A Borel equivalence relation on a Polish space is said to be countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in recursion theory are: recursive isomorphism, Turing equivalence, arithmetic equivalence, etc. There is a canonical hierarchy of complexity of countable Borel equivalence relations imposed by the notion of Borel reducibility. We will survey results and conjectures concerning the problem of identifying the place in this hierarchy of these equivalence relations from recursion theory and also discuss some of their implications.
We construct an increasing $ω$-sequence $(a_n)$ of Turing degrees which forms an initial segment of the Turing degrees, and such that each~$a_{n+1}$ is diagonally noncomputable relative to $a_n$. It follows that the~$\mathsf{DNR}$ principle of reverse mathematics does not imply the existence of Turing incomparable degrees.
Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of non-trivial automorphisms of the Turing degrees are indicated.
We discuss sufficiently fast-growing sequences of Turing degrees. The key result is that, assuming sufficient determinacy, if $ϕ$ is a formula with one free variable, and S and T are sufficiently fast-growing sequences of Turing degrees of length $ω_1$, then $ϕ(S) \iff ϕ(T)$. We also define degrees for subsets of $ω_1$ analogous to Turing degrees, and prove that under sufficient determinacy and CH, all sufficiently high degrees are also effectively indistinguishable.
Let $π$ be an automorphism of the Turing degrees induces by a homeomorphism $φ$ of the Cantor space $2^ω$ such that $φ$ preserves all Bernoulli measures. It is proved that $π$ must be trivial. In particular, a permutation of $ω$ can only induce the trivial automorphism of the Turing degrees.
We characterize the isomorphism types of principal ideals of the Turing degrees below 0' that are lattices as the lattices with a Sigma-0-3 presentation, by showing that each Sigma-0-3 presentable bounded upper semilattice is isomorphic to such a principal ideal. We get a similar result for the Turing degrees below any degree above 0".
The Turing degree of a real measures the computational difficulty of producing its binary expansion. Since Turing degrees are tailsets, it follows from Kolmogorov's 0-1 law that for any property which may or may not be satisfied by any given Turing degree, the satisfying class will either be of Lebesgue measure 0 or 1, so long as it is measurable. So either the \emph{typical} degree satisfies the property, or else the typical degree satisfies its negation. Further, there is then some level of randomness sufficient to ensure typicality in this regard. A similar analysis can be made in terms of …
The degree spectrum of a countable structure is the set of all Turing degrees of presentations of that structure. We show that every nonlow Turing degree lies in the spectrum of some differentially closed field (of characteristic 0, with a single derivation) whose spectrum does not contain the computable degree 0. Indeed, this is an equivalence, for we also show that every such field of low degree is isomorphic to a computable differential field. Relativizing the latter result and applying a theorem of Montalban, Soskova, and Soskov, we conclude that the spectra of countable differentially clo…
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 define the bounded jump of A by A^b = {x | Exists i <= x [phi_i (x) converges and Phi_x^[A|phi_i(x)](x) converges} and let A^[nb] denote the n-th bounded jump. We demonstrate several properties of the bounded jump, including that it is strictly increasing and order preserving on the bounded Turing (bT) degrees (also known as the weak truth-table degrees). We show that the bounded jump is related to the Ershov hierarchy. Indeed, for n > 1 we have X <=_[bT] 0^[nb] iff X is omega^n-c.e. iff X <=_1 0^[nb], extending the classical result that X <=_[bT] 0' iff X is omega-c.e. Finally, we prove th…
In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and order. Afterwards, we combine the results from our study of these sets as orders with a classical construction from Avraham to obtain a restriction about how non trivial automorphism of the Turing degrees (if they exist) interact with $1$-generic degrees.
The Turing degree spectrum of a countable structure $\mathcal{A}$ is the set of all Turing degrees of isomorphic copies of $\mathcal{A}$. The Turing degree of the isomorphism type of $\mathcal{A}$, if it exists, is the least Turing degree in its degree spectrum. We show there are countable fields, rings, and torsion-free abelian groups of arbitrary rank, whose isomorphism types have arbitrary Turing degrees. We also show that there are structures in each of these classes whose isomorphism types do not have Turing degrees.
We study functions from reals to reals which are uniformly degree-invariant from Turing-equivalence to many-one equivalence, and compare them "on a cone." We prove that they are in one-to-one correspondence with the Wadge degrees, which can be viewed as a refinement of the uniform Martin's conjecture for uniformly invariant functions from Turing- to Turing-equivalence. Our proof works in the general case of many-one degrees on $\mathcal{Q}^ω$ and Wadge degrees of functions $ω^ω\to\mathcal{Q}$ for any better quasi ordering $\mathcal{Q}$.
We initiate the computability-theoretic study of ringed spaces and schemes. In particular, we show that any Turing degree may occur as the least degree of an isomorphic copy of a structure of these kinds. We also show that these structures may fail to have a least degree.
We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees.
We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (\eg the Shore-Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Add…
The coarse similarity class $[A]$ of $A$ is the set of all $B$ whose symmetric difference with $A$ has asymptotic density 0. There is a natural metric $δ$ on the space $\mathcal{S}$ of coarse similarity classes defined by letting $δ([A],[B])$ be the upper density of the symmetric difference of $A$ and $B$. We study the resulting metric space, showing in particular that between any two distinct points there are continuum many geodesic paths. We also study subspaces of the form $\{[A] : A \in \mathcal U\}$ where $\mathcal U$ is closed under Turing equivalence, and show that there is a tight conn…
Given a finite lattice $L$ that can be embedded in the recursively enumerable (r.e.) Turing degrees $\mathcal{R}_{\mathrm{T}}$, it is not known how one can characterize the degrees $\mathbf{d}\in\mathcal{R}_{\mathrm{T}}$ below which $L$ can be bounded. Two important characterizations are of the $L_7$ and $M_3$ lattices, where the lattices are bounded below $\mathbf{d}$ if and only if $\mathbf{d}$ contains sets of ``\emph{fickleness}'' $>ω$ and $\geqω^ω$ respectively. We work towards finding a lattice that characterizes the levels above $ω^2$, the first non-trivial level after $ω$. We considere…
We consider a seemingly weaker form of $Δ^1_1$ Turing determinacy. Let $2 \leq ρ< ω_1^{\textrm{CK}}$, $\textrm{Weak-Turing-Det}_ρ(Δ^1_1)$ is the statement: Every $Δ^1_1$ set of reals cofinal in the Turing degrees contains two Turing distinct, $Δ^0_ρ$-equivalent reals. We show in $\textrm{ZF}^-$: $\textrm{Weak-Turing-Det}_ρ(Δ^1_1)$ implies: for every $ν< ω_1^{\textrm{CK}}$ there is a transitive model: $M \models \textrm{ZF}^- + \aleph_ν\textrm{ exists}$. As a corollary: If every cofinal $Δ^1_1$ set of Turing degrees contains both a degree and its jump, then for every $ν< ω_1^{\textrm{CK}}$, the…
We consider graph Turing machines, a model of parallel computation on a graph, in which each vertex is only capable of performing one of a finite number of operations. This model of computation is a natural generalization of several well-studied notions of computation, including ordinary Turing machines, cellular automata, and parallel graph dynamical systems. We analyze the power of computations that can take place in this model, both in terms of the degrees of computability of the functions that can be computed, and the time and space resources needed to carry out these computations. We furt…
We consider the question whether there is an infinitary analogue of the Church-Turing-thesis. To this end, we argue that there is an intuitive notion of transfinite computability and build a canonical model, called Idealized Agent Machines ($IAM$s) of this which will turn out to be equivalent in strength to the Ordinal Turing Machines defined by P. Koepke.
We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $\bmΔ^1_2$ in a satisfying way.
Generic computability has been studied in group theory and we now study it in the context of classical computability theory. A set A of natural numbers is generically computable if there is a partial computable function f whose domain has density 1 and which agrees with the characteristic function of A on its domain. A set A is coarsely computable if there is a computable set C such that the symmetric difference of A and C has density 0. We prove that there is a c.e. set which is generically computable but not coarsely computable and vice versa. We show that every nonzero Turing degree contain…
We prove that, for every theory $T$ which is given by an ${\mathcal L}_{ω_1,ω}$ sentence, $T$ has less than $2^{\aleph_0}$ many countable models if and only if we have that, for every $X\in 2^ω$ on a cone of Turing degrees, every $X$-hyperarithmetic model of $T$ has an $X$-computable copy. We also find a concrete description, relative to some oracle, of the Turing-degree spectra of all the models of a counterexample to Vaught's conjecture.
We investigate the complexity of isomorphisms of computable structures on cones in the Turing degrees. We show that, on a cone, every structure has a strong degree of categoricity, and that degree of categoricity is $\bf{0^{(α)}}$ for some $α$. To prove this, we extend Montalbán's $η$-system framework to deal with limit ordinals in a more general way. We also show that, for any fixed computable structure, there is an ordinal $α$ and a cone in the Turing degrees such that the exact complexity of computing an isomorphism between the given structure and another copy $\mathcal{B}$ in the cone is a…
A set $A$ is coarsely computable with density $r \in [0,1]$ if there is an algorithm for deciding membership in $A$ which always gives a (possibly incorrect) answer, and which gives a correct answer with density at least $r$. To any Turing degree $\mathbf{a}$ we can assign a value $Γ_T(\mathbf{a})$: the minimum, over all sets $A$ in $\mathbf{a}$, of the highest density at which $A$ is coarsely computable. The closer $Γ_T(\mathbf{a})$ is to $1$, the closer $\mathbf{a}$ is to being computable. Andrews, Cai, Diamondstone, Jockush, and Lempp noted that $Γ_T$ can take on the values $0$, $1/2$, and …
Recently we have introduced a new model of infinite computation by extending the operation of ordinary Turing machines into transfinite ordinal time. In this paper we will show that the infinite time Turing machine analogue of Post's problem, the question whether there are supertask degrees between 0 and the supertask jump 0^jump, has in a sense both positive and negative solutions. Namely, in the context of the reals there are no degrees between 0 and 0^jump, but in the context of SETS of reals, there are; indeed, there are incomparable semi-decidable supertask degrees. Both arguments employ …
We show that, from the topological point of view, 2-tape Büchi automata have the same accepting power as Turing machines equipped with a Büchi acceptance condition. The Borel and the Wadge hierarchies of the class RAT_omega of infinitary rational relations accepted by 2-tape Büchi automata are equal to the Borel and the Wadge hierarchies of omega-languages accepted by real-time Büchi 1-counter automata or by Büchi Turing machines. In particular, for every non-null recursive ordinal $α$, there exist some $Σ^0_α$-complete and some $Π^0_α$-complete infinitary rational relations. And the supremum …
