We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the computably enumerable sets with trivial initial segment prefix-free complexity. A computably enumerable set $A$ is $K$-trivial if and only if the family of sets with complexity bounded by the complexity of $A$ is uniformly computable from the halting problem.
We study computably enumerable equivalence relations (ceers) on N and unravel a rich structural theory for a strong notion of reducibility among ceers.
We study the degree spectra and reverse-mathematical applications of computably enumerable and co-computably enumerable partial orders. We formulate versions of the chain/antichain principle and ascending/descending sequence principle for such orders, and show that the former is strictly stronger than the latter. We then show that every $\emptyset'$-computable structure (or even just of c.e.\ degree) has the same degree spectrum as some computably enumerable (co-c.e.)\ partial order, and hence that there is a c.e.\ (co-c.e.)\ partial order with spectrum equal to the set of nonzero degrees.
As suggested by the title, this paper is a survey of recent results and questions on the collection of computably enumerable sets under inclusion. This is not a broad survey but one focused on the author's and a few others' current research.
We show that there are Turing complete computably enumerable sets of arbitrarily low non-trivial initial segment prefix-free complexity. In particular, given any computably enumerable set $A$ with non-trivial prefix-free initial segment complexity, there exists a Turing complete computably enumerable set $B$ with complexity strictly less than the complexity of $A$. On the other hand it is known that sets with trivial initial segment prefix-free complexity are not Turing complete. Moreover we give a generalization of this result for any finite collection of computably enumerable sets $A_i, i<k$…
We prove that every computably enumerable (c.e.) random real is provable in Peano Arithmetic (PA) to be c.e. random. A major step in the proof is to show that the theorem stating that "a real is c.e. and random iff it is the halting probability of a universal prefix-free Turing machine" can be proven in PA. Our proof, which is simpler than the standard one, can also be used for the original theorem. Our positive result can be contrasted with the case of computable functions, where not every computable function is provably computable in PA, or even more interestingly, with the fact that almost …
Our focus will be on the computably enumerable (c.e.) sets and trivial, non-trivial, Friedberg, and non-Friedberg splits of the c.e. sets. Every non-computable set has a non-trivial Friedberg split. Moreover, this theorem is uniform. V. Yu. Shavrukov recently answered the question which c.e. sets have a non-trivial non-Friedberg splitting and we provide a different proof of his result. We end by showing there is no uniform splitting of all c.e. sets such that all non-computable sets are non-trivially split and, in addition, all sets with a non-trivial non-Friedberg split are split accordingly.
The goal of this paper is to show there is a single orbit of the c.e. sets with inclusion, $\mathcal{E}$, such that the question of membership in this orbit is $Σ^1_1$-complete. This result and proof have a number of nice corollaries: The Scott rank of $\mathcal{E}$ is $ω^{CK}_1+1; Not all orbits are elementarily definable; There is no arithmetic description of all orbits of $\mathcal{E}$; For all finite $α\geq 9$, there is a properly $Δ^0_α$ orbit (from the proof). April 6, 2007, minor changes Nov 20, 2007, minor changes
The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, $\E$, such that the question of membership in this orbit is $Σ^1_1$-complete. This result and proof have a number of nice corollaries: the Scott rank of $\E$ is $\wock +1$; not all orbits are elementarily definable; there is no arithmetic description of all orbits of $\E$; for all finite $α\geq 9$, there is a properly $Δ^0_α$ orbit (from the proof). A few small corrections made in this version
We investigate what collections of c.e.\ Turing degrees can be realised as the collection of elements of a separating $Π^0_1$ class of c.e.\ degree. We show that for every c.e.\ degree $\mathbf{c}$, the collection $\{\mathbf{c}, \mathbf{0}'\}$ can be thus realized. We also rule out several attempts at constructing separating classes realizing a unique c.e.\ degree. For example, we show that there is no \emph{super-maximal} pair: disjoint c.e.\ sets $A$ and $B$ whose separating class is infinite, but every separator of c.e.\ degree is a finite variant of either $A$ or $\overline{B}$.
We show that given any non-computable left-c.e. real $α$ there exists a left-c.e. real $β$ such that $α\neq β+γ$ for all left-c.e. reals and all right-c.e. reals $γ$. The proof is non-uniform, the dichotomy being whether the given real $α$ is Martin-Loef random or not. It follows that given any universal machine $U$, there is another universal machine $V$ such that the halting probability of U is not a translation of the halting probability of V by a left-c.e. real. We do not know if there is a uniform proof of this fact.
A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and Y does not compute X. A real X is relatively simple and above if there is a real Y <_T X such that X is c.e.(Y) and there is no infinite subset Z of the complement of X such that Z is c.e.(Y). We prove that every nonempty Pi^0_1 class contains a member which is not relatively c.e. and that every 1-generic real is relatively simple and above.
We study connections between classical asymptotic density and c.e. sets. We prove that a c.e. Turing degree d is not low if and only if d contains a c.e. set A of density 1 which has no computable subsets of density 1, giving a natural characterization of non-low c.e. degrees. In contrast, we prove that every nonzero c.e. degree contains a set which is generically computable but not coarsely computable. There is a very close connection between the computational complexity of a set and the computational complexity of its density as a real number where we measure complexity of real numbers as th…
We work with the structure consisting of all computably enumerable (c.e.) sets ordered by set inclusion. The question we will partially address is which c.e.\ sets are autormorphic to low (or low$_2$ sets. Using work of Miller, we can see that every set with semilow complement is $Δ^0_3$ automorphic to a low set. While it remains open whether every set with semilow complement is effectively automorphic to a low set, we show that there are sets without semilow complement that are effectively automorphic to low sets. We also consider other lowness notions such as having a semilow$_{1.5}$ complem…
The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of investigation in computable model theory. In this paper, we focus on the lattice structure of computably enumerable substructures of a given canonical computable structure. In particular, for a Turing degree $\mathbf{d}$, we investigate the groups of $\mathbf{d}$ -computable automorphisms of the lattice of $\mathbf{d}$-computably enumerable vector spaces, o…
We prove an algebraic extension theorem for the computably enumerable sets, $\mathcal{E}$. Using this extension theorem and other work we then show if $A$ and $\hat{A}$ are automorphic via $Ψ$ then they are automorphic via $Λ$ where $Λ\restriction Ł^*(A) = Ψ$ and $Λ\restriction \E^*(A)$ is $Δ^0_3$. We give an algebraic description of when an arbitrary set $\Ahat$ is in the orbit of a \ce set $A$. We construct the first example of a definable orbit which is not a $Δ^0_3$ orbit. We conclude with some results which restrict the ways one can increase the complexity of orbits. For example, we show …
We investigate conditions under which a co-computably enumerable set in a computable metric space is computable. Using higher-dimensional chains and spherical chains we prove that in each computable metric space which is locally computable each co-computably enumerable sphere is computable and each co-c.e. cell with co-c.e. boundary sphere is computable.
We investigate conditions under which a co-computably enumerable closed set in a computable metric space is computable and prove that in each locally computable computable metric space each co-computably enumerable compact manifold with computable boundary is computable. In fact, we examine the notion of a semi-computable compact set and we prove a more general result: in any computable metric space each semi-computable compact manifold with computable boundary is computable. In particular, each semi-computable compact (boundaryless) manifold is computable.
We consider the arithmetic complexity of index sets of uniformly computably enumerable families learnable under different learning criteria. We determine the exact complexity of these sets for the standard notions of finite learning, learning in the limit, behaviorally correct learning and anomalous learning in the limit. In proving the $Σ_5^0$-completeness result for behaviorally correct learning we prove a result of independent interest; if a uniformly computably enumerable family is not learnable, then for any computable learner there is a $Δ_2^0$ enumeration witnessing failure.
(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal co-computably enumerable equivalence relation. (3) Each c.e. truth-table degree contains the word problem of a finitely generated group of computable permutations.
Institute of Discrete Mathematics and Geometry, TU Wien（维也纳工业大学离散数学与几何研究所）；Department of Humanistic Research and Innovation, University of Bari（巴里大学人文研究与创新系）
