Relative to a Random Oracle, NP Is Not Small

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1 Relative to a Random Oracle, NP Is Not Small Steven M. Kautz * Department of Mathematics Randolph- Macon Woman s College 2500 Rivermont Avenue Lynchburg, VA Peter Bro Miltersen t Computer Science Department University of Warwick Coventry CV4 7AL U.K. Abstract Resource-bounded measure as originated by Lutz is an extension of classical measure theory which provides a probabilistic means of describing the relative sires of complexity classes. Lutz has proposed the hypothesis that NP does not have measure zero an the class E2 = DTIME(2poynomid), meaning loosely that NP contains a non-negligible subset of exponential time. This hypothesis implies a strong separation of P from NP and is supported by a growing body of plausible consequences which are not known to follow from the weaker assertion P # NP. It is shown in this paper that relative to a random omcle, NP does not have measure zero in Ez, improving the earlier result of Bennett and Gill that P # NP relative to a random oracle. Several new techniques are introduced; in particular the proof exploits the independence properties of algorithmically random sequences, and a strong independence result is shown: if A is an algorithmically random sequence and a subsequence A0 is chosen by means of a bounded Kolmogorov-Loveland place selection, then the sequence A of unselected bits is random relative to Ao, i.e., A0 and A are independent. A bounded Kolmogorov-Loveland place selection is a very general type of recursive selection rule which may be interpreted as the sequence of oracle queries of a timebounded Turing machine, so the methods used may be applicable to other questions involving random omdes. *Much of this author s research was performed while visiting Iowa State University, supported by National Science Foundation Grant CCR , with matching funds from Rockwell International and Microware Systems Corporation. Supported by a grant of the Danish Science Research Council (through Aarhu University) and partidy supported by the ESPRIT I Basic Research Actions Program of the European Community under contract No. 74 (project ALCOM ). Introduction The conjecture P # NP is a reasonable working hypothesis because of the plausibility of its consequences and the body of empirical evidence supporting it. Lutz has proposed a stronger hypothesis, that NP does not have measure zero within the class E2 = DTIME(2po ~nomi ). This hypothesis has greater explanatory power than P # NP and is supported by a number of credible consequences, some of which are summarized later in this section. The main result of this paper is that Lutz hypothesis holds relative to a random oracle, strengthening the previous result of Bennett and Gill [l] that P # NP relative to a random oracle. The meaning of measure zero is in terms of resource-bounded measure, an extension of classical measure or probability theory due to Lutz [2]. The aspects of resource-bounded measure needed for the present results are introduced in Section 3. Intu- itively, NP does not have measure zero in Ez, written p(np I Ez) # 0, means that NP comprises a nonnegligible subset of Ea, or very loosely, a random language in Ez has nonzero probability of being in NP. %source-bounded measure provides a meaningful answer to the question of what it means for a language in E2 to be random. Since it is known that the class P does have measure zero in Ez, the hypothesis p(np I Ez) # 0 implies that P # NP. Lutz and Mayordomo cite evidence in [3] for the plausibility of the hypothesis p(np I Ez) # 0. In particular, its negation would imply the the existence of a betting algorithm for efficiently predicting membership in NP languages, a consequence which turns out to be intuitively quite unlikely. We discuss this in greater detail in Section 3 after giving the relevant definitions. The hypothesis p(np I Ez) # 0 also has a number of plausible consequences which are not known /94 $03.00 Q 994 IEEE 62

2 to follow from the weaker assertion P # NP (although the examples mentioned here actually follow from the weaker NP does not have pmeasure zero ; see Section 3). In particular in [3] Lutz and Mayordomo prove that if p(np I E2) # 0, then the Cook versus Karp-Levin (CvKL) conjecture holds for NP, that is, there is a language which is $complete but not <:-complete for NP. Evidence for the plausibil- ity of the CvKL conjecture as cited in [3] includes the following facts: The CvKL conjecture holds for E = DTIME(2ine ) (KO and Moore, [4]) and for NE (Watanabe [5], Buhrman, Homer, and Torenvliet [SI). Under certain additional hypotheses it holds for PSPACE (Watanabe and Tang [7]). If E # NE the CvKL coqjecture holds for NP U co-np and if E # NEnco-NE it holds for NP (Selman [SI). At this time the CvKL conjecture is not known to be a consequence of the assertion P # NP. This fact and the plausibility of the CvKL conjecture itself suggest that a stronger class separation such as p(np I Ez) # 0 is likely to be true. Other consequences of the hypothesis p(np I E2) # 0 cited in [3] include the following (here EE denotes the doubly-exponential class UEo DTIME(22n+c) and NEE denotes the corresponding nondeterministic class):. E # NE and EE # NEE (Mayordomo [9], Lutz and Mayordomo [3]). 2. There exist NP search problems which are not reducible to the corresponding decision problems (this follows from item above and a result of Bellare and Goldwasser [lo]). 3. Every <:-complete language for NP contains a dense exponential complexity core (Juedes and Lutz [ll]). 4. For every real number a <, every <:a-,,-hard language for NP is dense (Lutz and Mayordomo [2). Our main result is that p(np I E2) # 0 holds relative to a random oracle; we also show that relative to a random oracle, NP does not have measure zero in the class E = DTIME(2lineW). It is difficult at this point to assess the exact meaning of a random oracle separation such as Bennett and Gill s [l] or the stronger result proved here. In [l], Bennett and Gill proposed the random omcle hypothesis, i.e., if a property holds relative to almost every oracle, then it must hold in unrelativized form. The random oracle hypothesis was first shown to be false in general by Kurtz [3], and more recently it has been shown that IP = PSPACE ([4]) but IPA # PSPACEA for a random oracle A ([5], [IS]). We do not suggest that the present result should be interpreted as evidence for the hypothesis p(np I E2) # 0, only that it is an interesting, related result. The philosophical view of the first author is that random oracle results are useful largely for the insight they give into the properties of algorithmically random sequences and into the nature of probabilistic computation. In any event the present paper is a nontrivial improvement of [l] and introduces techniques which may be useful in other contexts. In the course of the proof we develop a method for dealing with the situation in which a betting algorithm, by means of querying an oracle, is able to gain a slight probabilistic advantage in predicting a language or sequence. The idea, very loosely, is that playing an unbiased game with an slight advantage should somehow be equivalent to playing without an advantage in a slightly biased game. Given a situation in which an algorithm has an advantage in predicting a language, we transform it into the dual situation in which the algorithm has no advantage but the odds on membership in the language are biased. The latter case, it turns out, is much more amenable to analysis, and by determining the extent of the bias one is able to determine whether the algorithm actually predicts the language successfully. A second new technique we introduce is the exploitation of the strong independence properties, investigated by van Lambalgen [7] and Kautz [8], of sequences which are algorithmically random (in the sense of Martin-Lof, Solovay, or Chaitin; see Section 2). Roughly, two subsequences A0 and A of an infinite random sequence A are independent if each is random relative to the other. For example, it is shown in [7] and in [8] that if A is algorithmically random and Ao, AI denote the even and odd bits of A, respectively, then A0 and AI are independent. Independence results for a number of other kinds of subsequences are given in [8]. Our results include a proof that if the subsequence A0 is chosen from A according to a bounded Kolmogomv-Loveland place selection, and A denotes the nonselected bits, then A0 and A are independent, A Kolmogorov-Loveland place selection, described in [9], is a very general kind of recursive selection rule in which the nth bit selected may depend adaptively upon the n - l previous bits in the subsequence as well as upon previously examined bits which are not necessarily included in the subsequence. The sequence of oracle queries of a Turing machine is an example of a subseqence chosen according to 63

3 this type of rule; our result applies to a slight restriction of Kolmogorov-Loveland place selections which is nonetheless useful for dealing with time-bounded comput at ions. In the next section we review some of the terminology and notation to be used, and in Section 3 we cover the pertinent notions of resource-bounded measure. Section 4 contains the proof of our main theorem. 2 Preliminaries Let IN = {0,,2,...} denote the natural numbers and IR the real numbers. A string is an element of {O,l}*; for z E {O,l}*, 2 denotes the length of z, and A is the unique string of length 0. If 2: E {0,}* and j,k E IN with 0 5 j 5 k < 2, z[k] is the kth bit of z and zcj..k] is the string consisting of the jth through kth bits oft (note that the "first" bit of z is the 0th). For strings 2 and y, cy is the concatenation of z and y, and 2 c y means z is an initial segment of y, i.e., x 5 Iyl and z[k] = y[k] for all E E IN with 0 5 k < IyI. For an infinite binary sequence A E (0, }0, the notations A[k], ACj..k], and z c A are defined analogously. denotes the bitwise complement of A. Fix a standard enumeration of {0,}*, so = A, SI = 0, s2 =, s3 = 00, s4 = 0,... A language is a sub- set of {O,l}*; a language A will be identified with its characteristic sequence XA E (0, }0, defined by sy E A e XA[Y] = for y E IN. We will consistently write A for XA. It is convenient to use strings over the alphabet {O,, I}* as well as over {O,}* to represent partially defined languages, where I denotes an undefined bit. These strings will typically be represented by lowercase greek letters. For U E (0, l,i}* we regard U, U Ik, and U IOo as essentially the same object. For 0, E ~ {0,, L}*, U E r means that if ~[k] is defined, then 7[k] is also defined and ~[k] = 7[k]. Likewise for A E (0, }0, U A means ~[k] = A[k] whenever bit ~[k] is defined. When Q E {0,, I}* and 7 E (0, l}*, the notation Q 7 ( ''T inserted into a'') is defined by (a.)[.i = a[z] if a[.] is defined rb] if z is the jth undefined position in Q and j < 7 I otherwise. For A, E E (0, }0, A/E is the subsequence of A selected by E, i.e., if yo, y,... are the positions of the -bits of B in increasing order, then (A/E)[z] = A[y,]. Note that A/B is a finite string if E contains only finitely many 's. For U,T E (0, l}*, U/. may be defined analogously. E = E denotes the class DTIME(2ine") and E2 denotes the class DTIME(2Po'Ynomi"). Given a function f : {O,l}* -, {O,l]*, we say f is in the class p = p if f(z) is computable in time polynomial in 2, and f is in the class p2 if f(z) is computable in time We assume there is a fixed pairing function on strings so that, for example, a function on IN x {0,}* can be interpreted as a function on {0,}* (where the numeric input is represented as a unary string). A string U defines the subset Ext(a) = {A E {0,}0 : U C A} of (O,l}w, called an interval. Likewise if S is a set of strings, Ext(S) denotes Ext(a). By a measure we simply mean a probability distribution on (0, l}", and for our present purposes it is sufficient to consider the uniform distribution (each bit is equally likely to be a zero or a one), known as Lebesgue measure. Thus for any subset & of {0,}0, the measure of &, denoted Pr(&), can be interpreted as the probability that a sequence produced by tossing a fair coin is in the set &. Thus the measure or probability of an interval Ext(u), abbreviated Pr(a), is just ($)l"l. For S a set of strings, we abbreviate Pr(Ext(S)) by Pr(S). Consider a recursively enumerable (r.e.) set of strings; associated with any r.e. set is an index (e.g., the code of a program for enumerating it), and a sequence {Si} of r.e. sets is uniform if there is a recursive function g such that g(i) = index of Si. A constructive null cover is a uniform sequence of r.e. sets {Si} such that Pr(Si) 5 2-i. A sequence A E (0, l}w is algorithmically random if it is not contained in any constructive null cover, that is, for every constructive null cover {Si}, A $! ni Ext(Si). If A, B E (0, }0, we also say that A is algorithmically random relative to B, or that A is independent of B, if A is not contained in any constructive null cover relative to B (i.e., the enumerations of all the sets Si have access to the oracle B). The above definition of algorithmic randomness is due to Martin-Lof [20]. 3 Resource-bounded measure Itesource-bounded measure theory, as formulated by Lutz [2], is a form of effective measure theory which provides a means of describing the measure or probability of sets of languages within complexity classes and of defining the random languages (i.e., the pseudorandom sequences) within a complexity class. The 64

4 formulation of Lutz is extremely general-for example, the classical theory of Lebesgue measure is a special case-and is presented in [2] in terms of the powerful notion of n-dimensional density Systems, but its origins may be traced back to the work of Schnorr on matingales [2]. Our presentation here is highly abbreviated, covering just those aepects needed for the proofs at hand, that is, to describe measure in the classes E and E2 and we use the language Of martingales rather than density systems. The reader is encouraged to consult [2] for the complete story. Definition 3. A martingale is a function d : (0,}* + IR such that for all U E (0, }*, d(a) = d(u0) + d(a) 2 () A martingale may be intuitively understood as a strategy for betting on the values of successive bits of a binary sequence. We picture the space (0, }0 of all possible sequences as a tree, the value d(a) at the root as the gambler s initial capital, and the value d(a) at node U as the amount of capital she would possess after the initial sequence of outcomes a. The bet at node U on i = 0 or is just the amount B, 0 5 B 5 d(a), for which d(ai) = d(a) + B and d(o( - i)) = d(a) - B. The condition () merely asserts that the game is fair. The sequences of particular interest are those for which the capital becomes unbounded as the game progresses. Definition 3.2 A martingale d succeeds on a sequence A E (0, }O0 if limsupd(a[o..n]) = 00. n-m Thus for a martingale d to succeed on a sequence A, it must be able to make a good prediction of the (n+l)st bit of A from the first n bits, and must do so often enough to win an infinite amount of money. Intuitively it is not surprising that this hardly ever happens; given a martingale d, if a sequence A is generated by repeatedly tossing a fair coin, then d succeeds on A with probability zero. This is a consequence of the following standard result, known as Kolmogomv s inequality for martingales (see [7, p. 7 or [22, p. 242). Theorem 3.3 Let d be a martingale and a > 0; then Pr(a E {0,}* : d(u) > a} < h. In particular if d (or some close approximation to d) is a recursive function, the sequence {Si} defined by Si = {U E {0,}* : d(a) > 2 ) is a constructive null cover. Defmitiqn 3.4 A computation of a martingale d is a, function d : INx {0,}* -+ {0,}* (where the value of d is interpreted as the binary representation of a dyadic rational number in [o, 3) su& that - d(,)l 5 2- for a a E {O, I* and E IN. For = Or 2, d is a Pi-comPtatzon of d if d E pi; then we also refer to as a pi-marlingale. The measure structure of the class Ei is then defined in terms of pi-martingales. The key definition is the following. Definition 3.5 Let i = or 2. A set X s (0,}0 has pi-measure zero, written pp,(x) = 0, if there is a pi-martingale which succeeds on every sequence A E X. Likewise X C (0, }0 has pi-measure one, written ppi(x) =, if the complement Xc of X has pi-measure zero. The class X C (0, }0 has measure zem in Ei, written p(x I Ei) = 0, if ppi(x n Ei) = 0; X has measure one in Ei if pp,(x fl Ei) =. We also write ppi(x) # 0 to indicate that X does not have pi-measure zero; note that this does not imply that pp,(x) has some nonzero value, since X may not be pi-measureable. If p(x I Ej) = 0, we say that X is a negligibly small part of Ej; if p(x I Ei) =, then almost every lan- guage A E Ei is in X. Lutz [2] has proved a number of results justifying the use of this terminology, e.g., the measure zero sets in Ei behave set-theoretically like small sets and the measure one sets behave settheoretically like Klarge sets. It is also not difficult to show that p(ei I Ei) # 0 and that p(p I Ei) = 0. We are interested in the size of NP in the classes E and Ez. The table below, adapted from [3], summarizes some of the known relationships among nonsmallness conditions on NP. P(NP I Ed # 0 P(NP I E) # 0 $ U PPm? # 0 PpW) # 0 U P#NP Lutz has suggested that the conditions above be investigated as scientific hypotheses, i.e., evaluated in terms of explanatory power and intrinsic plausibility. In Section we discussed some of the consequences of the hypothesis p(np I Ez) # 0, although actually the examples cited follow from the weaker assertion pp(np) # 0; we conclude this section with a brief intuitive argument for the intrinsic plausibility of the hypothesis NP is not small as originally given in [3]. (We refer specifically to the condition pp(np) # 0 but 66

5 the reasoning applies to any of conditions in the table above.) The condition pp(np) = 0 would imply that there exists a single pcomputable martingale which succeeds on every language A E NP. This means that there is a fited polynomial zc such that for every NP language A, given the first z bits of A, d has time tc = 2=" to compute its bet on whether s, E A, where n = Isz I. However, for arbitrarily large k, there are NP languages A for which determining whether s, E A apparently requires checking 2'" potential witnesses (possible nondeterministic computation paths). Thus an individual in possession of the algorithm d could successfully bet on all NP languages while only examining the fraction 2M/2kn of the search space of potential witnesses. Since c is fixed and E is arbitrarily large, the fraction 2cn/2kn = 2c'k is arbitrarily small; thus it seems extremely unlikely that such an algorithm could exist. 4 Main result Theorem 4. If A E (0, l}m is algorithmically random, then (i) &NPA n EA) # 0. (iz) pf2(npa n E$) # 0. The proofs of (i) and (ii) are almost identical, so we will present both together and remark on the differences where appropriate. Theorem 4. follows from Theorems 4.7 and 4.9 below; before we can formally state Theorems 4.7 and 4.9 we need to develop some preliminary results. Remark 4.2 We make some simplifying assumptions about martingales. Let d be a martingale and let d = d, denote a pi-computation of 4 with some fixed r, for i = or 2. We assume that d(x) is always. We assume that the values d(u0) and d(u) are produced sjmultaneously, e.g., as a pair, and we use the notation d(u0) to denote this pair of values. The computation of d(u0) always begins by precisely duplicating the computations of d(u[~..i]) for i = 0,,...,. -, in order. Associated with d is a function f E pi such that on an input sequence U E (0,}+ of length m, d(u0) runs for exactly f (m) steps. We assume that each step includes exactly one oracle query and that no bit of the oracle is-queried more than once during the computation of d(u0). It is important to note that these restrictions do not affect which sets have pi-measure zero. Throughout the discussion below, let i = 0: i = 2 be fixed, let be a pi-martingale and let d denote the fuyction dl in a pi-computation of d, so that Id(u) - d(u)i 5!j for all Q E (0,}+. Let f denote the time bound function for d^ as in Remark 4.2 above. We first define the construction of a lan- guage LA E NPA fl E? depending in a uniform way on A E (0, l}m; the idea will be to eventually show that when A is algorithmically random, d does not succeed on LA using oracle A. If y = lul, then we think of d(u0) as first producing the value of the capital at node U, and then determin- ing how to bet on the next bit LA[y], i.e., on whether sy E LA. To make its decision, d has time f(y); note that since the yth bit of LA represents a string sy of length n = logy, when expressed in terms of n the time bound on d is of the form 2t(n), where t(n) = cn iff E p and t(n) = nc iff E pa. Let U(.) = t(n) + 2n +. Let i(n) be the real-valued function defined by and define v(n) = lg(n>l Now given A E (0, }O0, we partition A into independent blocks of contiguous bits and let each block determine a single bit of LA. For each y, the block corresponding to sy will consist of U(.). ~ (n) bits of A, where n = Isyl. Specifically let bo = 0, by = c U(I~2)V(ISrI). Z<Y We will refer to A[$..b,+l - 3 as the yth block of A. The yth block of A determines the bit LA[^] according to the mapping 9 defined below. Definition 4.3 Let U, U, and by be as defined above, let y E IN, and let A E (0, }O0. = w (33 < U(.) [A[by + t. U(.) + j ] = 0 for 0 5 j < U( n)], and Qy(A) = 0 otherwise. Then LA is the language defined by sy E La 9 Qy(A) =. That is, LA[Y] = just if for some t, the tth group of U(.) bits within the yth block of A is all zeros. Such 66

6 an z will be called a witness for LA[Y] =, and we say that dfinds a witness for LA[y] = if dqueries dl U(.) bits in the zth group (for some x) and determines that all are zeros. We verify in Lemma 4.4 that the function v is has been defined so that approximately half of the possible configurations of A[by..by+l - correspond to LA[^] = 0. We also verify that LA E NPA fl E?. Lemma 4.4 (i) Let y E IN and n = Isy]; then which establishes the desired inequality. (ii) It is clear that LA E NPA since verifying sy E LA requires checking only U(.) bits of A. Then note that since from below, (2;) LA E NPA rl E?. Proof (i) For any fixed x, 0 5 x L v(n), and so Pr(z is a witness for LA[y] = ) = Pr((Vj < u(n))a[by + x. U(.) + j] = 0) - 24") - Pr(z is not a witness for LA[^] = ) = Pr((3j < u(n))a[b, + x u(n) + jl # 0) It follows that pr(l~[y] = 0) = Pr((Vx < v(n))(3j < u(n))a[by + x. U(.) + j ] # 0) By the definition of 6(n) and the fact that v(n) 5 G(n) < v(n) +, we have Hence we know that for all n. Given y E IN, let n = Isy/; to determine whether sy E LA requires examining the yth block of A, which consists of u(n). v(n) bits. For i =, U(.) =, (c + 2). +, so by (4), U(.). v(n) is of the form 2near; likewise for i = 2, u(n) = ne + 2n +, so U(.). v(n) is of the form 2p0ynomid. 0 Our object in defining LA is to show that for random A, d cannot succeed on LA. However, since LA is obtained deterministically from A, d can get information about LA by querying the oracle A. For any U,. = y and n = Isyl, the bit LA[^] de- pends on U(.). v(n) bits of A, so there are v(n) po- tential witnesses for LA[y] =. During the computation of ;(CO), i.e., in deciding how to bet on LA[p], 2 may make 2t(n) queries. In particular this means that d can examine at most 2(") potential witnesses, and since v(n) is of roughly the same order as 2'4") = 2t(n)+2n+, this is a very small fraction of the possible witnesses. The lemma below confirms that d^ can gain only a very slight advantage by querying A. Lemma 4.5 For each y E IN, let n = Isyl, k. 5 2'("), and let XI, 22,..., xk denote any sequence of k natuml numbers with xj < v(n). There exists a sequence { E ~ } with &cy < cm such that for all y 6 IN, (2) the pm ba bilrty Pr(one of XI,...,Xk is a witness for LA[g] = ) is bounded by cy, and (3) (ii) - 5 Pr(LA[y] = 0 x,...,xk 2 not witnesses for LA[Y] = ) 67

7 PmoJ First note that since = (I-&)) k monotonically from below and the left-hand-side is equal to a when z =, -<(l-$) <- for all z > 0. It follows that and so Pr(one of z,..., Xk is a witness for LA [y] = ) Similarly since 2" 2-= -- < (;). (5) Let Note that and again the limit is monotonic from below, 22- < +-. 2" for all z > 0. Then we can write (6) I Then for (ii), n using (2) we see that Pr(LA [y] = 0 I 2,..., zk are not witnesses for LA[y] = ) 4"I-k = ( - 3 ). Since w(n) - k. < w(n) 5 ij(n), 24") u(n)-k For the second inequality, ( - = (;)2-2n and so (since U(.) = t(n) + 2n + ) > -- by (5), 22" (7) < + & by (6). (8) 2 = 5 +cy, Then for (i), using (2) we have where Pr(z,..., zk are not witnesses for LA[y] = ) 2 Y = - 2u(n) + 22n+l+ 2"0+2" 68

8 and n = Isvl. Clearly ci is summable in y as for above. To complete the proof we take 0 2 cy = max(cy,cy). Using an effective form of the Borel-Cantelli lemma (see [2] and [23]), it follows from part (i) that if A is random, d can actually find a witness for LA[Y] = only finitely often. Part (ii) then asserts that the information d gains by not finding a witness gives it only a very slight advantage which is bounded by a rapidly decreasing sequence cy. The difficulty is to show that the slight advantage d^ gains by querying A is not enough to enable d to succeed on LA. We will first define qa, the bounded query sequence fop A, to consist of just those bits in the yth block of A which d was able to query before having to decide the value of d(uu), where y = 0,,2,..., and U = LA[O..~ -. Informally, for any z E IN let y be the integer for which by 5 z < by+l, and define B[z] = just if A[z] is queried during the computation of d(uo), where U = LA[O..Y -. Then QL will consist of the bits of A/B, not in their natural order in A, but in the order queried by d. We will also define NA = A/B, the nonselected bits of A. For the purpose of proving Theorem 4.7 below, we will use the following formal definition. Definition 4.6 Fix an oracle A E {O,l}m. For - an input string U with IuI = y, define a function F, : {O,l}* IN as follows: if ( is the sequence of responses to the first [(I < f(y) oracle queries in the computation of d(u0) relative to A, then Fu([) is the position of the next bit to be queried. For an infinite sequence C E (0, }O0 let Fc(() = Fu((), where U = C[O..y - for the least y such that < f(y). Define sequences of strings (0 C ( C... and po C pi C... such that 50 = po = A, (j+ = (ja[f((j)], and pj+l = pjg((j). Let Q = limj (, and R = limj pj. Then the bounded query sequence for C is the sequence Q* = &/RI and in particular we let QI denote the bounded query sequence for C = LA. We also define B E (0, }03 by B[z] = if and only if for some j, F~~(<jj) = z and G(<j) = ; then NA = A/B is the sequence of nonselected bits of A. Qfi includes a relatively small part of the yth block of A-at most 2t((n) bits-and moreover by Lemma 4.5(ii), if we look at the bits of NA within the yth block (i.e., everything in the yth block not included in Q;), approximately half the possible configurations correspond to LA[$/] = 0 and half to LA[^] =. Thus if d is successful in predicting whether LA [g] = 0 based on the partial information in QI, then d in effect has a great deal of information about the nonqueried bits in the yth block of A, which should be impossible unless A itself has some kind of internal regularity, i.e., is nonrandom. What we actually prove is that if d succeeds on LA, there is a martingale h which succeeds on NA, and that h can be approximated by a recursive function that uses Q; as oracle. Theorem 4.7 Let A E (0, l}, and let d, d, LA, QI, and NA be as defined above. Suppose that d succeeds on LA using oracle A, and that d finds a witness for LA[y] = for only finitely many y E IN. Then there as a, martingale h which succeeds on NA and a function d, recursive in the oracle PA, which appmzimates h in the sense that there is a constant K such that for infinitely many initial segments p c NA, a(p) > K. 2t + =% h(p) > 2t. It then follows easily using Kolmogorov s inequality (Theorem 3.3) that NA is not algorithmically random relative to QI : Corollary 4.8 Under the hypotheses of Theorem 4.7, NA is contained in a constructive null cover relative to &A. Proof of Theorem 4.7. The plan of the-proof is as follows. We first construct the function- d* and then define the martingale h. The function d* may be regarded more or less as an approximation to a partially defined martingale which is attemptipg to succeed on NA; thus we think of the inputs p to d* as possible initial segments of NA. The construction of d proceeds in stages; at stage y +, d* attempts to simulate the computation of d(u0) for strings U of length y. Consider steps f(y - ) through f(y) - in the computation of d(u0); if d queries bit z of the oracle, where z 2 by, the value of A[z] is available from Qfi. The information in Q; can then be used by to construct an approximation of A, that is, to fill in some of the bits of a string ap E {0,, I}* representing a belief about A associated with a given input p. However, during this part of the computation of d(uu), values for d(u[o..j]), j < y, have already been produced, so if a bit z < by is queried by d, the value A[z] is not part of the bounded query sequence Q;, i.e, it resides in 69

9 s:,[rl NA. What (2+ does is to use the input string p to fill in the values of bits z < b, of ap which are not provided Most input strings are not initial segments by va. of NA, of coutse, so most of the time the attempted simulation of d is incorrect, but there must be one sequence of inputs po c p c... whichaare true initial segments,of NA, and on these inputs 8 wil! correctly simulate d. We will then arrange to define d*(p) to be equal to an associated value d(up), where up is a true initial segment of LA, so that if lim supj d( LA [O..j]) = CO (whence limsup,j(la[o..j]) = CO), then limsupj &(N~[o..j]) = 00 also. may be regarded as an approximation of a martingale d', where the values of 8 are related tp the values of d in the safne way that the values of d' are related to those of d. However, d' is a biased martingale, and the extent to which it is biased-the extent to which the probability associated with each bit differs from +--is bounded by the sequence c, of Lemma 4.5. Thus instead of a strategy d which has an advantage cy in betting on LA, we have in a sense the "dual" problem of a strategy d' betting on a sequence in which the probabilities may be biased by cy. This is actually a fortuitous state of flairs, since the effect of the bias cy can now be much more readily analyzed. The idea is then to define the (unbiased) martingale h by making careful adjustments in the values of d* so that condition () can be satisfied, and to do so in such a way that the limsup is preserved. The key to being able to do this is the fact that the sequence cy is rapidly decreasing. Note that while will be com- putable relative to oracle Q:, d* will not necessarily be computable at all; it will be used only to define the martingale h. We next give a formal description of the construction of d'; the function 6c will be defined simultaneously. The construction takes place in stages. If #(p) is defined during stage y, we will say that the node p is active during stage y +. Associated with each active node p at stage y + is the following cast of characters : ap : belief about A at node p. ap is a string over {0,,-I-}*, but all bits z < by are defined. up : belief about LA corresponding to ap, i.e., upb] Y* for j = 0,...,y-. up is astringof length rp : an integer representing the next bit of Q> to be queried by &. tp : sequence of all query responses produced by the simulation of (i(ap). - ) (where we define f(-) = 0). &, is a string of length f ( ~ Construction at stage 0: Let &(A) = d (~) =, and ax = Im, ax = A, I'x = 0, and <A = A. Construction at stage y + : For each active node p, let a = cyp, U = up, r = rpl and < = tp. Note that y =. by construction. We simulate the computation of d(a0) on steps f(y-) through f(y)-, extending a and < as follows: Then let for j = f(y - ) to f(y) - do + FOK) if z 2 by then else < lilt Si [rl 4. + rcr+i rbl Simulate the jth step in the computation of i(ao), using <b] for the oracle response. b and k Q A = i(uo), = i(ul), = number of undefined bits in ' ~[b,..b~+~ -. For each string r E (0, l}e define & on node pr by: At the same time let the actual values of d, and define d' Finally let Upr = (9) a = d(4, (0) 6 = d(al), a0 0 r) = 0 if ay(" r) =, on node pr by: 70

10 apt = a 37 rpr = J?,and (PT = <* This completes stage y +. There is a unique sequence of strings X = po C p C p2 C. such that for each y, py c NA and py is active at stage y +. It can be shown by induction that at stage y +, ff Pv c A, bpv = La[O..y-, and &Y) = 4%). Let cy = d(l~[o..y- 3) and ty = d(la[o..y-l]); then d*(py) = 2,. If d succeeds on LA, then limsupy cy = 00, and since (cy - tyl 5 4, limsup, ty = 00 also. It follows that limsupci.(py) = 00. () Y We next define the martingale h. Let p be an active node at any stage y +, and let a, U, a, b, and 6 be as in (9) and (0). We will adjust the value of d* at each node so the condition can be satisfied (note that (2) is just the obvious extension of ()). First we need the following definition: Define p to be bad if ay(& 4 r) = for every 7 E (0, l}k. That is, a node is bad if the bits of the oracle queried by d actually form a block of zeros witnessing that L~[yl =. Otherwise the node p is good. Note that the goodness or badness of a node p is evident at the stage where p is active. Now if p is good, we define the quantity qp = I{. E {0,}' : s(p.) = Q } I if Q # b, and qp = 4 if Q = b. That is, qp is the proportion of extensions pr of p, r = k, for which d*(pr) = a, or equivalently, for which QY(a r) = 0. The crucial fact about qp is that by Lemma 4.5(ii), 2-2 Let ma = ; if p is good let 2k - < qp < - + y. I (3) and if p is bad let C mpr = - b for every string r E (0, l}k, where c = d*(p) = d(u). Note that if d succeeds on a sequence C and p C C, neither b nor c can be zero. Moreover since (a+b)/2 = c, it is always the case that c- l a ---+->b 2 2b-2 Now for any y and any node p active at stage y +, let X = po C p C p2 C -. c py = p be the nodes preceding p such that p, is active at stage j +. Define let h(a) = d*(x) = and define We can show that h satisfies (2): again let p be an active node and let A = po C p C C py = p denote its active predecessors. Let Q, b, and k be as in (9) and (lo), and let c = d*(p) = d(up). If p is good, we have c = - a+b 2 since 2kqp is the number of extensions pr for which d*(p~) = a, and 2k( - qp) is the number for which d*(p~) = b. If p is bad, then d*(pr) = b for all 7 E (0, ilk, so again Thus in either case if d*(pr) = Q if d * W = b, 7

11 Then the domain of h can be extended to all strings simply by requiring Thus for any py in the sequence, for all p E {O,l]*. (We have not defined d^ or d* on strings other than the active nodes, but it is not necessary to do so.) Next we need to prove the following fact: Let po c p C p2 C... be a sequence of nodes such that pj is active at stage j + and such that only finitely many of the pj are bad. Let B denote the number of bad nodes in the sequence. Then there is a constant J such that MPj > 2-J2-B for each node p, in the sequence. To see this, let {cy} be the sequence defined in Lemma 4.5. Since - ( + 2 ~~)-~, e < 4, ( + 2cy) < 42ty, and hence so that log( + 2 y) < 4 y Clog( + 2fY) < J < 00 for some constant J. Thus Y Hence >, 2-J2-B. and so for any t E IN, 2 d^ (py) > 2J+B2t + - * S(py) > 2J+B2t 3 hby) > 2 (4) Then if po C p c p2 C. is the sequence of active nodes with py c NA, by hypothesis only finitely many of the py are bad, so by the argument above there are J and B such that (4) holds for this sequence of nodes. Let and!+2r, >2-J. If py is a good node, then f L q p, < f + cy by (U), so 5 2qp, < + 2cy and For good nodes p we always have and for bad nodes p it is always the case that mpr L -* 2 It follows from () and (4) that h succeeds on NA. This completes the proof of Theorem Proof of Corollary 4.8. Let K and B be as defined in (5). For t E IN, define the set St as the enumeration of those strings p satisfying the following conditions: (i) p is active at some stage in the construction, (ii) at most B of the active nodes p c p are bad, (iii) a(p) > K.2t + f. Since d^ is recursive in PA, the sets St are uniformly r.e. relative to Q;. By (ll), St contains some initial segment of NA, SO NA E next(si). 72

12 To show that {St} is a constructive null cover, we need to show that Pr(St) 5 2-t for each t E IN. This is where we use the fact that h is a martingale; define the sets s; = {p : h(p) > 2t}. By Kolmogorov s inequality (Theorem 3.3), Pr(S:) < 2-t. Then (4) implies that p E st =2 h(p) > P. (i) H is nondecreasing, i.e., if c e then H(() 5 HK L (ii) H is unbounded, i.e., if (j and pj are defined for all j then limj H(<j) = 00, and (iii) G is determined by H according to the rule so St S:. Thus Pr(St) < 2-$. 0 Note that if d finds a witness for LA[^] = for infinitely many y E IN, then we know that A is not algorithmically random by Lemma 4.5(i) and the Borel- Cantelli lemma; see the remarks following Lemma 4.5. Corollary 4.8 establishes that if the pi-martingale d succeeds on LA, but only finitely often finds a witness for LA[Y] =, then NA is not algorithmically random relative to Q;. We now exploit the fact that this can only occur if A itself is not algorithmically random. Theorem 4.9 Let A E {O,l}Oo and let Q; and NA be as in definition 4.6. If A is algorithmically random, then NA is algorithmically random relative to qa. The proof of Theorem 4.9 is based on techniques developed in [7] and [MI, where a number of general independence properties are established for subsequences of random sequences. In particular, the method is an extension of techniques used for Lemma.3.8 in [8]. It will be helpful to first place Theorem 4.9 in a slightly more general context. Definition 4.0 A Kolmogomv-Loveland place selection [9] is a pair of partial recursive functions F : {0,}* -, IN and G : {O,l}* -, {O,l}. Let A E (0, l}oo; F and G select a subsequence Q* from A as follows. First define sequences of strings <O C ( C... and po c p c... such that to = po = A, <j+l = <ja[f((j)], and pj+l = pjg((j) (with the proviso that t,+l is undefined if F(&) = F(&) for some i < j or if either F or G fails to converge). If (j and pj are defined for all j let Q = limj (, and R = limj pj. Thus Q represents the sequence of all bits of A examined by F, in the order examined. The idea is that a given bit Q[j] = A[F((j)] is included in the subsequence Q just if G(&) =, i.e. F determines which bits of A to examine, and G determines which ones to include in the sequence Q. Formally we define Q = Q/R. A bounded Kolmogorov-Loveland place selection also includes a partial recursive function H : (0,l) + IN in addition to F and G such that It is also useful to define a sequence B by B[z] = if and only if for some j, F((,) = z and G(<j) =, so that N = A/B consists of the nonselected bits of A, in their natural order. It is not difficult to see that the bounded query sequence of Definition 4.6 is an instance of a sequence selected according to a rule of this form. While the selection function FL, of Definition 4.6 apparently depends on LA, note that for a given string (, FL,(() depends only on an initial segment of LA of length y, where f(y - ) 5 ( < f(y). Since LA[O..Y - is determined by A[O..b,-l-, and since no bit of A to the left of by-l can thereafter be added to the bounded query sequence, it is possible to define a p.r. function F which, on an input < of length = f(y - l), examines all previously unexamined bits of A to the left of by-ll determines the string LA[O..Y - 3, and then simulates FL, (thus the value of H(() will just be the appropriate value of by- ). The selected subsequence Q is then the bounded query sequence of Definition 4.6. We can then state Theorem 4.9 in the slightly more general form: Theorem 4.9* Let F, G, and H be partial recursive functions determining a bounded Kolmogorov- Loveland place selection, let A E (0, l}oo, and let N and & be as in Definition 4.0. If A is algorithmically random and N is infinite, then N is algorithmically random relative to Q*. Acknowledgments The authors would like to acknowledge the contributions of Jack Lutz and Elvira Mayordomo, each of whom has discussed the problem at length with one or both of the authors. The first author would also like to thank Jack Lutz for providing the moral and financial support for the author s visit to Iowa State University, where much of this work took place. 73

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