Jekyll2022-05-04T18:25:01+00:00https://drossegger.github.io/feed.xmlDino Rossegger’s homepage:triangular_ruler: Jekyll theme for building a personal site, blog, project documentation, or portfolio.Dino Rosseggerdino@math.berkeley.eduTuring degrees that compute HYP but can not compute Kleene’s _O_.2022-04-20T00:00:00+00:002022-04-20T00:00:00+00:00https://drossegger.github.io/degreescomputinghyp<script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script>
<p>Let \(HYP\) be the set of hyperarithmetic degrees, i.e. \(HYP=\{ \mathbf d:\exists (\alpha<\omega_1^{\mathrm{CK}}) \mathbf d< \mathbf 0^{(\alpha)}\}.\)
Jockusch and Simpson [1] showed that there is a degree \(\mathbf d\) that is minimal over \(HYP\) (i.e., if \(\mathbf c<\mathbf d\), then \(\mathbf c\in HYP\)) and such that \(\mathbf d^{(3)}=deg(\mathcal O)\) where \(\mathcal O\) is Kleene’s \(\mathcal O\), or, up to m-equivalence, the set of indices of computable well-orderings.</p>
<p>I recently thought about the Turing degree structure of the degrees that compute every hyperarithmetic sets and could not find much besides the results of Jockusch and Simpson. However, after talking to a few people, I realized that quite a bit is known. Here is an attempt to summarize a few things. I apologize for skipping a lot of the details but hope that anyone can figure those out with the references provided.</p>
<p><strong>Theorem 1.</strong> Let \(\mathbf d> HYP\). Then \(\mathbf d\) computes a copy of \(\omega_1^{\mathrm{CK}}\) if and only if \(\mathbf d^{(3)}\geq deg(\mathcal O)\).</p>
<p><em>Proof sketch.</em> Assume that \(\mathbf d^{(3)}\geq deg(\mathcal O)\). Then \(\mathbf d^{(3)}\) can compute a copy of \(\omega_1^{\mathrm{CK}}\). One can show that \(\mathbf d\) then computes a copy of \(\omega^2\cdot \omega_1^{\mathrm{CK}}\cong \omega_1^{\mathrm{CK}}\) (see Ash and Knight [Theorem 9.11, 2]). For the other direction assume that \(\mathbf d\) computes a copy of \(\omega_1^{\mathrm{CK}}\). We will show that \(d^{(3)}\) can compute the set of indices of computable well-orderings. One can show that if \(L\) is a well-order and \(o(L)\leq \omega^{\alpha}\), then \(L\) is uniformly \(\Delta^0_{2\alpha}\) categorical, i.e., for \(L\) computable, \(\mathbf 0^{(2\alpha)}\) (\(\mathbf 0^{(2\alpha-1)}\) if \(\alpha\) is finite) can compute an isomorvarphism between any two computable copies of \(L\), and this is uniform. Let \(C\) be \(\mathbf d\)’s copy of \(\omega_1^{\mathrm{CK}}\). All computable well-orders have order-type less than \(C\), so the set of indices of computable well-orders is \(O=\{e:\exists a \varphi_e \cong C_{<a}\}\) where \(C_{<a}\) is the initial segment of \(C\) up to the element \(a\). Then \(O\) can be defined by</p>
\[e\in O\Leftrightarrow \exists i\exists (a\in C) \varphi_i^{H^{[a]}}: \varphi_e \cong C_{< a}\]
<p>where \(H\) denotes a jump hierarchy on \(C\). Note that it mus not be the case that \(\mathbf d\) computes a jump hierarchy, after all it only computes all hyperarithmetic sets, but it does not necessarily do this uniformly. However, creating a jump hierarchy in this case is \(\Sigma^0_3\) (see Montalbán [3] for more on jump hierarchies). Checking whether \(\varphi^{H^{[a]}}_i\) is an isomorvarphism is \(\Pi^0_2\). So \(\mathbf d^{(3)}\geq deg(O)=deg(\mathcal O)\).</p>
<p>We have seen, that if we restrict our degrees to degrees that compute \(HYP\) and a copy of \(\omega_1^{\mathrm{CK}}\) then we can not be far below Kleene’s O. What if we can not compute \(\omega_1^{\mathrm{CK}}\)? It turns out we can be quite far away in that case.</p>
<p><strong>Theorem 2.</strong> There is a degree \(\mathbf d>HYP\) that is hyperarithmetically low, i.e., \(\omega_1^{\mathbf d}=\omega_1^{\mathrm{CK}}\).</p>
<p><em>Proof sketch.</em> Consider the set</p>
\[X=\left\{ (e,H) : \begin{aligned}\varphi_e \text{ is a linear order,}\\ H \text{ is a jump hierarchy on }\varphi_e,\\ \text{ and } \varphi_e\text{ does not have hyperarithmetic descending sequences}\end{aligned}\right\}\]
<p>This set is \(\Sigma^1_1\) so by the Gandy basis theorem it must contain a set \(Y=(e_0,H_0)\) such that \(\omega_1^{Y}=\omega_1^{\mathrm{CK}}\). Thus there is no \(\alpha\) such that \(Y^{(\alpha)}\equiv_T \mathcal O\). See Montalbán [3] for more on Gandy’s basis theorem and jump hierarchies in this context.</p>
<p>Thanks to Antonio Montalbán and Ted Slaman for showing me this.</p>
<p>[1] Jockusch Jr, Carl G., and Stephen G. Simpson. 1976. “A Degree-Theoretic Definition of the Ramified Analytical Hierarchy.” Annals of Mathematical Logic 10 (1): 1–32.</p>
<p>[2] Ash, Christopher J., and Julia F. Knight. 1990. “Pairs of Recursive Structures.” Annals of Pure and Applied Logic 46 (3): 211–34.</p>
<p>[3] Montalbán, Antonio. 2021. Computable Structure Theory: Beyond the Arithmetic. draft. https://math.berkeley.edu/~antonio/CSTpart2_DRAFT.pdf.</p>Dino Rosseggerdino@math.berkeley.eduLet \(HYP\) be the set of hyperarithmetic degrees, i.e. \(HYP=\{ \mathbf d:\exists (\alpha<\omega_1^{\mathrm{CK}}) \mathbf d< \mathbf 0^{(\alpha)}\}.\) Jockusch and Simpson [1] showed that there is a degree \(\mathbf d\) that is minimal over \(HYP\) (i.e., if \(\mathbf c<\mathbf d\), then \(\mathbf c\in HYP\)) and such that \(\mathbf d^{(3)}=deg(\mathcal O)\) where \(\mathcal O\) is Kleene’s \(\mathcal O\), or, up to m-equivalence, the set of indices of computable well-orderings.Positive enumerable functors2021-02-09T00:00:00+00:002021-02-09T00:00:00+00:00https://drossegger.github.io/news/positiveenumerablefunctors<script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script>
<p>We just submitted a new article titled “Positive enumerable functors” and you can find the preprint <a href="/assets/files/positiveenumerablefunctors.pdf">here</a> or on <a href="https://arxiv.org/abs/2011.14160">arxiv</a>. This article is joint work with Barbara Csima and Daniel Yu who proved most of the results presented there in an undergraduate research project he did with Barbara and me over the summer.</p>
<p>We investigate different effectivizations of functors using Turing operators and enumeration operators and compare these notions. The most promising of these notions are <em>positive enumerable functors</em>. A positive enumerable functors from \(\mathcal A\) to \(\mathcal B\) consists of two enumeration operators \(\Psi\) and \(\Psi_*\) such that when given an enumeration of the positive diagram \(P(\hat{\mathcal A})\) of a structure isomorphic to \(\hat{\mathcal A}\cong \mathcal A\), \(\Phi\) enumerates the positive diagram of \(F(\hat{\mathcal A})\). The operator \(\Phi_*\) will take care of the isomorphisms by enumerating the graph of \(F(f:\hat{\mathcal A}\to\tilde{\mathcal A})\) given \(P(\hat{\mathcal A})\oplus Graph(f)\oplus P(\tilde{\mathcal A})\).</p>
<p>We show that if \(\mathcal A\) is positive enumerable bi-transformable to \(\mathcal B\) (if there are positve enumerable functors \(F:\mathcal A\to\mathcal B\) and \(G:\mathcal B\to \mathcal A\) such that \(F\) and \(G\) are “enumerable pseudo-inverses”) then \(\mathcal A\) and \(\mathcal B\) have the same enumeration degree spectrum and that positive enumerable functors are independent from the already established enumerable and computable functors.</p>
<p>We submitted the article to a conference and hope to expand it in the future, adding more preservation results and investigating syntactic equivalences.</p>Dino Rosseggerdino@math.berkeley.eduNew preprint on arXiv!2020-10-05T00:00:00+00:002020-10-05T00:00:00+00:00https://drossegger.github.io/news/spectra-analyticcomp<script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script>
<p>My newest preprint <a href="https://arxiv.org/abs/2010.00755">Degree spectra of analytic complete equivalence relations</a> just appeared on arXiv.</p>
<p>This article investigates the bi-embeddability and elementary bi-embeddability relation on countable structures both from the viewpoint of descriptive set theory and computable structure theory. Two structures are <em>bi-embeddable</em> if either is isomorphic to an substructure of the other. They are <em>elementary bi-embeddable</em> if either is isomorphic to a elementary substructure of the other.</p>
<p>Our main results are the following.</p>
<p><strong>Theorem 1.</strong> <em>The elementary bi-embeddability relation on graphs is a \(\pmb \Sigma^1_1\) complete equivalence relation under Borel reducibility.</em></p>
<p>The proof of this result uses Marker extensions with so called minimal models. A structure \(\mathcal A\) is <em>minimal</em> if it does not contain an elementary substructure.</p>
<p>Let \(\sim\) be an equivalence relation on a class of structures. Then given
a structure \(\mathcal A\) we define the \(\sim\) spectrum of \(\mathcal A\) as the
set of all sets Turing equivalent to a structure \(\sim\) equivalent to \(\mathcal A\), i.e.,
\[ DgSp_\sim(\mathcal A)=\{ X\subseteq \omega: \exists \mathcal B \sim \mathcal A \ X\equiv_T \mathcal A\}.\]</p>
<p><strong>Theorem 2.</strong> <em>Every bi-embeddability spectrum of a graph is the jump spectrum of an elementary bi-embeddability spectrum of a graph, i.e., for every \(\mathcal G\), there is a graph \(\mathcal G'\) such that</em>
\[ DgSp_\approx(\mathcal G’)=\{ X: X’\in DgSp_\approxeq(\mathcal G)\}.\]</p>
<p>This result is proven by analyzing the reduction given to prove Theorem 1. We show that this reduction induces a computable bi-transformation with a suitable class of structures.</p>Dino Rosseggerdino@math.berkeley.eduMy newest preprint Degree spectra of analytic complete equivalence relations just appeared on arXiv.New homepage online2020-08-06T00:00:00+00:002020-08-06T00:00:00+00:00https://drossegger.github.io/news/homepage-online<p>I just created this new homepage with all my professional information. I hope that you find it interesting and useful. I appreciate feedback, just write me an e-mail.</p>Dino Rosseggerdino@math.berkeley.eduI just created this new homepage with all my professional information. I hope that you find it interesting and useful. I appreciate feedback, just write me an e-mail.