Effective Descriptive Set Theory Research Articles

Effective aspects of Hausdorff and Fourier dimension

In this paper, we study Hausdorff and Fourier dimension from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity. Working in the hyperspace K ( X ) of compact subsets of X, with X = [ 0 , 1 ] d or X = R d , we characterize the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. This, in turn, allows us to show that family of all the closed Salem sets is Π 3 0 -complete. One of our main tools is a careful analysis of the effectiveness of a classical theorem of Kaufman. We furthermore compute the Weihrauch degree of the functions computing Hausdorff and Fourier dimension of closed sets.

Descriptive complexity of [formula omitted]-spaces

We survey current stage of effective descriptive set theory which was fast evolving in the last decade. Most attention will be given to (effective) quasi-Polish spaces and their important subclasses (Polish spaces, ω-continuous domains) but also some important non-countably-based spaces (like the Kleene-Kreisel functionals or spaces of polynomials) will be discussed. Along with effective DST, also classical DST will be considered, which is the “limit” of the effective DST relative to arbitrary oracles. We will discuss not only classical hierarchies of sets and functions but also versions of (effective) Wadge hierarchy where a considerable progress was recently achieved.

Computability Theory

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 journey through computability, topology and analysis

AbstractThis thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and (effective) descriptive set theory. We will especially focus on Weihrauch reducibility as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of problems arising from theorems that lie at the higher levels of the reverse mathematics hierarchy.We first analyze the strength of the open and clopen Ramsey theorems. Since there is not a canonical way to phrase these theorems as multi-valued functions, we identify eight different multi-valued functions (five corresponding to the open Ramsey theorem and three corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility.We then discuss some new operators on multi-valued functions and study their algebraic properties and the relations with other previously studied operators on problems. In particular, we study the first-order part and the deterministic part of a problem f, capturing the Weihrauch degree of the strongest multi-valued problem that is reducible to f and that, respectively, has codomain $\mathbb {N}$ or is single-valued.These notions proved to be extremely useful when exploring the Weihrauch degree of the problem $\mathsf {DS}$ of computing descending sequences in ill-founded linear orders. They allow us to show that $\mathsf {DS}$ , and the Weihrauch equivalent problem $\mathsf {BS}$ of finding bad sequences through non-well quasi-orders, while being very “hard” to solve, are rather weak in terms of uniform computational strength. We then generalize $\mathsf {DS}$ and $\mathsf {BS}$ by considering $\boldsymbol {\Gamma }$ -presented orders, where $\boldsymbol {\Gamma }$ is a Borel pointclass or $\boldsymbol {\Delta }^1_1$ , $\boldsymbol {\Sigma }^1_1$ , $\boldsymbol {\Pi }^1_1$ . We study the obtained $\mathsf {DS}$ -hierarchy and $\mathsf {BS}$ -hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level.Finally, we work in the context of geometric measure theory and we focus on the characterization, from the point of view of descriptive set theory, of some conditions involving the notions of Hausdorff/Fourier dimension and Salem sets. We first work in the hyperspace $\mathbf {K}([0,1])$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a $\boldsymbol {\Pi }^0_3$ -complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in $\mathbf {K}([0,1]^d)$ . We also generalize the results by relaxing the compactness of the ambient space and show that the closed Salem sets are still $\boldsymbol {\Pi }^0_3$ -complete when we endow $\mathbf {F}(\mathbb {R}^d)$ with the Fell topology. A similar result holds also for the Vietoris topology.We conclude by analyzing the same notions from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity, and show that the complexities remain the same also in the lightface case. In particular, we show that the family of all the closed Salem sets is $\Pi ^0_3$ -complete. We furthermore characterize the Weihrauch degree of the functions computing Hausdorff and Fourier dimension of closed sets.Abstract prepared by Manlio Valenti.E-mail:manliovalenti@gmail.com

Turing degrees in Polish spaces and decomposability of Borel functions

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore–Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture.

Isomorphism and classification for countable structures

We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and computable structure theory, extending concepts from the latter beyond computable structures to examine the isomorphism problem on arbitrary countable structures. We give examples using specific classes of fields and of trees, illustrating how the new concepts can yield classifications that reveal differences between seemingly similar classes. Finally, we use a computable homeomorphism to define a measure on the space of isomorphism types of algebraic fields, and examine the prevalence of relative computable categoricity under this measure.

RANDOMNESS VIA INFINITE COMPUTATION AND EFFECTIVE DESCRIPTIVE SET THEORY

AbstractWe study randomness beyond${\rm{\Pi }}_1^1$-randomness and its Martin-Löf type variant, which was introduced in [16] and further studied in [3]. Here we focus on a class strictly between${\rm{\Pi }}_1^1$and${\rm{\Sigma }}_2^1$that is given by the infinite time Turing machines (ITTMs) introduced by Hamkins and Kidder. The main results show that the randomness notions associated with this class have several desirable properties, which resemble those of classical random notions such as Martin-Löf randomness and randomness notions defined via effective descriptive set theory such as${\rm{\Pi }}_1^1$-randomness. For instance, mutual randoms do not share information and a version of van Lambalgen’s theorem holds.Towards these results, we prove the following analogue to a theorem of Sacks. If a real is infinite time Turing computable relative to all reals in some given set of reals with positive Lebesgue measure, then it is already infinite time Turing computable. As a technical tool towards this result, we prove facts of independent interest about random forcing over increasing unions of admissible sets, which allow efficient proofs of some classical results about hyperarithmetic sets.

JUMP OPERATIONS FOR BOREL GRAPHS

AbstractWe investigate the class of bipartite Borel graphs organized by the order of Borel homomorphism. We show that this class is unbounded by finding a jump operator for Borel graphs analogous to a jump operator of Louveau for Borel equivalence relations. The proof relies on a nonseparation result for iterated Fréchet ideals and filters due to Debs and Saint Raymond. We give a new proof of this fact using effective descriptive set theory. We also investigate an analogue of the Friedman-Stanley jump for Borel graphs. This analogue does not yield a jump operator for bipartite Borel graphs. However, we use it to answer a question of Kechris and Marks by showing that there is a Borel graph with no Borel homomorphism to a locally countable Borel graph, but each of whose connected components has a countable Borel coloring.

A recursion theoretic characterization of the Topological Vaught Conjecture in the Zermelo‐Fraenkel set theory

AbstractWe prove a recursion theoretic characterization of the Topological Vaught Conjecture in the Zermelo‐Fraenkel set theory by using tools from effective descriptive set theory and by revisiting the result of Miller that orbits in Polish G‐spaces are Borel sets.

Intuitionism and effective descriptive set theory

Our very eloquent charge from Jan van Mill was to “draw a line to Brouwer from descriptive set theory, but this proved elusive: in fact there are few references to Brouwer, in Lusin (1928) [36] and Lusin (1930), none of them substantial; and even though Brouwer refers to Borel, Lebesgue and Hadamard in his early papers, it does not appear that he was influenced by their work in any substantive way.11Cf. Michel(2008) [40], who traces the origin and dismisses the significance of terms like “semi-” or “pre-” intuitionists.We have not found any references by him to more developed work in descriptive set theory, after the critical Lebesgue (1905). So instead of looking for historical connections or direct influences (in either direction), we decided to identify and analyze some basic themes, problems and results which are common to these two fields; and, as it turns out, the most significant connections are between intuitionistic analysis and effective descriptive set theory, hence the title.⇒ We will outline our approach and (limited) aims in Section 1, marking with an arrow (like this one) those paragraphs which point to specific parts of the article. Suffice it to say here that our main aim is to identify a few, basic results of descriptive set theory which can be formulated and justified using principles that are both intuitionistically and classically acceptable; that we will concentrate on the mathematics of the matter rather than on history or philosophy; and that we will use standard, classical terminology and notation.This is an elementary, mostly expository paper, broadly aimed at students of logic and set theory who also know the basic facts about recursive functions but need not know a lot about either intuitionism or descriptive set theory. The only (possibly) new result is Theorem 6.1, which justifies simple definitions and proofs by induction in Kleene’s Basic System of intuitionistic analysis, and is then used inTheorems 7.1 and 7.2 to give in the same system a rigorous definition of the Borel sets and prove that they are analytic; the formulation and proof of this last result is one example where methods from effective descriptive set theory are used in an essential way.

A comparison of concepts from computable analysis and effective descriptive set theory

Computable analysis and effective descriptive set theory are both concerned with complete metric spaces, functions between them and subsets thereof in an effective setting. The precise relationship of the various definitions used in the two disciplines has so far been neglected, a situation this paper is meant to remedy.As the role of the Cauchy completion is relevant for both effective approaches to Polish spaces, we consider the interplay of effectivity and completion in some more detail.

Choice free fixed point property in separable Banach spaces

We show that the standard approach of minimal invariant sets, which applies Zorn’s Lemma and is used to prove fixed point theorems for non-expansive mappings in Banach spaces, can be applied without any reference to the full Axiom of Choice when the given Banach space is separable. Our method applies results from classical and effective descriptive set theory.

Ambiguity of {\omega}-Languages of Turing Machines

An {\omega}-language is a set of infinite words over a finite alphabet X. We consider the class of recursive {\omega}-languages, i.e. the class of {\omega}-languages accepted by Turing machines with a B\"uchi acceptance condition, which is also the class {\Sigma}11 of (effective) analytic subsets of X{\omega} for some finite alphabet X. We investigate here the notion of ambiguity for recursive {\omega}-languages with regard to acceptance by B\"uchi Turing machines. We first present in detail essentials on the literature on {\omega}-languages accepted by Turing Machines. Then we give a complete and broad view on the notion of ambiguity and unambiguity of B\"uchi Turing machines and of the {\omega}-languages they accept. To obtain our new results, we make use of results and methods of effective descriptive set theory.

Bad Wadge-like reducibilities on the Baire space

We consider various collections of functions from the Baire space X into itself naturally arising in (effective) descriptive set theory and general topology, including computable (equivalently, recursive) functions, contraction mappings, and functions which are nonexpansive or Lipschitz with respect to suitable complete ultrametrics on X (compatible with its standard topology). We analyze the degree-structures induced by such sets of functions when used as reducibility notions between subsets of X, and we show that the resulting hierarchies of degrees are much more complicated than the classical Wadge hierarchy; in particular, they always contain large infinite antichains, and in most cases also infinite descending chains.

Effective compactness and sigma-compactness

Using the Gandy-Harrington topology and other methods of effective descriptive set theory, we prove several theorems about compact and σ-compact sets. In particular, it is proved that any Δ11-set A in the Baire space N either is an at most countable union of compact Δ11 -sets (and hence is σ-compact) or contains a relatively closed subset homeomorphic to N (in this case, of course, A cannot be σ-compact).

Об эффективной компактности и сигма-компактности

Using the Gandy -- Harrington topology and other methods of effective descriptive set theory, we prove several theorems on compact and sigma-compact pointsets. In particular we show that any $\Sigma^1_1$ set $A$ of the Baire space $N^N$ either is covered by a countable union of compact $\Delta^1_1$ sets, or $A$ contains a subset closed in $N^N$ and homeomorphic to $N^N$ (and then $A$ is not covered by a sigma-compact set, of course).

Classical descriptive set theory as a refinement of effective descriptive set theory

The (effective) Suslin–Kleene Theorem is obtained as a corollary of a standard proof of the classical Suslin Theorem, by noticing that it is mostly constructive and applying to it a naive realizability interpretation.

A dichotomy result for a pointwise summable sequence of operators

Let X be a separable Banach space and Q be a coanalytic subset of X N × X . We prove that the set of sequences ( e i ) i ∈ N in X which are weakly convergent to some e ∈ X and Q ( ( e i ) i ∈ N , e ) is a coanalytic subset of X N . The proof applies methods of effective descriptive set theory to Banach space theory. Using Silver’s Theorem [J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970) 60–64], this result leads to the following dichotomy theorem: if X is a Banach space, ( a i j ) i , j ∈ N is a regular method of summability and ( e i ) i ∈ N is a bounded sequence in X , then there exists a subsequence ( e i ) i ∈ L such that either (I) there exists e ∈ X such that every subsequence ( e i ) i ∈ H of ( e i ) i ∈ L is weakly summable w.r.t. ( a i j ) i , j ∈ N to e and Q ( ( e i ) i ∈ H , e ) ; or (II) for every subsequence ( e i ) i ∈ H of ( e i ) i ∈ L and every e ∈ X with Q ( ( e i ) i ∈ H , e ) the sequence ( e i ) i ∈ H is not weakly summable to e w.r.t. ( a i j ) i , j ∈ N . This is a version for weak convergence of an Erdös–Magidor result, see [P. Erdös, M. Magidor, A note on Regular Methods of Summability, Proc. Amer. Math. Soc. 59 (2) (1976) 232–234]. Both theorems obtain some considerable generalizations.

Randomness via effective descriptive set theory

1.1 Informal introduction to Π1 relations Let 2N denote Cantor space. • A relation B ⊆ N × (2N)r is Π1 if it is obtained from an arithmetical relation by a universal quantification over sets. • If k = 1, r = 0 we have a Π1 set ⊆ N. • If k = 0, r = 1 we have a Π1 class ⊆ 2N. • The relation B is∆1 if both B and its complement are Π1. There is an equivalent representation of Π1 relations where the members are enumerated at stages that are countable ordinals. For Π1 sets (of natural numbers) these stages are in fact computable ordinals, i.e., the order types of computable well-orders.

Towards a descriptive set theory for domain-like structures

This is a survey of results in descriptive set theory for domains and similar spaces, with the emphasis on the ω -algebraic domains. We try to demonstrate that the subject is interesting in its own right and is closely related to some areas of theoretical computer science. Since the subject is still in its beginning, we discuss in detail several open questions and possible future development. We also mention some relevant facts of (effective) descriptive set theory.