1 s 2 ::: of f0 1g,we (independently of all other strings) toss a A-1

15 special coin, whose proaility is i of coming up heads, in which case s i 2 A, and 1 ; i of coming up tails, in which case s i 62 A. Denition. A proaility measure on C is positive if, for all w 2f0 1g, (w) > 0. Denition. If is a positive proaility measure and u v 2f0 1g, then the conditional -measure of u given v is 8 >< 1 if u v v (u) (ujv) = if v v u (v) >: 0 otherwise. Note that (ujv) is the conditional proaility that A 2 C u, given that A 2 C v, when A 2 C is chosen according to the proaility measure. Denition. A proaility measure on C is strongly positive if ( is positive and) there is aconstant >0 such that,for all w 2f0 1g and 2f0 1g, (wjw). Denition. A sequence of iases ~ =( :::)isstrongly positive if there is a constant >0such that, for all i 2 N, i 2 [ 1 ; ]. We next review the well-known notion of a martingale over a proaility measure. Computale martingales were used y Schnorr [20, 21, 22, 23] in his investigations of randomness, and have more recently een used y Lutz [15] in the development of resourceounded measure. Denition. Let e a proaility measure on C. Then a -martingale is a function d : f0 1g ;! [0 1) such that, for all w 2f0 1g, d(w)(w) =d(w0)(w0) + d(w1)(w1): If ~ is a sequence of iases, then a ~ -martingale is simply called a ~ -martingale. A - martingale is even more simply called a martingale. (That is, when the proaility measure is not specied, it is assumed to e the uniform proaility measure.) A-2

16 Denition. A -martingale d succeeds on a language A 2 C if The success set of a -martingale d is the set lim sup d( A [0::n ; 1]) = 1: n;!1 S 1 [d] =fa 2 C j d succeeds on Ag : The strong success set of a -martingale d is the set S 1 str[d] = A 2 C j lim sup d(a[0::n ; 1]) = 1 n!1 : Denition. Let e a proaility measure on C. 1. A p--martingale is a -martingale that is p-computale. 2. Ap 2 --martingale is a -martingale that is p 2 -computale. Ap- ~ -martingale is called a p- ~ -martingale, a p--martingale is called a p-martingale, and similarly for p 2. We now come to the fundamental ideas of resource-ounded -measure. Denition. Let e a proaility measure on C, and let X C. 1. X has p--measure 0, and we write p (X) = 0, if there is a p--martingale d such that X S 1 [d]. 2. X has p--measure 1, and we write p (X) =1,if p (X c )=0,whereX c = C ; X. The conditions p2 (X) =0and p2 (X) = 1 are dened analogously. Denition. Let e a proaility measure on C, and let X C. 1. X has -measure 0 in E, and we write (XjE) = 0, if p (X \ E) =0. A-3

17 2. X has -measure 1 in E, and we write (XjE) = 1, if (X c je) = X has -measure 0 in E 2, and we write (XjE 2 )=0,if p2 (X \ E 2 )=0. 4. X has -measure 1 in E 2, and we write (XjE 2 )=1,if(X c je 2 )=0. Denition. Let e a positive proaility measure on C, let A f0 1g, and let i 2 N. Then the i th conditional -proaility along A is A (i +1ji) =( A [0::i] j A [0::i ; 1]): Denition. Two positive proaility measures and 0 on C are summaly equivalent, and we write t 0, if for every A f0 1g, 1X i=0 j A (i +1ji) ; 0 A(i +1ji)j < 1: Denition. 1. AP-sequence of iases is a sequence ~ =( :::)ofiases i 2 [0 1] for which there is a function ^ : N N ;! Q \ [0 1] with the following two properties. (i) For all i r 2 N, j ^(i r) ; i j2 ;r. (ii) There is an algorithm that, for all i r 2 N, computes ^(i r) in time polynomial in js i j + r (i.e., in time polynomial in log(i +1)+r). 2. A P-exact sequence of iases is a sequence ~ = ( :::) of (rational) iases i 2 Q \ [0 1] such that the function i 7;! i is computale in time polynomial in js i j. Denition. If ~ and ~ are sequences of iases, then ~ and ~ are summaly equivalent, and we write~ t ~,if P 1 i=0 j i ; i j < 1. It is clear that ~ t ~ if and only if ~ t ~. A-4

18 Lemma A.1 (Breutzmann and Lutz [5]). For every P-sequence of iases, ~ there is a P-exact sequence of iases ~ 0 such that ~ t ~ 0. A-5

19 Appendix B. Truth-Tale Reductions A truth-tale reduction (riey,a tt -reduction) is an ordered pair (f g) of total recursive functions such that for each x 2f0 1g, there exists n(x) 2 Z + such that the following two conditions hold. (i) f(x) is (the standard encoding of) an n(x)-tuple (f 1 (x) ::: f n(x) (x)) of strings f i (x) 2 f0 1g, which are called the queries of the reduction (f g) on input x. We use the notation Q (f g) (x) =ff 1 (x) ::: f n(x) (x)g for the set of such queries. (ii) g(x) is (the standard encoding of) an n(x)-input, 1-output Boolean circuit, called the truth tale of the reduction (f g) on input x. We identify g(x) with the Boolean function computed y this circuit, i.e., g(x) :f0 1g n(x) ;!f0 1g : A truth-tale reduction (f g) induces the function F (f g) : C ;! C F (f g) (A) = x 2f0 1g j g(x) ; [[f 1 (x) 2 A] [[f n(x) (x) 2 A]] =1 : If A and B are languages and (f g) isa tt -reduction, then (f g) reduces B to A, and we write B tt A via (f g) if B = F (f g) (A). More generally, ifa and B are languages, then B is truth-tale reducile (riey, tt -reducile) to A, and we write B tt A, if there exists a tt -reduction (f g) such thatb tt A via (f g). If (f g) is a tt -reduction, then the function F (f g) : C ;! C dened aove induces a corresponding function F (f g) : f0 1g ;!f0 1g [ C dened as follows. (It is standard practice to use the same notation for these two functions, and no confusion will result from this practice here.) Intuitively, if A 2 C and w v A, then F (f g) (w) is the largest prex of F (f g) (A) such that w answers all queries in this prex. Formally, let w 2f0 1g, and let A w = s i 0 i<jwj and w[i] =1 : A-6

20 If Q (f g) (x) fs 0 :::s jwj;1 g for all x 2f0 1g,then F (f g) (w) =F (f g) (A w ): Otherwise, F (f g) (w) = F(f g)(a w)[0::m ; 1] where m is the greatest nonnegative integer such that m;1 [ i=0 Q (f g) (s i ) s 0 ::: s jwj;1 Now let (f g) e a tt -reduction, and let z 2 f0 1g. Then the inverse image of the cylinder C z under the reduction (f g) is F ;1 (f g) (C z) = A 2 C j F (f g) (A) 2 C z = A 2 C j z v F (f g) (A) : The following well-known fact is easily veried. Lemma B.1. If is a proaility measure on C and (f g) is a tt -reduction, then the function (f g) : f0 1g ;! [0 1] is also a proaility measure on C. (f g) (z) =(F ;1 (f g) (C z)) The proaility measure (f g) of Lemma B.1 is called the proaility measure induced y and (f g). In this paper, we use the following special type of tt -reduction. Denition. A tt -reduction (f g) is orderly if, for all x y u v 2 f0 1g, if x < y, u 2 Q (f g) (x), and v 2 Q (f g) (y), then u<v. That is, if x precedes y (in the standard ordering of f0 1g ), then every query of (f g) on input x precedes every query of (f g) on input y. A-7

21 Appendix C. Proof of Martingale Contraction Theorem Let w 2 f0 1g, and let y = F ;1 (w) Note that for any v y, jvj = step(jyj) and (f g) d either F (v) =w0 orf (v) =w1. Let l = step(jyj) ;jyj. We have (f g) d(w) = d(y) = X vy d(v)(vjy) = X vy F (v)=w0 X vy F (v)=w0 = [(f g) = [(f g) = (f g) d(v)(vjy) + X d(v)(vjy) vy F (v)=w1 [min v d(v)](vjy) + X vy d(w0)] d(w0)] X vy F (v)=w0 F (v)=w1 (vjy) + [(f g) [min v d(v)](vjy) X x2f0 1g l F (yx)=w0 (yxjy) + [(f g) d(w0) (f g) (w0jw) + (f g) d(w0)] d(w0)] X vy F (v)=w1 X (vjy) x2f0 1g l F (yx)=w1 d(w1) (f g) (w1jw): (yxjy) The penultimate step follows from the fact that (f g) is an orderly tt -reduction, and the last step is Lemma 6.4 of [5]. This shows that (f g) d is a ( f g)-supermartingale. To see that (f g) d satises the desired success condition, let A, e a language such that F ;1 (fag) (f g) S1 str[d]. If A 62 range F (f g), then F ;1 (w) is undened for all suciently (f g) d long prexes w of A, whence it is clear that A 2 S 1 [(f g) d]. If A 2 range F (f g), then F ;1 ;1 ;1 (A[0::n ; 1]) is dened for all n and F (A) 2 F (fag), so (f g) d (f g) d (f g) lim sup(f g) n!1 whence we again have A 2 S 1 [(f g) d(a[0::n ; 1]) = lim sup F ;1 (A[0::n ; 1]) (f g) d n!1 lim inf n!1 F ;1 (A[0::n ; 1]) (f g) d d]. lim inf n!1 F ;1 (f g) d = 1 (A)[0::n ; 1] A-8

22 Appendix D. Proof of Positive Bias Reduction Theorem Let e the coin toss distriution specied y iases 0 1 ::: 2 [ 1 ; ], and let 0 2 [ 1 ; ] and>0egiven. We want to construct a formula of the form C 0 = a 1 ^ j 1 =1 x 1j1 1 A ^ 1_ k 1 =1 y k < A a 2 ^ : j 2 =1 x 2j2 1 A ^ 20 4@ 1_ k 2 =1 y k = A _ 5 A : (2) We suppose that the inputs to this circuit are random and independent, and that Pr(z = 1) = i i =0 1 2 :::, if z, ranging over all x's and y's, appears i th in the formula aove. Under this hypothesis, we want j Pr(C 0 =1); 0 j <and that the numer of inputs to C 0 e at most O(lg(1=)). For example, if a 1 = a 2 = = 2 and 1 = 2 = =3,wehave: (and (and z0 z1) (or (or z2 z3 z4) (and (and z5 z6) (or (or z7 z8 z9) (and (and z10 z11) (or (or z12 z13 z14))))))) and Pr(z i =1)= i. In pictures, we'd have A-9

23 ^ \\ ^ _ jj \\ _ ^ jjj \\ ^ _ jj \\ _ ^ jjj \\ ^ _ jj _ jjj For real numers x y 2 [0 1], let x y denote 1 ; (1 ; x)(1 ; y). Thus, for independent A and B, Pr(A) Pr(B) = Pr(A _ B). Note that is monotonically increasing in its arguments, that L n k=1 x k is monotonically increasing in n, and that the empty, L 0 k=1 x k, is 0. We need to determine the a's and 's in Formula (2). The algorithm, on input 0, ::: 2 [ 1 ; ], and tolerance, is as follows: If > 1 return the constant false circuit. Also do the right thing if 0 is 0 or 1. Otherwise continue... Determine a so that Put A = Q a j=1 j. Determine so that Put B = L a+ k=a+1 k: A a+1 Y j=1 Ma+ k=a+1 j < 0 ay j=1 k < 0 A j : a++1 M k=a+1 k : Determine 00 so that 0 = A(B 00 ), i.e., 00 = 0 ;AB A(1;B). Inductively nd a formula C 00 of the top-level shape whose proaility of acceptance is 00. Use tolerance =(A(1 ; A-10

24 B)). Put C 0 = a^ j 1 =1 x 1j1 1 A ^ _ k 1 =1 y k1 1 1 A _ C A 00 : Now we analyze the algorithm. First, the formula generated has at most O(lg(1= 0 )) inputs, where 0 is the initial value of. Note that each recursive call increases the tolerance y at least the factor 1=(A(1 ; B)) = 1=(1 ; ) a+ it follows that will grow to e at least 1 for P (a i + i ) lg 0 lg(1;). Next, the algorithm is correct, i.e., produces a circuit with proaility of acceptance in the range 0. Clearly this is the case if the algorithm returns immediately (when >1). Otherwise, suppose inductively that C 00 has proaility 00 =(A(1 ; B)). It follows that C has acceptance proaility in A B 00 = A 1 ; (1 ; B) 1 ; 00 A(1 ; B) ; A(1 ; B) = A 1 ; (1 ; B) 1 ; 00 A(1 ; B) A(1 ; B) = A(B 00 ) : A-11