We study computably enumerable equivalence relations (ceers), under the reducibility $R \le S$ if there exists a computable function f such that $x\,R\,y$ if and only if $f\left( x \right)\,\,S\,f\left( y \right)$ , for every $x,y$ . We show that the degrees of ceers under the equivalence relation generated by $\le$ 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 $R\prime \le R$ , where $R\prime$ 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 ${\rm{\Sigma }}_3^0$ -complete (the former answering an open question of Gao and Gerdes).

WSPC Proceedings -9in x 6inBrendleBrookeNgNies˙Cichon˙recursion˙theory page 2 2 We present an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong. While this analogy was first studied explicitly by Rupprecht (Effective correspondents to cardinal characteristics in Cichoń's diagram, PhD thesis, University of Michigan, 2010), many prior results can be viewed from this perspective. After a comprehensive survey of the analogy for characteristics from Cichoń's diagram, we extend it to Kurtz randomness and the analogue of the Specker-Eda number.

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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 byHere, f is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and f must be computable. We show that there is a Π 0 1 -complete equivalence relation, but no Π 0 k -complete for k ≥ 2. We show that Σ 0 k preorders arising naturally in the above-mentioned areas are Σ 0 k -complete. This includes polynomial time m-reducibility on exponential time sets, which is Σ 0 2 , almost inclusion on r.e. sets, which is Σ 0 3 , and Turing reducibility on r.e. sets, which is Σ 0 4 . 859 860 EGOR IANOVSKI, RUSSELL MILLER, KENG MENG NG, AND ANDRÉ NIES so that we can actually learn something from the reduction. In our example, one should ask how hard it is to compute the dimension of a rational vector space. It is natural to restrict the question to computable vector spaces over Q (i.e., those where the vector addition is given as a Turing-computable function on the domain of the space). Yet even when its domain D E such vector spaces, computing the function which maps each one to its dimension requires a 0 -oracle, hence is not as simple as one might have hoped. (The reasons why 0 is required can be gleaned from [7] or [8].) 1.1. Effective reductions. Reductions are normally ranked by the ease of computing them. In the context of Borel theory, for instance, a large body of research is devoted to the study of Borel reductions (the standard book reference is [17]). Here, the domains D E and D F are the set 2 or some other standard Borel space, and a Borel reduction f is a reduction (from E to F , these being equivalence relations on 2 ) which, viewed as a function from 2 to 2 , is Borel. If such a reduction exists, one says that E is Borel reducible to F , and writes E ≤ B F . A stronger possible requirement is that f be continuous, in which case we have (of course) a continuous reduction. In case the reduction is given by a Turing functional from reals to reals, it is a (type-2) computable reduction.A further body of research is devoted to the study of the same question for equivalence relations E and F on , and reductions f : → between them which are computable. If such a reduction from E to F exists, we say that E is computably reducible to F , and write E ≤ c F , or often just E ≤ F . These reductions will be the focus of this paper. Computable reducibility on equivalence relations was perhaps first studied by Ershov [12] in a category theoretic setting.The main purpose of this paper is to investigate the complexity of equivalence relations under these reducibilities. In certain cases we will generalize from equivalence relations to preorders on . We restrict most of our discussion to relatively low levels of the hierarchy, usually to Π 0 n and Σ 0 n with n ≤ 4. One can focus more closely on very low levels: Such articles as [3,5,18], for instance, have dealt exclusively with Σ 0 1 equivalen...

Abstract. Consider a Martin-Löf random ∆ 0 2 set Z. We give lower bounds for the number of changes of Zs n for computable approximations of Z. We show that each nonempty Π 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 Zs n changes more than c2 n times.

We study computable Polish spaces and Polish groups up to homeomorphism. We prove a natural effective analogy of Stone duality, and we also develop an effective definability technique which works up to homeomorphism. As an application, we show that there is a $\Delta ^0_2$ Polish space not homeomorphic to a computable one. We apply our techniques to build, for any computable ordinal $\alpha $, an effectively closed set not homeomorphic to any $0^{(\alpha )}$-computable Polish space; this answers a question of Nies. We also prove analogous results for compact Polish groups and locally path-connected spaces.

We use ideas and machinery of effective algebra to investigate computable structures on the space C[0, 1] of continuous functions on the unit interval. We show that (C[0, 1], sup) has infinitely many computable structures non-equivalent up to a computable isometry. We also investigate if the usual operations on C[0, 1] are necessarily computable in every computable structure on C[0, 1]. Among other results, we show that there is a computable structure on C[0, 1] which computes + and the scalar multiplication, but does not compute the operation of pointwise multiplication of functions. Another unexpected result is that there exists more than one computable structure making C[0, 1] a computable Banach algebra. All our results have implications for the study of the number of computable structures on C[0, 1] in various commonly used signatures.

Abstract. 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 Π 0 1 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, 1 then almost all points in X recur in a set of positive measure:. 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 n (x) ∈ E for infinitely many n.