RANDOMNESS IN THE HIGHER SETTING

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1 RANDOMNESS IN THE HIGHER SETTING C. T. CHONG AND LIANG YU Abstract. We study the strengths of various notions of higher randomness: (i) strong Π -ML-randomness is separated from Π -ML-randomness; (ii) the hyperdegrees of Π -random reals are closed downwards (except for the trivial degree); (iii) the reals z in NCR Π are precisely those satisfying z L ω z ; and (iv) lowness for -randomness is strictly weaker than that for Π -randomness.. Introduction Randomness in the higher setting refers to the study of algorithmic randomness properties of reals from the point of view of effective descriptive set theory. Until recently, the study of algorithmic randomness has been focused on reals in the arithmetical hierarchy. The only exception was a paper by Martin-Löf [3], in which he showed the intersection of a sequence of -sets of reals to be Σ (Sacks [9] introduced the notion of Π and -randomness in two exercises). The first systematic study of higher randomness appeared in Hjorth and Nies [0] where the notion of Π -Martin-Löf randomness was defined and the key properties investigated. The paper also studied the stronger notion of Π -randomness and showed the existence of a universal test for Π -random reals. In Chong, Nies and Yu [2] the authors examined the relative strengths of Π -Martin-Löf randomness, Π and -randomness, as well as their associated notions of lowness. Effective descriptive set theory offers a natural and different platform for the study of algorithmic randomness. Since the Gandy-Spector Theorem injects a new perspective to Π -sets of natural numbers, viewing them as Σ -definable subsets of L ω CK and therefore recursively enumerable (r.e.) in the larger universe, the tools of hyperarithmetic theory are readily available for the investigation of random reals in the generalized setting. Just as arithmetical randomness has drawn new insights into the structure of Turing degrees below 0 (n) (for n < ω), the study of higher randomness properties has enhanced our understanding of hyperdegrees and Π -sets of reals, a point which we hope results presented in this paper will convey. We consider several basic notions of randomness (see the next section for the definitions). In [2] it was shown that Π -Martin-Löf randomness, Π and -randomness 200 Mathematics Subject Classification. 03D30,03D32,68Q30. We thank the referee for helpful comments and suggestions, as well as for a very careful reading of the manuscript. We also thank André Nies for useful discussions and comments. Most of the results in the paper were presented in Logic blog 202 maintained by Nies. Yu s research was partially supported by National Natural Science Fund of China, No. 074 and Chong s research was partially supported by NUS Grant C

2 2 C. T. CHONG AND LIANG YU are equivalent for reals x if and only if ω x = ω CK. In [5], Nies introduced another notion called strong Π -Martin- Löf randomness which is an analog of weak 2-randomness in the literature. We prove (Theorem 3.5) that every hyperdegree greater than or equal to the hyperdegree of Kleene s O contains a real that is Π but not strongly Π - Martin-Löf random, thus separating these two notions of randomness. In Theorem 4.4, we show that every nontrivial hyperdegree below the hyperdegree of a Π -random real contains a Π -random real. Such a downward closure property is not shared by weaker notions such as Π -Martin-Löf randomness. In fact, every nontrivial real below a Π -random is Π -random relative to a measure (Corollary 4.5), so that such reals are still essentially random. This result is strengthened in Theorem 5.: We characterize the class NCR Π of reals x which are not Π -random relative to any representation of a continuous measure to be precisely those which satisfy x L ω x. Our final result (Theorem 6.5) separates the notion of low for -randomness from that of low for Π -randomness. To obtain this, we prove a general theorem about hyperdegrees (Theorem 6.3) which states that any two uncountable Σ -set of reals generate the cone of hyperdegrees with base the hyperdegree of Kleene s O. The latter has its root in a result of Martin [2] that every uncountable -set of reals contains a member of each hyperdegrees greater than or equal to the degree of O. The paper concludes with a list of questions. 2. Preliminaries We assume that the reader is familiar with hyperarithmetic theory and randomness theory. For a general reference, refer to [6], [5], [9] or [3]. The notations adopted are standard. Reals are denoted x, y, z,... A tree T is a subset of 2 <ω or ω <ω. [T ] denotes the set of infinite paths on T. By abuse of notation, we also write x T (or x U for U 2 <ω ) if the context is clear. We use k n to express the fact that the number k is much bigger than n. If λ is a measure on the Cantor space 2 ω, and σ 2 <ω, denote λ(σ) to be the measure of λ on the basic open set {x σ x}. We also let [σ] denote the set of binary strings extending σ. Definition 2.. Given a measure λ on 2 ω, a real ˆλ represents λ if for any σ 2 <ω and rational numbers p, q, σ, p, q ˆλ p < λ(σ) < q. Given a representation ˆλ of a measure λ, one may define the notion of a ˆλ-Martin- Löf test as usual. More details can be found in [6]. We use µ to denote the Lebesgue measure throughout this paper. Definition 2.2. (i) A Π -ML-test is a sequence {U m } m ω of uniformly Π -open sets such that m(µ(u m ) < 2 m ). We say that x is Π -ML random if x / m U m for every such collection {U m }, i.e. if x passes every Π -ML test. (ii) ([5, Problem 9.2.7]) {U m } is a Π -generalized ML test if {U m } is a sequence of uniformly Π -open sets and lim m µ(u m ) = 0. We say that x is strongly Π -MLrandom if x passes every generalized Π -ML-test. Definition 2.2 (ii) is an analog of the notion of weak-2-randomness for reals, where Π is replaced by r.e. One may refine Definition 2.2 (i) as follows. A -ML-test is

3 RANDOMNESS IN THE HIGHER SETTING 3 obtained when Π in the definition is replaced by. Indeed, if {U n } n ω is a - ML-test, then there is a recursive ordinal α such that {U n } n ω is uniformly (α) -r.e. We call such a test a (α) -ML-test. A real x is -ML-random if it passes every - ML-test. If x is not -ML-random, then there is an α < ω CK and an (α) -ML-test in which x fails. This fact will be used in Section 4. Definition 2.3. (Hjorth and Nies in [0]) A real x is Π -random if it does not belong to any null Π -set of reals. Clearly m ω U m is Π for any uniformly Π -collection of Π -open sets, so that Π - randomness implies strong Π -ML-randomness. We say that a real x is -dominated if every function hyperarithmetic in x is dominated by a hyperarithmetic function. As usual, ω x is the least ordinal which is not an x-recursive ordinal, and Church-Kleene ω is ω which is always denoted ω CK. By a result in [2], we have the following proposition. Proposition 2.4 (Chong, Nies and Yu). If ω x = ω CK, then x is Π -ML-random if and only if it is Π -random. Moreover, each Π -random real is -dominated. The Gandy Basis Theorem plays an important role in our present study: Theorem 2.5 (Gandy [8]). If A 2 ω is a nonempty Σ -set, then there is an x A such that ω x = ω CK. Let L α be the Gödel constructibility hierarchy at level α. The following is a settheoretic characterization of Π -sets. Theorem 2.6 (Spector[2], Gandy [9]). A set A 2 ω is Π if and only if there is a Σ -formula ϕ such that x A L ω x [x] = ϕ. We use h to denote hyperarithmetic reduction. A(ω CK, x) is the structure for the ramified analytical hierarchy relative to x. For more details concerning the ramified analytical hierarchy, see [9]. If T is a tree that is Π (L ω CK )-definable, then there is an effective enumeration over L ω CK of the nodes not in T. For any γ < ω CK, let T [γ] be the -tree which is an approximation of T at stage γ. Then T = γ<ω T [γ]. CK In Nies [5], Problem asks 3. Strong Π -ML-randomness Question 3.. Is strong Π -ML-randomness equivalent to Π -ML-randomness? The question was motivated by the following consideration. In the standard argument separating weak 2-randomness from ML-randomness, one exploits the fact that the rate of convergence of µ(u n ) to 0 can be coded by the size of the space available to U n, where {U n } n ω is a test designed to exhibit the separation (the technical details can be found in [5]). Such an approach is no longer possible in the present setting, since U n is now enumerated in ω CK, instead of ω, -many stages. The following result leads to a negative solution.

4 4 C. T. CHONG AND LIANG YU Theorem 3.2. If x is the leftmost path of a Σ -closed set of reals, then x is not strongly Π -ML-random. The proof is measure-theoretic. More than separating the two notions of randomness, a measure-theoretic proof extracts useful information about the distribution of strong Π -ML random reals in the hyperdegrees. We first give a criterion for a uniformly Π -sequence of open sets to be a generalized Π -Martin-Löf test. This lemma will also be applied to show Theorem 3.5. Lemma 3.3. Suppose that {U n } n ω is a uniformly Π -sequence of open sets. If there is a Σ (L ω CK ) enumeration {Ûn,γ} n<ω,γ<ω CK of the sequence with two numbers k and m such that for every n, U n = γ<ω Û CK n,γ and for every γ < ω CK : (a) Ûn+,γ Ûn,γ and each string in Ûn has length at least k n, (b) σ 2 k n m (µ(ûn,γ [σ]) < 2 +m k n ), and (c) For γ < ω CK and any real z, if z Ûn,<γ \ Ûn,γ, where Ûn,<γ = β<γ Ûn,β, then z Ûn,β for any β γ. Then {U n } n ω is a generalized Π -ML-test. Proof. Note that by (c) the enumeration {Ûn,γ} of U n is not cumulative. Assume µ( n ω U n) > 0 for a contradiction. We will exhibit an infinite descending sequence of ordinals {γ n } n<ω for a contradiction. First of all, by the Lebesgue Density Theorem, the assumption implies that there is a σ 0 such that µ( n ω U n [σ 0 ]) > 2 σ 0 ( 2 3 ) = σ 0. Moreover, we may assume that k divides σ 0 + m. Let n 0 = σ 0 +m. Then there is a k least γ 0 < ω CK such that By (b), µ(ûn 0, γ 0 [σ 0 ]) > σ 0. µ((ûn 0,<γ 0 \ Ûn 0,γ 0 ) [σ 0 ]) > 2 σ 0 ( ) = σ 0. By (a) and (c), the set of strings which appear after the ordinal γ 0 is contained in the complement of (Ûn 0,<γ 0 \ Ûn 0,γ 0 ) [σ 0 ], and so µ( n>n 0 Û n,<γ0 [σ 0 ]) > ( ) 2 σ0 = 4 2 σ0. Hence there is a σ σ 0 such that µ( n>n 0 Û n,<γ0 [σ ]) > σ0. Bienvenu, Greenberg and Monin [] have a shorter proof of this theorem.

5 RANDOMNESS IN THE HIGHER SETTING 5 We may assume that k divides σ + m and σ σ 0. Let n = σ +m k n 0. Then there is a least γ < γ 0 such that µ(ûn, γ [σ ]) > σ. Repeating the argument, we obtain an infinite descending sequence γ 0 > γ >, which is not possible. Proof. (of Theorem 3.2). Let T 2 <ω be a Σ -tree. For any n < ω and γ < ω CK, let and Define Û n,γ = {z n + z is the leftmost path in T [γ]}. Û n,<γ = β<γ Û n,β U n = The following facts are immediate. γ<ω CK Û n,γ. () For any n and γ < ω CK, Ûn+,γ Ûn,γ and every string in Ûn has length at least n; (2) σ 2 n (µ(ûn,γ [σ]) 2 n < 2 n ); (3) For any n, γ < ω CK and real z, if z Ûn,<γ \ Ûn,γ, then z Ûn,β for any β γ. Clearly {U n } n ω is uniformly Π. By () (3) and setting k = m = in Lemma 3.3, i {U n } n ω is a generalized Π -ML-test. Obviously x n ω U n. We conclude that x is not strongly Π -ML-random. Corollary 3.4. Π -ML-randomness is strictly weaker than strong Π -ML-randomness. Proof. By a result in [0], there is a Σ -tree T such that [T ] is not empty and consists entirely of Π -ML-random reals. According to Theorem 3.2, the leftmost path in T is not strongly Π -ML-random. We give another application of Lemma 3.3. The following theorem may be proved by combining results in [] and [0]. We give a direct proof here. Theorem 3.5. For any real x h O, there is a Π -ML-random y h x which is not strongly Π -ML-random. Proof. Given a tree T, let T (T ) be the smallest subtree of T such that T (T ), and For σ T (T ), let V σ = {ν ν σ ν = σ + 4 [ν] T is infinite}. If τ is the leftmost or rightmost string in V σ, then τ T (T ). Now let T 2 <ω be a Σ -tree of positive measure so that [T ] consists entirely of Π -ML-random reals. Note that T has no isolated infinite paths.

6 6 C. T. CHONG AND LIANG YU and For any γ < ω CK, let Define Û n,γ = σ T (T [γ]) σ =4n+4 Û n,<γ = β<γ Û n,β U n = The following facts are immediate. γ<ω CK ([σ] T (T [γ])). Û n,γ. () For any n and γ < ω CK, Ûn+,γ Ûn,γ and every string in Ûn,γ has length at least 4n (in fact 4n + 4); (2) σ 2 4n (µ(ûn,γ [σ]) 2 2 4n 4 < 2 4n ); (3) For any n, γ < ω CK and real z, if z Ûn,<γ \ Ûn,γ, then z Ûn,β for any β γ. By () (3) and Lemma 3.3 by setting k = 4 and m = 0, {U n } n<ω is a generalized Π -ML-test. It is obvious that n ω U n contains a perfect subset of [T ]. Furthermore, O hyperarithmetically computes a perfect tree S with [S] n ω U n so that no path in S is strongly Π -ML-random. Hence no path in S is Π -random and by Proposition 2.4, any y [S] satisfies ω y > ω CK and so O h y. Such a y exists in every hyperdegree above the degree of O. Theorem 3.5 is proved. 4. Hyperdegrees of Π -random reals While the hyperdegrees of -random reals cover the cone of hyperdegrees above the hyperjump, it is not difficult to see that the situation is quite different outside this cone: Proposition 4.. If x is -random and ω x = ω CK, then there is a real y h x with ω y = ω CK whose hyperdegree contains no -random real. Proof. Suppose that x is -random and ω x = ω CK. Let H(x) = {y y T x f T y g h x(g is dominated by f)}. By Theorem 2.6, H(x) is Σ (x). Since O x H(x), H(x) is not empty. Relativizing Gandy s Basis Theorem 2.5 to x, there is a real y H(x) with ω y = ω x = ω CK. Thus y is not -dominated and so by Proposition 2.4, no real z h y is -random. By contrast, the hyperdegrees of Π -random reals are downward closed. Lemma If x is Π -random and y h x, then there is a recursive ordinal γ such that y T x (γ). 2 The lemma was also proved by Bienvenu, Greenberg and Monin [] independently.

7 RANDOMNESS IN THE HIGHER SETTING 7 Proof. Suppose that x is Π -random and y h x. Then ω x formula ϕ(ẋ, n) with rank α 0 < ω CK such that n y A(ω CK, x) = ϕ(x, n). = ω CK and there is a Recall that for a ranked sentence ψ, the relation µ({z A(ω CK, z) = ψ) > 0 is Π (Theorem.3.IV of [9]). Hence by the admissibility of ω CK, there is a recursive ordinal β > α 0 such that A α0 = { ψ ψ has rank at most α 0 µ({z A(ω CK, z) = ψ}) > 0} is recursive in (β). Then there is a recursive α β such that for any natural number i and formula ψ of rank at most β, there is a formula ψ of rank at most α such that {z A(ω CK, z) = ψ } is a Π 0 ( (α) )-subset of {z A(ω CK, z) = ψ} and the difference in measure between these two sets is less than 2 i. Repeating this, we obtain a -definable ω-sequence of ordinals α 0 < α < in L ω CK whose supremum γ = i<ω α i satisfies the following two properties: for any β < γ, (i) The set A β = { ϕ ϕ has rank at most β µ({z A(ω CK, z) = ϕ}) > 0} is recursive in (γ) ; and (ii) For any natural number i and formula ψ with rank at most β, there is a formula ψ of rank less than γ such that for some β < γ, {z A(ω CK, z) = ψ } is a Π 0 ( (β ) )-subset of {z A(ω CK, z) = ψ} and the difference in measure between these two sets is less than 2 i. Note that by Π -randomness, for any ranked formula ψ, if x P ψ = {z A(ω CK, z) = ψ}, then P ψ has positive measure. By Proposition 2.4, x is -dominated and so there is a hyperarithmetic function f : ω ω such that for any n O with n < γ and any e for which Φ ( n ) e computes a tree T e,n, if x [T e,n ], then x f( e, n ) T e,n. This allows us to implement the following construction. Recursively in x (γ) f, first find a ψ 0 with rank less than γ such that P 0 = {z A(ω CK, z) = ψ 0 } contains x, has positive measure, and is a closed subset of either {z A(ω CK, z) = ϕ(z, 0)} or {z A(ω CK, z) = ϕ(z, 0)}. Since x is Π -random, by (ii), such a ψ 0 exists. Note that x (γ) f is able to decide if x P 0. In general, for any n recursively in x (γ) f choose the formula ψ n+ with rank less than γ such that P n+ = {z A(ω CK, z) = ψ n+ } contains x, has positive measure, and is a closed subset of either P n {z A(ω CK, z) = ϕ(z, n)} or P n {z A(ω CK, z) = ϕ(z, n)}. Since x is Π -random, by (ii) there is such a ψ n+. Thus y T x (γ) f. Without loss of generality, we may assume that f T (γ). Then y T x (γ). Corollary 4.3. For any Π -random x and y h x, there is a recursive ordinal α, a function f T (α) and an oracle function Φ such that for every n, y(n) = Φ x (α) f(n) (n)[f(n)]. In other words, x (α) Turing computes y via the function Φ with both use and time bounded by f.

8 8 C. T. CHONG AND LIANG YU Proof. Suppose that x is Π -random and y h x. By Lemma 4.2, there is a recursive ordinal γ and an oracle function Φ such that for every n, y(n) = Φ x (γ) (n). Let g < h x such that for every n, y(n) = Φ x (γ) g(n) (n)[g(n)]. Since x is -dominated, there is a hyperarithmetic h such that for all n, h(n) > g(n). Hence there is a recursive ordinal α γ such that h is many-one reducible to (α). Then it is not difficult to define an f T (α) and an oracle function Ψ such that for every n, y(n) = Ψ x (α) f(n) (n)[f(n)]. Theorem 4.4. If x is Π -random and < h y h x, then there is a Π -random z h y. Proof. Suppose that x is Π -random and y h x is not hyperarithmetic. Then there is a recursive ordinal α, a nondecreasing function f T (α) and an oracle function Ψ such that lim n f(n) = and for every n, y(n) = Ψ x (α) f(n) (n)[f(n)]. We use a technique which is essentially due to Demuth [4]. For any τ 2 <ω, let C(τ) = {σ σ 2 f( τ ) Ψ σ (α) f( τ ) [f( τ )] τ}. In other words, C(τ) is the clopen set of strings of length f( u ), which output extensions of τ via Ψ. For strings τ and u, let τ < l u mean τ is to the left of u. Define (α) -recursive functions: l(u) = ( 2 σ ) and τ 2 u τ< l u σ C(τ) r(u) = l(u) + σ C(u) 2 σ. Intuitively l(u) is the measure of the set of strings of length f( u ) which output strings at the left of u and r(u) is the l(u) plus the measure of the set of strings of length f( u ) which outputs extensions of u. One may view σ C(τ) 2 σ as a measure of τ, see Demuth [4]. For each n, let l n = l(y n), and r n = r(y n). Then l n l n+ r n+ r n for every n. Since y is not hyperarithmetic, then by Sacks s result in [8] that the hyperarithmetic degree upper cone of any nonhyperarithmetic real is null, we have that lim n r n = 0. Hence there is a unique real z = n ω(l n, r n ). Obviously z T y (α). To prove z h y, note that for any n, (α) -recursively find a string u of length n such that z lies in the interval (l(u), r(u)). Then it must be the case that u = y n. So y T z (α). And thus z h y. We claim that z is -random.

9 RANDOMNESS IN THE HIGHER SETTING 9 Suppose otherwise. Then there is a recursive ordinal β < ω CK and a (β) -ML-test {V n } n ω such that z n ω V n. Let ˆV n = {u ν k(ν is the k-th string in V n p, q Q(p < l(u) < r(u) < q [ν] (p, q) q p < 2 n k ν ))}. Since z V n, we have y ˆV n for every n. Note that { ˆV n } n ω is (β++α) -r.e. 3 Let U n = {σ τ ˆV n ( σ = f( τ ) Φ σ (α) f( τ ) [f( τ )] τ)} = u ˆV n C(u). Then {U n } n ω is (β++α) -r.e and x n ω U n. Note that for every n, µ(u n ) 2 τ = r(u) l(u) µ(v n )+ 2 n k 2+ < 2 n +2 n = 2 n+. u ˆVn τ C(u) u ˆV n k ω Then {U n+ } n ω is a (β++α) -ML-test. So x is not a -random, a contradiction. If we define λ(τ) = σ C(τ) 2 σ, then λ can be viewed as a continuous measure over 2 ω. Then the following is an immediate consequence of the proof of Theorem 4.4: Corollary 4.5. For any Π -random x, if < h y h x then y is Π -random relative to the hyperarithmetic continuous measure λ. A further result on continuous measures is discussed in Theorem 5.. Remark: Together with the argument used in the proof of Proposition 2.7 in [6], one can show that the converse of the Corollary 4.5 is also true. In other words, for any x, x is hyperarithmetically equivalent to a Π -random real if and only if x is Π -random relative to the hyperarithmetic continuous measure λ. 5. On NCR Π This section is inspired by the work of Reimann and Slaman in [6] and [7], where they investigated reals not Martin-Löf random relative to any continuous measure. They prove that NCR, the collection of such reals, is countable. In fact their proof shows that for any recursive ordinal α, the collection NCR α of reals not (α) -MLrandom relative to any continuous measure is countable. Hence a natural question to ask is how far the countability property extends. We set an upper limit for this by proving Theorem 5.. Given a representation ˆλ of a measure λ over 2 ω, define a real x to be Π -random relative to ˆλ if it does not belong to a λ-null set which is Π (ˆλ). Define 3 If let λ(τ) = σ C(τ) 2 σ, then λ can be viewed as a measure over 2 ω. Then λ( ˆV n+ ) = u ˆV n+ σ C(u) 2 σ µ(v n+ ) + k ω 2 n k 2+ 2 n + 2 n 2 n. In other words, { ˆV n } n is a (β) -ML-test relative to λ.

10 0 C. T. CHONG AND LIANG YU NCR Π = {x x is not Π -random relative to any representation ˆλ of a continuous measure}. Let C = {x 2 ω x L ω x }. It is known that C is the largest Π -thin set. Theorem 5.. NCR Π = C. We decompose the proof of Theorem 5. into a sequence of lemmas. Lemma 5.2. NCR Π does not contain a perfect subset. Proof. The proof is essentially due to Reimann and Slaman [6]. Suppose that there is a perfect tree T 2 <ω such that every member of [T ] is NCR Π. Define a measure λ as follows: λ( ) =, and { λ([σ]) If σ λ([σ i]) = ( i) T ; λ([σ]) Otherwise. 2 Then λ is a continuous measure so that λ([t ]) =. Π -random relative to any representation ˆλ of λ. Lemma 5.3. NCR Π is a thin Π -set, and hence NCR Π C. Thus [T ] must contain a Proof. By Lemma 5.2, NCR Π does not contain a perfect subset. Relative to any representation ˆλ of a continuous measure λ, we may perform the same proofs as in [8] so that all the results remain valid upon replacing Lesbegue measure µ by ˆλ. Then the set {z ω z ˆλ > ωˆλ } is Π (ˆλ) and λ-null. Hence as in [2], there is a Π set Q (2 ω ) 2 such that for each real ˆλ representing a continuous measure, the set Qˆλ = {y (ˆλ, y) Q} is the largest Π (ˆλ) λ-null set. Then, as in Reimann and Slaman [7], z NCR Π ˆλ(ˆλ represents a continuous measure z Qˆλ). Thus NCR Π is Π. Lemma 5.4. If x L ω x and z h x, then z x h O z. Proof. Suppose that x L ω x and z h x. Then ω z < ω. x So ω x z > ω. z Thus z x h O z. Lemma 5.5. If x C, then x NCR Π. Proof. Let λ be a continuous measure with representation ˆλ and x C. If x h ˆλ, then x is clearly not Π -random relative to ˆλ. Otherwise, by Lemma 5.4, x ˆλ h Oˆλ. But {z z ˆλ Oˆλ} is a Π (ˆλ) λ-null set. This implies that x is not Π -random relative to ˆλ.

11 RANDOMNESS IN THE HIGHER SETTING 6. Separating lowness for higher randomness notions In [2], Chong, Nies and Yu investigated lowness properties for and Π -randomness. It is unknown whether there is a nonhyperarithmetic real low for Π -random. However, there is a characterization of reals which are low for Π -randomness. Proposition 6. (Harrington, Nies and Slaman [2]). Being low for Π -randomness is equivalent to being low for -randomness and not cuppable above O by a Π -random. We may apply Proposition 6. to separate lowness for -randomness from lowness for Π -randomness. Recall that given a class of sets of reals Γ, a real x is Γ-Kurtz random if it does not belong to any Γ-closed null set. In [], Kjos-Hanssen, Nies, Stephan and Yu investigated lowness for -Kurtz randomness and lowness for Π -Kurtz randomness. They proved that lowness for Π - Kurtz randomness implies lowness for -randomness. We show that the implication cannot be reversed. In [22], Yu gave a new proof of the following theorem. Theorem 6.2 (Martin [2] and Friedman). Every Σ -tree T with uncountably many infinite paths has a member of each hyperdegree h O as a path. We apply the technique introduced in [22] to prove the following result. Theorem 6.3. Let A 0 and A be uncountable Σ -sets of reals. For any z h O, there are reals x 0 A 0 and x A such that x 0 x h z. Proof. Fix a real z h O and two uncountable Σ -sets A 0 and A. Then there are two recursive trees T 0, T 2 <ω ω <ω such that for i, A i = {x f n(x n, f n) T i }. We may assume that neither A 0 nor A contains a hyperarithmetic real. Also assume that if (σ, τ) T i, i, then σ = Dom(τ). Let T 2 ω <ω be recursive so that [T 2 ] is uncountable and does not contain a hyperarithmetic infinite path. Let f O be the leftmost path in T 2. Then f O h O. For i, let [T i ] = {(x, f) n((x n, f n) T i )}. Our plan is to define x i A i such that z h x 0 x. To this end, a procedure of coding z and f O into x 0 and decoding them from x 0 x will be introduced. Construction of x i will be carried out in L ω CK [z] on the recursive tree T i, hence hyperarithmetically in z (since z h O). Since A i is Σ, x i will code in the function f i which is the leftmost path in the second component of [T i ] serving as a witness to x i being in A i (i.e. (x i, f i ) [T i ] and for any f, if (x i, f) [T i ] then f(n) f i (n) for all n). Since A i has no hyperarithmetic member, for any (σ, τ) T i, if σ x and τ f for some (x, f) [T i ], then there exist incompatible extensions σ and σ of σ, and (possibly compatible) extensions τ, τ of τ, so that (σ, τ ) and (σ, τ ) both have extensions in [T i ]. This splitting property of [A i ] allows the coding of z, f and f O in x 0 and the coding of f 0 in x. More specifically, the branch to be selected by x 0 at a splitting node when z(s) is to be coded (at stage s + of the construction) will follow the value of z(s), so that a left turn is taken if z(s) = 0 and a right turn is taken if z(s) =. The coding at stage s + of τ,s, which denotes the initial segment of f defined at the end of stage s, is accomplished by taking τ,s (t)-many consecutive left turns at splitting nodes, for

12 2 C. T. CHONG AND LIANG YU each t Dom(τ,s ). The coding of f O (s) at stage s + of the construction is carried out by taking left turns at f O (s)-many consecutive splitting nodes. For the purpose of decoding, one has to delineate different types of action taken during the coding phase. Since τ,s is a finite function, the end of coding the value τ,s (t) and the beginning of coding the value τ,s (t+), for t < Dom(τ,s ), is separated by a right turn at the splitting node between the two codings (of course since the construction is executed stage by stage, one may assume that at the beginning of stage s +, the coding of τ,s is already completed. This means that at stage s +, one only needs to code the values τ,s (t) for t Dom(τ,s ) \ Dom(τ,s )). A right turn is chosen at the next splitting node to signify the end of coding τ,s, and the beginning of the coding of f O (s). Finally, a right turn is taken at the next splitting node after coding f O (s) to mark the end of the coding action for x 0 at stage s +. This initial segment of x 0 coded at stage s + is denoted σ 0,s+. Then τ 0,s+ f 0 is a finite string in ω <ω such that Dom(τ 0,s+ ) = σ 0,s+, τ 0,s+ extends τ 0,s, and is the leftmost such string (meaning taking a left turn at every splitting node during coding). The coding of the initial segment τ 0,s+ of f 0 in x at stage s +, denoted σ,s+, proceeds in a similar fashion. The definition of τ,s+, an initial segment of f constructed at stage s +, follows the same format. We now describe the construction of x 0 and x and the associated strings in detail. For i and (σ, τ) T i, let T i (σ, τ) be the set of strings in T i compatible with (σ, τ), i.e. T i (σ, τ) = {(σ, τ ) T i (σ, τ ) (σ, τ) (σ, τ ) (σ, τ)}. We say that a string (or node) σ 2 <ω is splitting over (σ, τ) in T i if σ σ and for j, T i,σ j(σ, τ) = {(σ, τ ) σ σ j τ τ (σ, τ ) T i } contains an infinite path. T i,σ j(σ, τ) is the subtree of T i with root (σ j, τ) (note that σ σ). Since A i has no hyperarithmetic path, for each j, there is a string that splits over some (σ, τ ) in T i,σ j(σ, τ). Note that σ does not lie on T i but some pair (σ, τ ) does and we call (σ, τ ) a splitting node on T i. For each i, we construct a sequence (σ i,0, τ i,0 ) (σ i,, τ i, ) on T i and let x i = j σ i,j. Again, the idea is to apply a mutual coding technique so that x 0 codes the leftmost witness function f = s τ.s (in the Σ -definition) for x and x codes the leftmost witness function f 0 = s τ 0,s for x 0. In addition, we also assign x 0 the responsibility of coding z as well as f O. More precisely, for each s ω we use σ 0,s to code z(s), f O (s) and τ,s, and use σ,s to code τ 0,s. At stage 0, let (σ i,0, τ i,0 ) = (, ) for i. Without loss of generality, assume that (, ) is a splitting node in both T 0 and T. The construction at stage s + proceeds as follows: Substage (i). First let σ be the shortest splitting node over (σ 0,s, τ 0,s ) in T 0. Thus T 0,σ j(σ 0,s, τ 0,s ) contains an infinite path for j. Let σ0,s+ 0 be the leftmost splitting node over (σ 0,s, τ 0,s ) extending σ z(s) in T 0. Thus z(s) is coded here. Next we code τ,s. Let n 0 s+ = Dom(τ,s ) Dom(τ,s ) (let Dom(τ,s ) = 0 if s = 0). Inductively, for any k [, n 0 s+], let σ0,s+ k be the leftmost splitting node over (σ 0,s, τ 0,s ) extending (σ0,s+) k in T 0 so that there are τ,s (k + Dom(τ,s ) )-many

13 RANDOMNESS IN THE HIGHER SETTING 3 splitting nodes over (σ 0,s, τ 0,s ) in T 0 between σ0,s+ k and σ0,s+. k This completes the coding of τ,s. To code f O (s), let σ n0 s+ + 0,s+ be the leftmost splitting node extending (σ n0 0,s+) over (σ 0,s, τ 0,s ) in T 0 so that there are f O (s)-many splitting nodes in T 0 over (σ 0,s, τ 0,s ) between σ n0 s+ 0,s+ and σ n0 s+ + 0,s+. For j, let σ n0 s+ ++j+ 0,s+ be the next splitting node in T 0 over (σ 0,s, τ 0,s ) extending (σ n0 s+ ++j 0,s+ ). This coding tells us that the action at this substage for the x 0 side is completed. Define σ 0,s+ = σ n0 s+ +3 0,s+. Let τ 0,s+ extend τ 0,s so that Dom(τ 0,s+ ) = σ 0,s+ and τ 0,s+ is the leftmost node such that the tree T 0,(σ0,s+,τ 0,s+ ) has an infinite path. Substage (ii). Let σ 0,s+ = σ,s and n s+ = Dom(τ 0,s+ ) Dom(τ 0,s ). Inductively for any k [, n s+], let σ k,s+ be the leftmost splitting node over (σ,s, τ,s ) extending (σ k,s+) so that in T there are τ 0,s+ (k + Dom(τ 0,s ) )-many splitting nodes over (σ,s, τ,s ) between σ,s+ k and σ,s+. k Hence τ 0,s+ is coded. For j, let σ n s+ +j+,s+ be the next splitting node over (σ,s, τ,s ) in T extending (σ n s+ +j,s+ ). This coding tells us that the action of coding τ 0,s+ at this substage for the x side is completed. Define σ,s+ = σ n s+ +2,s+. Thus we have coded τ 0,s+ into σ,s+. Let τ,s+ extend τ,s so that Dom(τ,s+ ) = σ,s+ and τ,s+ is the leftmost finite string such that the tree T,(σ,s+,τ,s+ ) has an infinite path. This ends the construction at stage s +. Let x i = s<ω σ i,s for i. Note that the construction is carried out over L ω CK +[z] since O L ω CK + and whether (σ, τ) T i has an infinite path extension in T i is decided by stage L ω CK +. As z h O we have ω z > ω CK and so z h x 0 x. Now we use x 0 and x to decode the coding construction and hence O and z. The decoding is achieved via a finite injury method quite similar to that used in [22] to prove Theorem 6.2. We point out the key steps and leave the details to the reader. Construct a sequence of ordinals {α s } s<ω which is -definable in L x ω 0 x [x 0 x ] so that lim s ω α s = ω CK, and use it as a parameter to decode the reals z and f 0, thereby concluding that x 0 x h z. As in [22], we may fix a Σ -enumeration {T i [α]} i 2,α<ω CK over L ω CK such that for i 2, T i [0] = T i T i [α] T i [β] if α β T i [ω CK ] = α<ω T CK i [α] T i [ω CK ] has no dead end nodes, and A i = {x f n(x n, f n) T i [ω CK ]}. Since [T i ] does not contain a hyperarithmetic infinite path, we have [T i [ω CK ]] = [T i ] to be a perfect tree (which as noted earlier enabled the coding procedure to be implemented). Clearly (x i, f i ) is an infinite path in T i [α] for each α. As was the case for T i, i, for each α < ω CK one may define the notion of a string σ being a splitting node over (σ, τ) in T i [α]. In particular, for (σ, τ) T i [α] such that σ x i, it makes sense to say that x i n splits over (σ, τ) in T i [α]. This means that there

14 4 C. T. CHONG AND LIANG YU is a τ τ such that (x i n.τ ) T i [α] and there are numbers n 0, n so that (x i n 0, τ n 0 ), (x i n, τ n ) T i [α]. In this case we also say that (x i n, τ ) is a splitting node over (σ, τ) in T i [α]. Note that this splitting is local in the sense that there is no guarantee that (x i n j, τ n j ) has an infinite extension in T i [α[ for j = 0,. Since for each i and α < ω CK, x i, f i is a path on T i [α], one may use x 0 x to approximate the values of f O (s) and f i (s) by simulating the construction of x i in T i [α]. This is achieved by attempting to retrieve the sequences {(σ i,s, τ i,s )} s ω, i, using x 0 x. Of course, since it is yet to be established that z and f O are hyperarithmetic in x 0 x, the retrieval is only by way of approximating the original construction. Using x 0 x, one may mimic the construction of {(σ i,s, τ i,s )} s ω to define {(σ i,s [α], τ i,s [α])} s ω, for i, so that σ i,s [α] is an initial segment of x i (in ascending order of length) and (σ i,s+ [α], τ i,s+ [α]) is a splitting node over (σ i,s [α], τ i,s [α]) in T i [α]. Furthermore, for each i, τ i,s [α] codes an approximation of f i s at stage α. We leave the details of defining (σ i,s, τ i,s ) to the reader. The algorithm we adopt for decoding, applying the idea in [22], proceeds as follows. Let f O,0 (0) = f O (0) and α 0 = 0. Suppose t > 0. We say that x 0 x computes f O correctly up to j at stage α if the approximation to f O via T 2 at stage α is consistent with that via {σ 0,s [α]} s ω up to the decoding of f O j. Note that at any stage α, x 0 x always computes f O correctly up to 0. Similarly, z is correctly computed up to j between α and α if the decoding of z j in {σ 0,s [α]} s ω agrees with that in {σ 0,s [α ]} s ω. Let j(t ) = Dom(f O,t ) where f O,t is correctly computed by x 0 x at stage α t. If z is correctly computed up to j(t ) between α t and α t +, let α t = α t +. Increase the computation of f O,s by one bit. To do this, let f O,t (j) = f O,t (j) for j j(t ) and let f O,t (j(t ) + ) be the value of f O (j(t ) + ) as decoded by {σ 0,s [α t ]} s ω. Otherwise, initialize the values of f O and z defined at step t which are incorrect. Let j j(t ) be the least j that is initialized. Search for a stage α > α t such that x 0 x correctly computes f O up to j at stage α. Note that such an α and j must exist (in the worst case, j = 0). This completes the construction at step t. Let γ = t ω α t. Clearly γ ω CK and is a limit ordinal. Furthermore γ < ω x 0 x. If γ = ω CK, then ω x 0 x > ω CK and this would imply that O h x 0 x and then z h x 0 x since the values of z j can be read off from {σ 0,s [ω CK ]} s ω by checking the (valid) splitting nodes in T 0 [ω CK ] along x 0. Otherwise, γ < ω CK. Since neither x 0 nor x is hyperarithmetic, as in [22] one argues that lim t f O,t (j) exists for every j. If not, choose the (least) j such that f O,t (j) f O,t+ (j) at infinitely many α t s. Then it must be the case that lim t f O,t (j) =. This implies that infinitely many left turns were selected at splitting nodes in T 0 [α t ] during the decoding phase to approximate f O (j), guided by σ 0,t [α t ] as t (recall that σ 0,t [α t ] mimics substage (i) at stage t in the construction

15 RANDOMNESS IN THE HIGHER SETTING 5 of x 0 ). Then there exist σ, τ so that x 0 is the leftmost real in the set X defined as {x σ n β < γ β τ τ(β β < γ Dom(τ ) = n (x n, τ ) T 0 [β ])}. Now X is a closed subset of 2 ω, implying that x 0 is hyperarithmetic which is a contradiction. Hence γ = ω CK. It remains for us to decode f i using the τ i,t [α t ] s. This follow the same approach discussed above and the detail is omitted. Thus for any n, there is a t n such that x 0 x correctly computes f O up to n at any stage α α tn. In other words, f O,tn (n) = f O,t (n) for t t n. Hence f O is the leftmost path of T 2 [γ]. Since [T 2 [γ]] = [T 2 ], we have lim t f O,t = f O. Hence O h f O h x 0 x. With this, we conclude that z h x 0 x and this completes the proof of the Theorem. Let F be the collection of all finite subsets of ω. A real x is -traceable if for any function f h x, there is a -function g : ω F such that for every n, g(n) = n and f(n) g(n). Lemma 6.4. There is an uncountable Σ -set A in which every member is - traceable. Proof. This is precisely what was proved in Theorem 4.7 of [20]. By [2] and [], each -traceable real is low for -randomness and hence low for -Kurtz randomness. By [0], the Π -random reals form a Σ -set. Then by Lemma 6.4 and Theorem 6.3, there is an x which is low for -randomness and x y h O for some Π -random y. So y is a Π (x)-singleton. We thus conclude: Theorem 6.5. Lowness for -randomness does not imply lowness for Π -randomness. And lowness for -Kurtz-randomness does not imply lowness for Π -Kurtz-randomness. Remark. Theorem 6.3 may be used to answer Question 58 in [7] and Question 3 in [20], whose solutions were announced by Friedman and Harrington but have remain unpublished. We end this paper with two problems. It is still unknown whether strong Π -ML-randomness coincides with Π -randomness. To separate these two notions, one way is to investigate the Borel ranks of different notions of randomness. Obviously the collection of Π 0 -ML-random reals is 3 and it 0 can be shown that it is not Π 3 (see Part 2, [23]). Moreover, it is not hard to see that the collection of Π Σ 0 0 -random reals is neither 2 nor 2. Its exact Borel rank remains unknown. We have the following conjecture. Σ Π Conjecture The collection of Π -random reals is not Π The conjecture was refuted by Monin [4] recently by showing that the collection of Π -random reals is proper Π 0 3.

16 6 C. T. CHONG AND LIANG YU Also the question whether lowness for Π -randomness coincides with hyperarithmeticity remains open. In view of Theorem 6., we have the following question. Question 6.7. Is it true that for any nonhyperarithmetic x and uncountable Σ -set A 2 ω, there is a y A such that x y h O? References [] Laurent Bienvenu, Noam Greenberg, and Benoit. Monin. Some higher randomness results. communication., 3, 2 [2] C. T. Chong, Andre Nies, and Liang Yu. Lowness of higher randomness notions. Israel J. Math., 66:39 60, 2008., 2, 5, 6, 6., 6 [3] Chong CT and Yu L. Recursion Theory: The Syntax and Structure of Definability. yuliang/course.pdf. 2 [4] Osvald Demuth. Remarks on the structure of tt-degrees based on constructive measure theory. Comment. Math. Univ. Carolin., 29(2): , [5] Rod Downey, Andre Nies, Rebecca Weber, and Liang Yu. Lowness and Π 0 2 nullsets. J. Symbolic Logic, 7(3): , [6] Rodney G. Downey and Denis R. Hirschfeldt. Algorithmic randomness and complexity. Theory and Applications of Computability. Springer, New York, [7] Harvey Friedman. One hundred and two problems in mathematical logic. J. Symbolic Logic, 40:3 29, [8] R. O. Gandy. On a problem of Kleene s. Bull. Amer. Math. Soc., 66:50 502, [9] R. O. Gandy. Proof of Mostowski s conjecture. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 8:57 575, [0] Greg Hjorth and André Nies. Randomness via effective descriptive set theory. J. Lond. Math. Soc. (2), 75(2): , 2007., 2.3, 3, 6 [] Bjørn Kjos-Hanssen, André Nies, Frank Stephan, and Liang Yu. Higher Kurtz randomness. Ann. Pure Appl. Logic, 6(0): , , 6 [2] Donald A. Martin. Proof of a conjecture of Friedman. Proc. Amer. Math. Soc., 55():29, 976., 6.2 [3] Per Martin-Löf. On the notion of randomness. In Intuitionism and Proof Theory (Proc. Conf., Buffalo, N.Y., 968), pages North-Holland, Amsterdam, 970. [4] Benoit. Monin. Higher randomness and forcing with closed sets. Stacs 204, to appear. 4 [5] André Nies. Computability and randomness, volume 5 of Oxford Logic Guides. Oxford University Press, Oxford, 2009., 2, 2.2, 3 [6] Jan Reimann and Theodore A. Slaman. Measures and their random reals , 4, 5, 5 [7] Jan Reimann and Theodore A. Slaman. Randomness for continuous measures , 5 [8] Gerald E. Sacks. Measure-theoretic uniformity in recursion theory and set theory. Trans. Amer. Math. Soc., 42:38 420, , 5 [9] Gerald E. Sacks. Higher recursion theory. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 990., 2, 2, 4 [20] Stephen G. Simpson. Minimal covers and hyperdegrees. Trans. Amer. Math. Soc., 209:45 64, , 6 [2] Clifford Spector. Hyperarithmetical quantifiers. Fund. Math., 48:33 320, 959/ [22] Liang Yu. A new proof of Friedman s conjecture. Bull. Symbolic Logic, 7(3):455 46, 20. 6, 6, 6 [23] Liang Yu. Higher randomness. Logic Blog,

17 RANDOMNESS IN THE HIGHER SETTING 7 Department of Mathematics, Faculty of Science, National University of Singapore, Singapore address: chongct@math.nus.eud.sg Institute of Mathematical Science, Nanjing University, Jiangsu Province 20093, P. R. of China address: yuliang.nju@gmail.com