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The Posner-Robinson Theorem states that for any reals $Z$ and $A$ such that $Z oplus 0 leq_mathrm{T} A$ and $0 <_mathrm{T} Z$, there exists $B$ such that $A equiv_mathrm{T} B equiv_mathrm{T} B oplus Z equiv_mathrm{T} B oplus 0$. Consequently, any nonzero Turing degree $operatorname{deg}_mathrm{T}(Z)$ is a Turing jump relative to some $B$. Here we prove the hyperarithmetical analog, based on an unpublished proof of Slaman, namely that for any reals $Z$ and $A$ such that $Z oplus mathcal{O} leq_mathrm{T} A$ and $0 <_mathrm{HYP} Z$, there exists $B$ such that $A equiv_mathrm{T} mathcal{O}^B equiv_mathrm{T} B oplus Z equiv_mathrm{T} B oplus mathcal{O}$. As an analogous consequence, any nonhyperarithmetical Turing degree $operatorname{deg}_mathrm{T}(Z)$ is a hyperjump relative to some $B$.
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We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the dissipation function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of Turing degrees. In particular, we show that every $Delta^0_2$-degree contains an NCR element.
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 that the analogue of Shoenfield inversion holds for the bounded jump on the bounded Turing degrees. That is, for every X such that 0^b <=_[bT] X <=_[bT] 0^[2b], there is a Y <=_[bT] 0^b such that Y^b =_[bT] X.
A computable structure A is x-computably categorical for some Turing degree x, if for every computable structure B isomorphic to A there is an isomorphism f:B -> A with f computable in x. A degree x is a degree of categoricity if there is a computable A such that A is x-computably categorical, and for all y, if A is y-computably categorical then y computes x. We construct a Sigma_2 set whose degree is not a degree of categoricity. We also demonstrate a large class of degrees that are not degrees of categoricity by showing that every degree of a set which is 2-generic relative to some perfect tree is not a degree of categoricity. Finally, we prove that every noncomputable hyperimmune-free degree is not a degree of categoricity.
Clift and Murfet (2019) introduced a naive Bayesian smooth relaxation of Turing machines motivated by work in differential linear logic; this was subsequently used to endow spaces of program codes of bounded length with a smooth manifold structure via the staged-pseudo universal Turing machine introduced by Clift, Murfet and Wallbridge (2021). In this paper, we give a general construction for simulating n-tape Turing machines on a single tape Turing machine such that the (naive Bayesian) uncertainty is propagated in an equivalent manner. This result suggests a stronger kind of equivalence between single tape and n-tape Turing machines than that established by classical results, however, the clarification of these implications is open to future work. We then construct a pseudo universal Turing machine which similarly preserves the propagation of uncertainty in its simulations, and observe that this gives rise to a particularly natural smooth relaxation of the space of programs.
We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the cototal enumeration degrees.
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