20 research outputs found
Notions of Complexity Within Computable Structure Theory
This thesis covers multiple areas within computable structure theory, analyzing the complexities of certain aspects of computable structures with respect to different notions of definability.
In chapter 2 we use a new metatheorem of Antonio Montalb\'an's to simplify an otherwise difficult priority construction. We restrict our attention to linear orders, and ask if, given a computable linear order \A with degree of categoricity , it is possible to construct a computable isomorphic copy of \A such that the isomorphism achieves the degree of categoricity and furthermore, that we did not do this coding using a computable set of points chosen in advance. To ensure that there was no computable set of points that could be used to compute the isomorphism we are forced to diagonalize against all possible computable unary relations while we construct our isomorphic copy. This tension between trying to code information into the isomorphism and trying to avoid using computable coding locations, necessitates the use of a metatheorem. This work builds off of results obtained by Csima, Deveau, and Stevenson for the ordinals and , and extends it to for any computable successor ordinal .
In chapter 3, which is joint work with Alvir and Csima, we study the Scott complexity of countable reduced Abelian -groups. We provide Scott sentences for all such groups, and show some cases where this is an optimal upper bound on the Scott complexity. To show this optimality we obtain partial results towards characterizing the back-and-forth relations on these groups.
In chapter 4, which is joint work with Csima and Rossegger, we study structures under enumeration reducibility when restricting oneself to only the positive information about a structure. We find that relations that can be relatively intrinsically enumerated from such information have a definability characterization using a new class of formulas. We then use these formulas to produce a structural jump within the enumeration degrees that admits jump inversion, and compare it to other notions of the structural jump. We finally show that interpretability of one structure in another using these formulas is equivalent to the existence of a positive enumerable functor between the classes of isomorphic copies of the structures
5th International Open and Distance Learning Conference Proceedings Book = 5. Uluslararası Açık ve Uzaktan Öğrenme Konferansı Bildiri Kitabı
In celebration of our 40th anniversary in open and distance learning, we are happy and proud to organize the 5th International Open & Distance Learning Conference- IODL 2022, which was held at Anadolu University, Eskişehir, Türkiye on 28-30 September 2022. After the conferences in 2002, 2006, 2010, and 2019, IODL 2022 is the 5th IODL event hosted by Anadolu University Open Education System (OES)
Computable classifications of continuous, transducer, and regular functions
We develop a systematic algorithmic framework that unites global and local
classification problems for functional separable spaces and apply it to attack
classification problems concerning the Banach space C[0,1] of real-valued
continuous functions on the unit interval. We prove that the classification
problem for continuous (binary) regular functions among almost everywhere
linear, pointwise linear-time Lipshitz functions is -complete. We
show that a function is (binary)
transducer if and only if it is continuous regular; interestingly, this
peculiar and nontrivial fact was overlooked by experts in automata theory. As
one of many consequences, our -completeness result covers the class
of transducer functions as well. Finally, we show that the Banach space
of real-valued continuous functions admits an arithmetical
classification among separable Banach spaces. Our proofs combine methods of
abstract computability theory, automata theory, and functional analysis.Comment: Revised argument in Section 5; results unchange
Computability Theory (hybrid meeting)
Over the last decade computability theory has seen many new and
fascinating developments that have linked the subject much closer
to other mathematical disciplines inside and outside of logic.
This includes, for instance, work on enumeration degrees that
has revealed deep and surprising relations to general topology,
the work on algorithmic randomness that is closely tied to
symbolic dynamics and geometric measure theory.
Inside logic there are connections to model theory, set theory, effective descriptive
set theory, computable analysis and reverse mathematics.
In some of these cases the bridges to seemingly distant mathematical fields
have yielded completely new proofs or even solutions of open problems
in the respective fields. Thus, over the last decade, computability theory
has formed vibrant and beneficial interactions with other mathematical
fields.
The goal of this workshop was to bring together researchers representing
different aspects of computability theory to discuss recent advances, and to
stimulate future work
A Note on Computable Embeddings for Ordinals and Their Reverses
We continue the study of computable embeddings for pairs of structures, i.e.
for classes containing precisely two non-isomorphic structures. Surprisingly,
even for some pairs of simple linear orders, computable embeddings induce a
non-trivial degree structure. Our main result shows that although is computably embeddable in , the class is
\emph{not} computably embeddable in for any
natural number .Comment: 13 pages, accepted to CiE 202
Embedding Theorem for the automorphism group of the α-enumeration degrees
It is a theorem of classical Computability Theory that the automorphism group of the enumeration degrees D_e embeds into the automorphism group of the Turing degrees D_T . This follows from the following three statements:
1. D_T embeds to D_e ,
2. D_T is an automorphism base for D_e,
3. D_T is definable in D_e .
The first statement is trivial. The second statement follows from the Selman’s theorem:
A ≤e B ⇐⇒ ∀X ⊆ ω[B ≤e X ⊕ complement(X) implies A ≤e X ⊕ complement(X)].
The third statement follows from the definability of a Kalimullin pair in the α-enumeration degrees D_e and the following theorem:
an enumeration degree is total iff it is trivial or a join of a maximal Kalimullin pair.
Following an analogous pattern, this thesis aims to generalize the results above to the setting of α-Computability theory. The main result of this thesis is Embedding Theorem:
the automorphism group of the α-enumeration degrees D_αe
embeds into the automorphism group of the α-degrees D_α if α is an infinite regular cardinal and assuming the axiom of constructibility V = L. If α is a general admissible ordinal, weaker results are proved involving assumptions on the megaregularity.
In the proof of the definability of D_α in D_αe a helpful concept of α-rational numbers Q_α emerges as a generalization of the rational numbers Q and an analogue of hyperrationals. This is the most valuable theory development of this thesis with many potentially fruitful directions