Wadge Degrees Research Articles

Lipschitz and Wadge binary games in second order arithmetic

We present a detailed formalization of Lipschitz and Wadge games in the context of second order arithmetic and we investigate the logical strength of Lipschitz and Wadge determinacy, and the tightly related Semi-Linear Ordering principle, for the first levels of the Hausdorff difference hierarchy in the Cantor space. As a result, we obtain characterizations of WKL0 and ACA0 in terms of these determinacy principles.

Non-collapse of the effective Wadge hierarchy

We study the recently suggested effective Wadge hierarchy in effective spaces, concentrating on the non-collapse property. Along with hierarchies of sets, we study hierarchies of k-partitions which are interesting on their own. In particular, we establish sufficient conditions for the non-collapse of the effective Wadge hierarchy and apply them to some concrete spaces, including the discrete space of natural numbers, the Baire space, and the Cantor space. While the latter two cases are obtained by easy effectivization of the corresponding topological result by T. Kihara and A. Montalban, the first case needs a different technique and is certainly the main technical result of this paper.

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.

Wadge-like degrees of Borel bqo-valued functions

We unite two well known generalisations of the Wadge theory. The first one considers more general reducing functions than the continuous functions in the classical case, and the second one extends Wadge reducibility from sets (i.e., { 0 , 1 } \{0,1\} -valued functions) to Q Q -valued functions, for a better quasiorder Q Q . In this article, we consider more general reducibilities on the Q Q -valued functions and generalise some results of L. Motto Ros [J. Symbolic Logic 74 (2009), pp. 27–49] in the first direction and of T. Kihara and A. Montalbán [Trans. Amer. Math. Soc. 370 (2018), pp. 9025–9044] in the second direction: Our main result states that the structure of the Δ α 0 \mathbf {\Delta }^0_\alpha -degrees of Δ α + γ 0 \mathbf {\Delta }^0_{\alpha +\gamma } -measurable Q Q -valued functions is isomorphic to the Δ β 0 \mathbf {\Delta }^0_\beta -degrees of Δ β + γ 0 \mathbf {\Delta }^0_{\beta +\gamma } -measurable Q Q -valued functions, and these are isomorphic to the generalized homomorphism order on the γ \gamma -th iterated Q Q -labeled forests.

On the Expressive Power of Non-deterministic and Unambiguous Petri Nets over Infinite Words

We prove that ω-languages of (non-deterministic) Petri nets and ω-languages of (nondeterministic) Turing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of ω-languages of (non-deterministic) Petri nets are equal to the Borel and Wadge hierarchies of the class of ω-languages of (non-deterministic) Turing machines. We also show that it is highly undecidable to determine the topological complexity of a Petri net ω-language. Moreover, we infer from the proofs of the above results that the equivalence and the inclusion problems for ω-languages of Petri nets are ∏21-complete, hence also highly undecidable. Additionally, we show that the situation is quite the opposite when considering unambiguous Petri nets, which have the semantic property that at most one accepting run exists on every input. We provide a procedure of determinising them into deterministic Muller counter machines with counter copying. As a consequence, we entail that the ω-languages recognisable by unambiguous Petri nets are △30 sets.

THE DISCONTINUITY PROBLEM

AbstractMatthias Schröder has asked the question whether there is a weakest discontinuous problem in the topological version of the Weihrauch lattice. Such a problem can be considered as the weakest unsolvable problem. We introduce the discontinuity problem, and we show that it is reducible exactly to the effectively discontinuous problems, defined in a suitable way. However, in which sense this answers Schröder’s question sensitively depends on the axiomatic framework that is chosen, and it is a positive answer if we work in Zermelo–Fraenkel set theory with dependent choice and the axiom of determinacy $\mathsf {AD}$ . On the other hand, using the full axiom of choice, one can construct problems which are discontinuous, but not effectively so. Hence, the exact situation at the bottom of the Weihrauch lattice sensitively depends on the axiomatic setting that we choose. We prove our result using a variant of Wadge games for mathematical problems. While the existence of a winning strategy for Player II characterizes continuity of the problem (as already shown by Nobrega and Pauly), the existence of a winning strategy for Player I characterizes effective discontinuity of the problem. By Weihrauch determinacy we understand the condition that every problem is either continuous or effectively discontinuous. This notion of determinacy is a fairly strong notion, as it is not only implied by the axiom of determinacy $\mathsf {AD}$ , but it also implies Wadge determinacy. We close with a brief discussion of generalized notions of productivity.

The Wadge hierarchy on Zariski topologies

We study the relation of continuous reducibility, or Wadge reducibility, between subsets of an affine variety. We show that on any curve the relation of continuous reducibility is a bqo, though it may have large finite antichains. We determine the Wadge hierarchy on irreducible curves and on countable irreducible affine varieties of any dimension. Under a technical assumption of adequateness, we prove that on the class of finite differences of closed subsets of an irreducible affine variety the structure of the Wadge hierarchy depends only on the dimension of the affine variety and coincides with the difference hierarchy; we also show that this assumption of adequateness is satisfied by a large class of affine varieties. In contrast, we show that for large cardinalities the behaviour of the Wadge hierarchy outside the class of finite differences of closed sets can be much wilder, for example there may exist antichains of size the continuum.

Effective Wadge hierarchy in computable quasi-Polish spaces

We define and study an effective version of the Wadge hierarchy in computable quasi-Polish spaces which include most spaces of interest for computable analysis. Along with hierarchies of sets we study hierarchies of k-partitions which are interesting on their own. We show that levels of such hierarchies are preserved by the computable effectively open surjections, that if the effective Hausdorff-Kuratowski theorem holds in the Baire space then it holds in every computable quasi-Polish space, and we extend the effective Hausdorff theorem to k-partitions.

A Q-WADGE HIERARCHY IN QUASI-POLISH SPACES

AbstractThe Wadge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g., several Hausdorff–Kuratowski (HK)-type theorems in quasi-Polish spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q.

Forests describing Wadge degrees and topological Weihrauch degrees of certain classes of functions and relations

We introduce notions of continuous strong Weihrauch reducibility and of continuous Weihrauch reducibility for functions with range in a preordered set. Then we associate with such functions certain labeled forests and trees and show that Wadge reduci

Games for Functions: Baire Classes, Weihrauch Degrees, Transfinite Computations, and Ranks

Game characterizations of classes of functions in descriptive set theory have their origins in the seminal work of Wadge, with further developments by several others. In this thesis we study such characterizations from several perspectives. We define modifications of Semmes's game characterization of the Borel functions, obtaining game characterizations of the Baire class $\alpha$ functions for each fixed $\alpha < \omega_1$. We also define a construction of games which transforms a game characterizing a class $\Lambda$ of functions into a game characterizing the class of functions which are piecewise $\Lambda$ on a countable partition by $\Pi^0_\alpha$ sets, for each $0 < \alpha < \omega_1$. We then define a parametrized Wadge game by using computable analysis, and show how the parameters affect the class of functions that is characterized by the game. As an application, we recast our games characterizing the Baire classes into this framework. Furthermore, we generalize our game characterizations of function classes to generalized Baire spaces, show how the notion of computability on Baire space can be transferred to generalized Baire spaces, and show that this is appropriate for computable analysis by defining a representation of Galeotti's generalized real line and analyzing the Weihrauch degree of the intermediate value theorem for that space. Finally, we show how the game characterizations of function classes discussed lead in a natural way to a stratification of each class into a hierarchy, intuitively measuring the complexity of functions in that class. This idea and the results presented open new paths for further research.

On the structure of the Wadge degrees of bqo-valued Borel functions

In this article, we give a full description of the Wadge degrees of Borel functions from $\omega^\omega$ to a better quasi ordering $\mathcal{Q}$. More precisely, for any countable ordinal $\xi$, we show that the Wadge degrees of $\mathbf{\Delta}^0_{1+\xi}$-measurable functions $\omega^\omega\to\mathcal{Q}$ can be represented by countable joins of the $\xi$-th transfinite nests of $\mathcal{Q}$-labeled well-founded trees.

Continuous reducibility: functions versus relations

It is proved that the Tang-Pequignot reducibility (or reducibility by relatively continuous relations) on a second countable, T0 space X either coincides with the Wadge reducibility for the given topology, or there is no topology on X that can turn it into Wadge reducibility.

The uniform Martin’s conjecture for many-one degrees

We study functions from reals to reals which are uniformly degree-invariant from Turing-equivalence to many-one equivalence, and compare them a cone. We prove that they are in one-to-one correspondence with the Wadge degrees, which can be viewed as a refinement of the uniform Martin's conjecture for uniformly invariant functions from Turing- to Turing-equivalence. Our proof works in the general case of many-one degrees on $\mathcal{Q}^\omega$ and Wadge degrees of functions $\omega^\omega\to\mathcal{Q}$ for any better quasi ordering $\mathcal{Q}$.

Comparison game on trace ideal

In this paper, we consider some properties about a Wadge-like game called Comparison game. We prove that the Comparison game is coherent with Wadge game on a subclass of Borel ideals. Thus answering some questions from M. Hrušák and D.M. Alcántara.

Locally finite ω‐languages and effective analytic sets have the same topological complexity

Local sentences and the formal languages they define were introduced by Ressayre in . We prove that locally finite ω‐languages and effective analytic sets have the same topological complexity: the Borel and Wadge hierarchies of the class of locally finite ω‐languages are equal to the Borel and Wadge hierarchies of the class of effective analytic sets. In particular, for each non‐null recursive ordinal there exist some ‐complete and some ‐complete locally finite ω‐languages, and the supremum of the set of Borel ranks of locally finite ω‐languages is the ordinal , which is strictly greater than the first non‐recursive ordinal . This gives an answer to the question of the topological complexity of locally finite ω‐languages, which was asked by Simonnet and also by Duparc, Finkel, and Ressayre in . Moreover we show that the topological complexity of a locally finite ω‐language defined by a local sentence φ may depend on the models of the Zermelo‐Fraenkel axiomatic system . Using similar constructions as in the proof of the above results we also show that the equivalence, the inclusion, and the universality problems for locally finite ω‐languages are ‐complete, hence highly undecidable.

Towards a descriptive theory of cb0-spaces

The paper tries to extend some results of the classical Descriptive Set Theory to as many countably basedT0-spaces (cb0-spaces) as possible. Along with extending some central facts about Borel, Luzin and Hausdorff hierarchies of sets we also consider the more general case ofk-partitions. In particular, we investigate the difference hierarchy ofk-partitions and the fine hierarchy closely related to the Wadge hierarchy.

Infinite games specified by 2-tape automata

We prove that the determinacy of Gale–Stewart games whose winning sets are infinitary rational relations accepted by 2-tape Büchi automata is equivalent to the determinacy of (effective) analytic Gale–Stewart games which is known to be a large cardinal assumption. Then we prove that winning strategies, when they exist, can be very complex, i.e. highly non-effective, in these games. We prove the same results for Gale–Stewart games with winning sets accepted by real-time 1-counter Büchi automata, then extending previous results obtained about these games.1.There exists a 2-tape Büchi automaton (respectively, a real-time 1-counter Büchi automaton) A such that: (a) there is a model of ZFC in which Player 1 has a winning strategy σ in the game G(L(A)) but σ cannot be recursive and not even in the class (Σ21∪Π21); (b) there is a model of ZFC in which the game G(L(A)) is not determined.2.There exists a 2-tape Büchi automaton (respectively, a real-time 1-counter Büchi automaton) A such that L(A) is an arithmetical Δ30-set and Player 2 has a winning strategy in the game G(L(A)) but has no hyperarithmetical winning strategies in this game.3.There exists a recursive sequence of 2-tape Büchi automata (respectively, of real-time 1-counter Büchi automata) An, n≥1, such that all games G(L(An)) are determined, but for which it is Π21-complete hence highly undecidable to determine whether Player 1 has a winning strategy in the game G(L(An)). Then we consider the strengths of determinacy for these games, and we prove the following results.1.There exists a 2-tape Büchi automaton (respectively, a real-time 1-counter Büchi automaton) A♯ such that the game G(A♯) is determined iff the effective analytic determinacy holds.2.There is a transfinite sequence of 2-tape Büchi automata (respectively, of real-time 1-counter Büchi automata) (Aα)α<ω1CK, indexed by recursive ordinals, such that the games G(L(Aα)) have strictly increasing strengths of determinacy. We also show that the determinacy of Wadge games between two players in charge of infinitary rational relations accepted by 2-tape Büchi automata is equivalent to the (effective) analytic Wadge determinacy and thus also equivalent to the (effective) analytic determinacy.

WADGE HIERARCHY OF DIFFERENCES OF CO-ANALYTIC SETS

AbstractWe begin the fine analysis of nonBorel pointclasses. Working in ZFC + DET$\left( {_1^1 } \right)$, we describe the Wadge hierarchy of the class of increasing differences of co-analytic subsets of the Baire space by extending results obtained by Louveau ([5]) for the Borel sets.

A Wadge hierarchy for second countable spaces

We define a notion of reducibility for subsets of a second countable T 0 topological space based on relatively continuous relations and admissible representations. This notion of reducibility induces a hierarchy that refines the Baire classes and the Hausdorff–Kuratowski classes of differences. It coincides with Wadge reducibility on zero dimensional spaces. However in virtually every second countable T 0 space, it yields a hierarchy on Borel sets, namely it is well founded and antichains are of length at most 2. It thus differs from the Wadge reducibility in many important cases, for example on the real line $${\mathbb{R}}$$ or the Scott Domain $${\mathcal{P}\omega}$$ .