The descriptive theory of represented spaces

This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented spaces. Most material is from work-in-progress, and thus there may be a stronger focus on projects involving the author than an objective survey would merit.Comment: survey of work-in-progres

Solovay reduction and continuity

The objective of this study is a better understanding of the relationships between reduction and continuity. Solovay reduction is a variation of Turing reduction based on the distance of two real numbers. We characterize Solovay reduction by the existence of a certain real function that is computable (in the sense of computable analysis) and Lipschitz continuous. We ask whether there exists a reducibility concept that corresponds to H\"older continuity. The answer is affirmative. We introduce quasi Solovay reduction and characterize this new reduction via H\"older continuity. In addition, we separate it from Solovay reduction and Turing reduction and investigate the relationships between complete sets and partial randomness.Comment: 19 pages, 2 figure

Point degree spectra of represented spaces

We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees, and so on. The notion of point degree spectrum creates a connection among various areas of mathematics including computability theory, descriptive set theory, infinite dimensional topology and Banach space theory. Through this new connection, for instance, we construct a family of continuum many infinite dimensional Cantor manifolds with property $C$ whose Borel structures at an arbitrary finite rank are mutually non-isomorphic. This provides new examples of Banach algebras of real valued Baire class two functions on metrizable compacta, and strengthen various theorems in infinite dimensional topology such as Pol's solution to Alexandrov's old problem

Total Representations

Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space). Total representations make the well known analogy between numberings and representations closer, unify some terminology, simplify some technical details, suggest interesting open questions and new invariants of topological spaces relevant to computable analysis.Comment: 30 page

-Generic Computability, Turing Reducibility and Asymptotic Density

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 contains a set which is not coarsely computable. We prove that there is a c.e. set of density 1 which has no computable subset of density 1. As a corollary, there is a generically computable set A such that no generic algorithm for A has computable domain. We define a general notion of generic reducibility in the spirt of Turing reducibility and show that there is a natural order-preserving embedding of the Turing degrees into the generic degrees which is not surjective

Computability and Tiling Problems

In this thesis we will present and discuss various results pertaining to tiling problems and mathematical logic, specifically computability theory. We focus on Wang prototiles, as defined in [32]. We begin by studying Domino Problems, and do not restrict ourselves to the usual problems concerning finite sets of prototiles. We first consider two domino problems: whether a given set of prototiles $S$ has total planar tilings, which we denote $TILE$, or whether it has infinite connected but not necessarily total tilings, $WTILE$ (short for `weakly tile'). We show that both $TILE \equiv_m ILL \equiv_m WTILE$, and thereby both $TILE$ and $WTILE$ are $\Sigma^1_1$-complete. We also show that the opposite problems, $\neg TILE$ and $SNT$ (short for `Strongly Not Tile') are such that $\neg TILE \equiv_m WELL \equiv_m SNT$ and so both $\neg TILE$ and $SNT$ are both $\Pi^1_1$-complete. Next we give some consideration to the problem of whether a given (infinite) set of prototiles is periodic or aperiodic. We study the sets $PTile$ of periodic tilings, and $ATile$ of aperiodic tilings. We then show that both of these sets are complete for the class of problems of the form $(\Sigma^1_1 \wedge \Pi^1_1)$. We also present results for finite versions of these tiling problems. We then move on to consider the Weihrauch reducibility for a general total tiling principle $CT$ as well as weaker principles of tiling, and show that there exist Weihrauch equivalences to closed choice on Baire space, $C_{\omega^\omega}$. We also show that all Domino Problems that tile some infinite connected region are Weihrauch reducible to $C_{\omega^\omega}$. Finally, we give a prototile set of 15 prototiles that can encode any Elementary Cellular Automaton (ECA). We make use of an unusual tile set, based on hexagons and lozenges that we have not see in the literature before, in order to achieve this.Comment: PhD thesis. 179 pages, 13 figure

Computable dyadic subbases and Tω\mathbf{T}^\omega-representations of compact sets

We explore representing the compact subsets of a given represented space by infinite sequences over Plotkin's $\mathbb{T}$. We show that computably compact computable metric spaces admit representations of their compact subsets in such a way that compact sets are essentially underspecified points. We can even ensure that a name of an $n$-element compact set contains $n$ occurrences of $\bot$. We undergo this study effectively and show that such a $\mathbb{T}^\omega$-representation is effectively obtained from structures of computably compact computable metric spaces. As an application, we prove some statements about the Weihrauch degree of closed choice for finite subsets of computably compact computable metric spaces. Along the way, we introduce the notion of a computable dyadic subbase, and prove that every computably compact computable metric space admits a proper computable dyadic subbase

Logic Blog 2015f

The 2015 Logic Blog contains a large variety of results connected to logic, some of them unlikely to be submitted to a journal. For the first time there is a group theory part. There are results in higher randomness, and in computable ergodic theory

Well Quasiorders and Hierarchy Theory

We discuss some applications of WQOs to several fields were hierarchies and reducibilities are the principal classification tools, notably to Descriptive Set Theory, Computability theory and Automata Theory. While the classical hierarchies of sets usually degenerate to structures very close to ordinals, the extension of them to functions requires more complicated WQOs, and the same applies to reducibilities. We survey some results obtained so far and discuss open problems and possible research directions.Comment: 37 page

Extensions Theorems, Orbits, and Automorphisms of the Computably Enumerable Sets

We prove an algebraic extension theorem for the computably enumerable sets, $\mathcal{E}$. Using this extension theorem and other work we then show if $A$ and $\hat{A}$ are automorphic via $\Psi$ then they are automorphic via $\Lambda$ where \Lambda \restriction \L^*(A) = \Psi and \Lambda \restriction \E^*(A) is $\Delta^0_3$. We give an algebraic description of when an arbitrary set \Ahat is in the orbit of a \ce set $A$. We construct the first example of a definable orbit which is not a $\Delta^0_3$ orbit. We conclude with some results which restrict the ways one can increase the complexity of orbits. For example, we show that if $A$ is simple and $\hat{A}$ is in the same orbit as $A$ then they are in the same $\Delta^0_6$-orbit and furthermore we provide a classification of when two simple sets are in the same orbit.Comment: Comments as of Aug 31, 05: This is now the final final version of the paper. Another section, 5.3, was added to the paper. No other change were made. This section was added to allow a clean clear inferface with the sequel. Comments as of March 31, 05: This is now the final version of this paper. (Section 7 was rewritten. A few other lemmas were added.