Keng Meng Ng

UNIVERSAL COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

We study computably enumerable equivalence relations (ceers), under the reducibility R ≤ S if there exists a computable function f such that x R y if and only if f(x) S f(y), for every x, y. We show that the degrees of ceers under the equivalence relation generated by ≤ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if R′ ≤ R, where R′ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are Σ3-complete (the former answering an open question of Gao and Gerdes).

On strongly jump traceable reals

Abstract In this paper we show that there is no minimal bound for jump traceability. In particular, there is no single order function such that strong jump traceability is equivalent to jump traceability for that order. The uniformity of the proof method allows us to adapt the technique to showing that the index set of the c.e. strongly jump traceables is Π 4 0 -complete.

Martin-Löf random points satisfy Birkhoff’s ergodic theorem for effectively closed sets

We show that if a point in a computable probability space X satisfies the ergodic recurrence property for a computable measure-preserving T : X → X with respect to effectively closed sets, then it also satisfies Birkhoff’s ergodic theorem for T with respect to effectively closed sets. As a corollary, every Martin-Löf random sequence in the Cantor space satisfies Birkhoff’s ergodic theorem for the shift operator with respect to Π1 classes. This answers a question of Hoyrup and Rojas. Several theorems in ergodic theory state that almost all points in a probability space behave in a regular fashion with respect to an ergodic transformation of the space. For example, if T : X → X is ergodic, then almost all points in X recur in a set of positive measure: Theorem 1 (See [5]). Let (X,μ) be a probability space, and let T : X → X be ergodic. For all E ⊆ X of positive measure, for almost all x ∈ X, T(x) ∈ E for infinitely many n. Recent investigations in the area of algorithmic randomness relate the hierarchy of notions of randomness to the satisfaction of computable instances of ergodic theorems. This has been inspired by Kučera’s classic result characterising MartinLöf randomness in the Cantor space. We reformulate Kučera’s result using the general terminology of [4]. Definition 2. Let (X,μ) be a probability space, and let T : X → X be a function. Let C be a collection of measurable subsets of X. A point x ∈ X is a Poincaré point for T with respect to C if for all E ∈ C of positive measure for infinitely many n, T(x) ∈ E. The Cantor space 2 is equipped with the fair-coin product measure λ. The shift operator σ on the Cantor space is the function σ(a0a1a2 . . . ) = a1a2 . . . . The shift operator is ergodic on (2, λ). Received by the editors July 20, 2010 and, in revised form, April 5, 2011 and April 8, 2011. 2010 Mathematics Subject Classification. Primary 03D22; Secondary 28D05, 37A30. The second author was partially supported by the Marsden Grant of New Zealand. The third author was supported by the National Science Foundation under grants DMS0945187 and DMS-0946325, the latter being part of a Focused Research Group in Algorithmic Randomness. 1Recall that if (X,μ) is a probability space, then a measurable map T : X → X is measure preserving if for all measurable A ⊆ X, μ ( T−1A ) = μ(A). We say that a measurable set A ⊆ X is invariant under a map T : X → X if T−1A = A (up to a null set). A measure-preserving map T : X → X is ergodic if every T -invariant measurable subset of X is either null or conull. c ©2012 American Mathematical Society Reverts to public domain 28 years from publication 3623 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3624 J.N.Y. FRANKLIN, N. GREENBERG, J.S. MILLER, AND K.M. NG Theorem 3 (Kučera [7]). A sequence R ∈ 2 is Martin-Löf random if and only if it is a Poincaré point for the shift operator with respect to the collection of effectively closed (i.e., Π1) subsets of 2 . Building on work of Bienvenu, Day, Mezhirov and Shen [2], Bienvenu, Hoyrup and Shen generalised Kučera’s result to arbitrary computable ergodic transformations of computable probability spaces. Theorem 4 (Bienvenu, Hoyrup and Shen [3]). Let (X,μ) be a computable probability space, and let T : X → X be a computable ergodic transformation. A point x ∈ X is Martin-Löf random if and only if it is a Poincaré point for T with respect to the collection of effectively closed subsets of X. One of the most fundamental regularity theorems is due to Birkhoff (see [5]). Birkhoff’s Ergodic Theorem. Let (X,μ) be a probability space, and let T : X → X be ergodic. Let f ∈ L(X). Then for almost all x ∈ X,

Bounded randomness

We introduce some new variations of the notions of being Martin-Lof random where the tests are all clopen sets. We explore how these randomness notions relate to classical randomness notions and to degrees of unsolvability.

Lowness for Demuth Randomness

We show that every real low for Demuth randomness is of hyperimmune-free degree.

Counting the changes of random Δ20 sets

Consider a Martin-Lof random Δ20 set Z. We give lower bounds for the number of changes of Zs ↑n for computable approximations of Z. We show that each nonempty Π10 class has a low member Z with a computable approximation that changes only o(2n) times. We prove that each superlow ML-random set already satisfies a stronger randomness notion called balanced randomness, which implies that for each computable approximation and each constant c, there are infinitely many n such that Zs ↑n changes more than c2n times.

COMPLEXITY OF EQUIVALENCE RELATIONS AND PREORDERS FROM COMPUTABILITY THEORY

We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations R,S, a componentwise reducibility is defined by RS () 9f 8x,y (xRy

Counting the Changes of Random {\Delta^0_2} Sets

f(x)Sf(y)). Here f is taken from a suitable class of effective functions. For us the re- lations will be on natural numbers, and f must be computable. We show that there is a � 0-complete equivalence relation, but no � 0-complete for k � 2. We show that � 0 preorders arising naturally in the above- mentioned areas are � 0-complete. This includes polynomial time m- reducibility on exponential time sets, which is � 0, almost inclusion on r.e. sets, which is � 0, and Turing reducibility on r.e. sets, which is � 0.

Effective Packing Dimension and Traceability

Consider a Martin-Lof random \({\Delta^0_2}\) set Z. We give lower bounds for the number of changes of \(Z_s \upharpoonright n\) for computable approximations of Z. We show that each nonempty \({\Pi^0_1}\) class has a low member Z with a computable approximation that changes only o(2 n ) times. We prove that each superlow ML-random set already satisfies a stronger randomness notion called balanced randomness, which implies that for each computable approximation and each constant c, there are infinitely many n such that \(Z_s\upharpoonright n\) changes more than c 2 n times.

Finitary Reducibility on Equivalence Relations

The concern of this paper is with effective packing dimension. This concept can be traced back to the work of Borel and Lebesgue who studied measure as a way of specifying the size of sets. Caratheodory later generalized Lebesgue measure to the n-dimensional Euclidean space, and this was taken further by Hausdorff [Hau19] who generalized the notion of s-dimensional measure to include non-integer values for s in any metric space. In the Cantor space with the clopen topology, this can be viewed as a scaling of the usual Lebesgue measure by a factor of s, in the sense of μs([σ]) = 2−s|σ|, where [σ] is the clopen set generated by σ, and 0 ≤ s ≤ 1. This gave rise to the concept of classical Hausdorff dimension, which provided a way of classifying different sets of measure zero, based on the intuition that not all null sets are created equal. There appeared many other related classical notions of fractional dimensions, such as box-counting dimension and packing dimension. The study of effective notions of randomness and their relationship with the Turing degrees was initiated by the early work of de Leeuw, Moore, Shannon and Shapiro [dLMSS56]. The effective versions of these various notions of fractional dimensions have been studied in connection with randomness. The best known examples of such effective notions are the effective Hausdorff, and effective packing dimensions. Hausdorff measure talks about covering the set by open balls from the exterior, while packing measure considers filling up a set from the interior. One can effectivize these two notions by looking at covering with Σ1 open sets in the Cantor space with s-measure. This work took a new direction when various authors Lutz [Lut90, Lut03], Staiger [Sta93], Mayordomo [May02], Artheya et al [AHLM04], and Reimann [Rei04] showed that there were simple characterizations of effective Hausdorff and packing dimensions using Kolmogorov complexity. Indeed, the effective Hausdorff dimension of a real A can be written as