## Works by Bernard Anderson

8 found
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1. Degrees That Are Not Degrees of Categoricity.Bernard Anderson & Barbara Csima - 2016 - Notre Dame Journal of Formal Logic 57 (3):389-398.
A computable structure $\mathcal {A}$ is $\mathbf {x}$-computably categorical for some Turing degree $\mathbf {x}$ if for every computable structure $\mathcal {B}\cong\mathcal {A}$ there is an isomorphism $f:\mathcal {B}\to\mathcal {A}$ with $f\leq_{T}\mathbf {x}$. A degree $\mathbf {x}$ is a degree of categoricity if there is a computable structure $\mathcal {A}$ such that $\mathcal {A}$ is $\mathbf {x}$-computably categorical, and for all $\mathbf {y}$, if $\mathcal {A}$ is $\mathbf {y}$-computably categorical, then $\mathbf {x}\leq_{T}\mathbf {y}$. We construct a $\Sigma^{0}_{2}$ set whose degree (...)
We define the bounded jump of $A$ by $A^{b}=\{x\in \omega \mid \exists i\leq x[\varphi_{i}\downarrow \wedge\Phi_{x}^{A\upharpoonright \!\!\!\upharpoonright \varphi_{i}}\downarrow ]\}$ and let $A^{nb}$ denote the $n$th bounded jump. We demonstrate several properties of the bounded jump, including the fact that it is strictly increasing and order-preserving on the bounded Turing degrees. We show that the bounded jump is related to the Ershov hierarchy. Indeed, for $n\geq2$ we have $X\leq_{bT}\emptyset ^{nb}\iff X$ is $\omega^{n}$-c.e. $\iff X\leq_{1}\emptyset ^{nb}$, extending the classical result that $X\leq_{bT}\emptyset '\iff (...) Direct download (5 more) Export citation Bookmark 4 citations 3. Bounded low and high sets.Bernard A. Anderson, Barbara F. Csima & Karen M. Lange - 2017 - Archive for Mathematical Logic 56 (5-6):507-521. Anderson and Csima :245–264, 2014) defined a jump operator, the bounded jump, with respect to bounded Turing reducibility. They showed that the bounded jump is closely related to the Ershov hierarchy and that it satisfies an analogue of Shoenfield jump inversion. We show that there are high bounded low sets and low bounded high sets. Thus, the information coded in the bounded jump is quite different from that of the standard jump. We also consider whether the analogue of the Jump (...) No categories Direct download (2 more) Export citation Bookmark 2 citations 4. Automorphisms of the truth-table degrees are fixed on a cone.Bernard A. Anderson - 2009 - Journal of Symbolic Logic 74 (2):679-688. Let$D_{tt} $denote the set of truth-table degrees. A bijection π:$D_{tt} \to \,D_{tt} $is an automorphism if for all truth-table degrees x and y we have$ \leqslant _{tt} \,y\, \Leftrightarrow \,\pi (x)\, \leqslant _{tt} \,\pi (y)$. We say an automorphism π is fixed on a cone if there is a degree b such that for all$x \geqslant _{tt} b$we have π(x) = x. We first prove that for every 2-generic real X we have (...) Direct download (6 more) Export citation Bookmark 3 citations 5. No categories Export citation Bookmark 2 citations 6. (1 other version)Partitions of trees and $${{\sf ACA}^\prime_{0}}$$.Bernard A. Anderson & Jeffry L. Hirst - 2009 - Archive for Mathematical Logic 48 (3-4):227-230. We show that a version of Ramsey’s theorem for trees for arbitrary exponents is equivalent to the subsystem${{\sf ACA}^\prime_{0}}$of reverse mathematics. Direct download (3 more) Export citation Bookmark 7. Relatively computably enumerable reals.Bernard A. Anderson - 2011 - Archive for Mathematical Logic 50 (3-4):361-365. A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X \not\leq_T Y}$$\end{document}. A real X is relatively simple and above if there is a real Y (...) Direct download (4 more) Export citation Bookmark 1 citation 8. Reals n-Generic Relative to Some Perfect Tree.Bernard A. Anderson - 2008 - Journal of Symbolic Logic 73 (2):401 - 411. We say that a real X is n-generic relative to a perfect tree T if X is a path through T and for all$\Sigma _{n}^{0}(T)\$ sets S, there exists a number k such that either X|k ∈ S or for all σ ∈ T extending X|k we have σ ∉ S. A real X is n-generic relative to some perfect tree if there exists such a T. We first show that for every number n all but countably many reals (...)