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We present what is known about the reverse mathematical strength of weak theorems involving Borel sets., LIPIcs, Vol. 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021), pages 4:1-4:2

International audience; Analog and evolving recurrent neural networks are super-Turing powerful. Here, we consider analog and evolving neural nets over infinite input streams. We then characterize the topological complexity of their ω-languages as a function of the specific analog or evolving weights that they employ. As a consequence, two infinite hierarchies of classes of analog and evolving neural networks based on the complexity of their underlying weights can be derived. These results constitute an optimal refinement of the super-Turing expressive power of analog and evolving neural networks. They show that analog and evolving neural nets represent natural models for oracle-based infinite computation.

Evolving recurrent neural networks represent a natural model of computation beyond the Turing limits. Here, we consider evolving recurrent neural networks working on infinite input streams. The expressive power of these networks is related to their attractor dynamics and is measured by the topological complexity of their underlying neural \(\omega \)-languages. In this context, the deterministic and non-deterministic evolving neural networks recognize the (boldface) topological classes of \(BC(\varvec{\mathrm {\Pi }}^0_2)\) and \(\varvec{\mathrm {\Sigma }}^1_1\) \(\omega \)-languages, respectively. These results can actually be significantly refined: the deterministic and nondeterministic evolving networks which employ \(\alpha \in 2^\omega \) as sole binary evolving weight recognize the (lightface) relativized topological classes of \(BC(\mathrm {\Pi }^0_2)(\alpha )\) and \(\mathrm {\Sigma }^1_1(\alpha )\) \(\omega \)-languages, respectively. As a consequence, a proper hierarchy of classes of evolving neural nets, based on the complexity of their underlying evolving weights, can be obtained. The hierarchy contains chains of length \(\omega _1\) as well as uncountable antichains.

It is well known that the notion of measure and integral were released early enough in close connection with practical problems of measuring of geometric figures. Notion of measure was outlined in the early 20th century through H. Lebesgue's research, founder of the modern theory of measure and integral. It was developed concurrently a technique of integration of functions. Gradually it was formed a specific area today called the measure and integral theory. Essential contributions to building this theory were made by a large number of mathematicians: C. Carathodory, J. Radon, O. Nikodym, S. Bochner, J. Pettis, P. Halmos and many others. In the following we present several abstract sets, classes of sets, positive measure on a cr-algebra, Lebesgue measure on R and the particularly measure Lebesgue-Stieltjes measure on R.

In the following we present the most important properties of positive measures on Borel sets.

For a non-empty set X, the collection Top ( X ) of all topologies on X sits inside the Boolean lattice P ( P ( X ) ) (when ordered by set-theoretic inclusion) which in turn can be naturally identified with the Stone space 2 P ( X ) . Via this identification then, Top ( X ) naturally inherits the subspace topology from 2 P ( X ) . Extending ideas of Frink (1942), we apply lattice-theoretic methods to establish an equivalence between the topological closures of sublattices of 2 P ( X ) and their (completely distributive) completions. We exploit this equivalence when searching for countably infinite compact subsets within Top ( X ) and in crystallizing the Borel complexity of Top ( X ) . We exhibit infinite compact subsets of Top ( X ) including, in particular, copies of the Stone–Cech and one-point compactifications of discrete spaces.

International audience; We study the topological complexity of languages of Büchi automata on infinite binary trees. We show that such a language is either Borel and WMSO-definable, or Σ 1 1-complete and not WMSO-definable; moreover it can be algorithmically decided which of the two cases holds. The proof relies on a direct reduction to deciding the winner in a finite game with a regular winning condition.

International audience; We consider analog recurrent neural networks working on in nite input streams, provide a complete topological characterization of their expressive power, and compare it to the expressive power of classical in nite word reading abstract machines. More precisely, we consider analog recurrent neural networks as language recognizers over the Cantor space, and prove that the classes of !-languages recognized by deterministic and non-deterministic analog networks correspond precisely to the respective classes of 02 -sets and 11 -sets of the Cantor space. Furthermore, we show that the result can be generalized to more expressive analog networks equipped with any kind of Borel accepting condition. Therefore, in the deterministic case, the expressive power of analog neural nets turns out to be comparable to the expressive power of any kind of Buchi abstract machine, whereas in the non-deterministic case, analog recurrent networks turn out to be strictly more expressive than any other kind of Buchi or Muller abstract machine, including the main cases of classical automata, 1-counter automata, k-counter automata, pushdown automata, and Turing machines.

Let Y and Z be two Borel spaces. By B ( Y , Z ) we denote the set of all Borel maps of Y into Z. In Aumann (1961) [2] and Rao (1971) [10] the authors tried to generalize the results of R. Arens and J. Dugundji (see Arens and Dugundji (1951) [1] ) for Borel spaces. Unfortunately as R.J. Aumann observed in Aumann (1961) [2] , the results of Arens and Dugundji (1951) [1] are not true for Borel spaces, since for some of the simplest Borel spaces for example it is impossible to defined a Borel structure on the set B ( Y , Z ) such that the map e : B ( Y , Z ) × Y → Z with e ( f , y ) = f ( y ) for every f ∈ B ( Y , Z ) and y ∈ Y is Borel. Even if we consider the discrete structure on B ( Y , Z ) , then e will not in general be Borel. It is for this reason that in Aumann (1961) [2] and Rao (1971) [10] the authors studied subsets F of B ( Y , Z ) and Borel structures on F such that the restriction of the map e on F × Y is Borel. In this paper we study the above problem and try to generalize the results presented in Arens and Dugundji (1951) [1] for Borel spaces. More precisely in Section 1 the paper preliminaries are given. In Sections 2 and 3 we give and study Borel A -splitting and A -admissible structures on B ( Y , Z ) , where A is an arbitrary family of Borel spaces, and prove that there exists at most one Borel structure on B ( Y , Z ) which is both Borel splitting and admissible. When this structure exists, it coincides with the greatest Borel splitting structure, which always exists. We also present and study some special Borel structures on B ( Y , Z ) . In Section 4 some remarks for Borel structures on B ( Y , Z ) are stated. In Section 5 we define and study some relations between the Borel structures of the set B ( Y , Z ) and the Borel structures of the set B Z ( Y ) consisting of all subsets f − 1 ( B ) of Y, where f ∈ B ( Y , Z ) and B is an element of the Borel structure of Z, concerning the notions of Borel A -splitting and Borel A -admissible Borel structures. Finally, some open questions for Borel structures on the set of Borel mappings are posed.

A well known fact is that every Lebesgue measurable set is regular, i.e., it includes an Fσ set of the same measure. We analyze this fact from a metamathematical or foundational standpoint. We study a family of Muchnik degrees corresponding to measuretheoretic regularity at all levels of the effective Borel hierarchy. We prove some new results concerning Nies's notion of LR-reducibility. We build some ω-models of RCA0 which are relevant for the reverse mathematics of measure-theoretic regularity.

We present new characterizations of universally meager sets, shown in [P. Zakrzewski, Universally meager sets, Proc. Amer. Math. Soc. 129 (6) (2001) 1793–1798] to be a category analog of universally null sets. In particular, we address the question of how this class is related to another class of universally meager sets, recently introduced by Todorcevic [S. Todorcevic, Universally meager sets and principles of generic continuity and selection in Banach spaces, Adv. Math. 208 (2007) 274–298].

In this thesis we first introduce Borel hierarchy of sets in metric spaces and prove some of its properties. Then for special Borel subsets of special metric spaces (Euclidean space of real numbers and the hyperspace of compact subsets of a Polish space with Vietoris topology) we find out where they are in Borel hierarchy, i. e. we find out the class of Borel hierarchy, in which they are, and such that they are in no smaller class with respect to inclusion, which can be understood as an expression of its complexity. Finally we give an example of a coanalytic subset of the hyperspace of compact subsets of a Polish space, which is not Borel, with the proof of its coanalyticity. Powered by TCPDF (www.tcpdf.org)

The perfect image of a Tychonoff space X that is a result of the Souslin operation applied to the resolvable sets of F. Hausdorff (called also H -sets) in any Tychonoff space containing X is of the same descriptive type. This answers a question of R.W. Hansell since, within Tychonoff spaces, it says that perfect mappings preserve scattered- K -analytic spaces. We get also a new proof of an analogous fact for Cech-analytic spaces that was proved already by R.W. Hansell and S. Pan using another method. We show that various absolute Borel classes are preserved by perfect mappings generalizing a result of J.E. Jayne and C.A. Rogers who proved the same fact within metric spaces. In fact, more general absolute descriptive classes and their stability with respect to perfect mappings are investigated.

Following the path trodden by several authors along the border between Algebraic Geometry and Algebraic Combinatorics, we present some new results on the combinatorial structure of Borel ideals. This enables us to prove theorems on the shape of thesectional matrix of a homogeneous ideal, which is a new invariant stronger than the Hilbert function.

This paper consists essentially of the text of a series of four lectures given by the author in the Summer Conference on General Topology and Applications, Amsterdam, August 1994. Instead of attempting to give a general survey of the interrelationships between the two subjects mentioned in the title, which would be an enormous and hopeless task, we chose to illustrate them in a specific context, that of the study of Borel actions of Polish groups and Borel equivalence relations. This is a rapidly growing area of research of much current interest, which has interesting connections not only with topology and set theory (which are emphasized here), but also to ergodic theory, group representations, operator algebras and logic (particularly model theory and recursion theory). There are four parts, corresponding roughly to each one of the lectures. The first contains a brief review of some fundamental facts from descriptive set theory. In the second we discuss Polish groups, and in the third the basic theory of their Borel actions. The last part concentrates on Borel equivalence relations. The exposition is essentially self-contained, but proofs, when included at all, are often given in the barest outline.

We show that for an uncountable κ in a suitable Cohen real model for any family {Av}v of sets of reals there is a σ-homomorphism h from the σ-algebra generated by Borel sets and the sets Av, into the algebra of Baire subsets of 2κ modulo meager sets such that for all Borel B,The proof is uniform, works also for random reals and the Lebesgue measure, and in this way generalizes previous results of Carlson and Solovay for the Lebesgue measure and of Kamburelis and Zakrzewski for the Baire property.

International audience; We prove in this paper that there exists some infinitary rational relations which are Sigma^0_3-complete Borel sets and some others which are Pi^0_3-complete. This implies that there exists some infinitary rational relations which are Delta^0_4-sets but not (Sigma^0_3U Pi^0_3)-sets. These results give additional answers to questions of Simonnet and of Lescow and Thomas.

A dictionary is a set of finite words over some finite alphabet X. The omega-power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [Omega-powers and descriptive set theory, JSL 2005] the complexity of the set of dictionaries whose associated omega-powers have a given complexity. In particular, he considered the sets $W({\bf\Si}^0_{k})$ (respectively, $W({\bf\Pi}^0_{k})$, $W({\bf\Delta}_1^1)$) of dictionaries $V \subseteq 2^\star$ whose omega-powers are ${\bf\Si}^0_{k}$-sets (respectively, ${\bf\Pi}^0_{k}$-sets, Borel sets). In this paper we first establish a new relation between the sets $W({\bf\Sigma}^0_{2})$ and $W({\bf\Delta}_1^1)$, showing that the set $W({\bf\Delta}_1^1)$ is "more complex" than the set $W({\bf\Sigma}^0_{2})$. As an application we improve the lower bound on the complexity of $W({\bf\Delta}_1^1)$ given by Lecomte. Then we prove that, for every integer $k\geq 2$, (respectively, $k\geq 3$) the set of dictionaries $W({\bf\Pi}^0_{k+1})$ (respectively, $W({\bf\Si}^0_{k+1})$) is "more complex" than the set of dictionaries $W({\bf\Pi}^0_{k})$ (respectively, $W({\bf\Si}^0_{k})$) ., Comment: To appear in Mathematical Logic Quarterly

In this paper we study the property of having a countable cover by sets of small local diameter (SLD for short). We show that in the context of Banach spaces (JNR property) it implies that the Banach space is Cech-analytic. We also prove that to have the JNR property, to be σ- fragmentable and to have the same Borel sets for the weak and the norm topologies, they all are topological invariants of the weak topology. Finally, by means of “good” injections into c0 (Γ), we give a great class of Banach spaces with the JNR property., Research partially supported by a grant of Caja de Ahorros del Mediterraneo.

Final version, published in Logical Methods in Computer Science.; International audience; We prove the following surprising result: there exist a 1-counter Büchi automaton A and a 2-tape Büchi automaton B such that : (1) There is a model $V_1$ of ZFC in which the omega-language $L(A)$ and the infinitary rational relation $L(B)$ are ${\bf \Pi}_2^0$-sets, and (2) There is a model $V_2$ of ZFC in which the omega-language $L(A)$ and the infinitary rational relation $L(B)$ are analytic but non Borel sets. This shows that the topological complexity of an omega-language accepted by a 1-counter Büchi automaton or of an infinitary rational relation accepted by a 2-tape Büchi automaton is not determined by the axiomatic system ZFC. We show that a similar result holds for the class of languages of infinite pictures which are recognized by Büchi tiling systems. We infer from the proof of the above results an improvement of the lower bound of some decision problems recently studied in previous papers [Fin09a, Fin09b].

We consider the problem of a connection between the complexity of a set and of its complement provided both sets belong to the Borelian family built over certain compact (semicompact) family of sets.

The operation of taking the omega-power $V^omega$ of a language $V$ is a fundamental operation over finitary languages leading to omega-languages. Since the set $X^omega$ of infinite words over a finite alphabet $X$ can be equipped with the usual Cantor topology, the question of the topological complexity of omega-powers of finitary languages naturally arises and has been posed by Damian Niwinski (1990), Pierre Simonnet (1992), and Ludwig Staiger (1997). We investigate the topological complexity of omega-powers. We prove the following very surprising results which show that omega-powers exhibit a great opological complexity: for each non-null countable ordinal $xi$, there exist some $Sigma^0_xi$-complete omega-powers, and some $Pi^0_xi$-complete omega-powers. On the other hand, the Wadge hierarchy is a great refinement of the Borel hierarchy, determined by Bill Wadge. We show that, for each ordinal $xi$ greater than or equal to 3, there are uncountably many Wadge degrees of omega-powers of Borel rank $xi +1$. Using tools of effective descriptive set theory, we prove some effective versions of the above results.

Omega-powers of finitary languages are languages of infinite words (omega-languages) in the form V^omega, where V is a finitary language over a finite alphabet X. They appear very naturally in the characterizaton of regular or context-free omega-languages. Since the set of infinite words over a finite alphabet X can be equipped with the usual Cantor topology, the question of the topological complexity of omega-powers of finitary languages naturally arises and has been posed by Niwinski (1990), Simonnet (1992) and Staiger (1997). It has been recently proved that for each integer n > 0, there exist some omega-powers of context free languages which are Pi^0_n-complete Borel sets, that there exists a context free language L such that L^omega is analytic but not Borel, and that there exists a finitary language V such that V^omega is a Borel set of infinite rank. But it was still unknown which could be the possible infinite Borel ranks of omega-powers. We fill this gap here, proving the following very surprising result which shows that omega-powers exhibit a great topological complexity: for each non-null countable ordinal alpha, there exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers., To appear in the Proceedings of the 16th EACSL Annual Conference on Computer Science and Logic, CSL 2007, Lausanne, Switzerland, September 11-15, 2007, Lecture Notes in Computer Science, (c) Springer, 2007

In a zero-dimensional Polish space, the Borel sets are generated from the clopen sets by repeatedly applying the operations of countable disjoint union and complementation. Here we look at topologically restrictive versions of the general countable disjoint union of sets, and obtain ``construction principles'' for $\bdz{3}$, i.e., sets which are both $\cF_{\sigma\delta}$ and $\cG_{\delta\sigma}$.

A classical theorem of Luzin is that the separation principle holds for the Pi^0_alpha sets but fails for the Sigma^0_alpha sets. We show that for every Sigma^0_alpha set A which is not Pi^0_alpha there exists a Sigma^0_alpha set B which is disjoint from A but cannot be separated from A by a Delta^0_alpha set C. Assuming Pi^1_1-determancy it follows from a theorem of Steel that a similar result holds for Pi^1_1 sets. On the other hand assuming V=L there is a proper Pi^1_1 set which is not half of a Borel inseparable pair. These results answer questions raised by F.Dashiell. Latest version at: www.math.wisc.edu/~miller/, Comment: LaTex2e 16 pages

International audience; Omega-powers of finitary languages are omega languages in the form V^omega, where V is a finitary language over a finite alphabet X. Since the set of infinite words over X can be equipped with the usual Cantor topology, the question of the topological complexity of omega-powers naturally arises and has been raised by Niwinski, by Simonnet, and by Staiger. It has been recently proved that for each integer n > 0 , there exist some omega-powers of context free languages which are Pi^0_n-complete Borel sets, and that there exists a context free language L such that L^omega is analytic but not Borel. But the question was still open whether there exists a finitary language V such that V^omega is a Borel set of infinite rank. We answer this question in this paper, giving an example of a finitary language whose omega-power is Borel of infinite rank.

We show that the Generalized Continuum Hypothesis GCH (its appropriate part) implies that many natural algebras on $\mathbb{R}$, including the algebra $\mathcal{B}$ of Borel sets and the interval algebra $\Sigma$, are outer Marczewski-Burstin representable by families of non-Borel sets. Also we construct, assuming again an appropriate part of GCH, that there are algebras on $\mathbb{R}$ which are not MB-representable. We prove that some algebras (including $\mathcal{B}$ and $\Sigma$) are not inner MB-representable. We give examples of algebras which are inner and outer MB-representable, or are inner but not outer MB-representable.

We prove that in an at least three dimensional Euclidean space the Dynkin system generated by the family of all open balls with radii at least one (that is, the smallest collection containing the open balls with radii at least one and closed under complements and countable disjoint unions) does not contain all Borel sets. We also give a simple characterization of the sets of this Dynkin system.

In \cite{T13}, Brian S. Thomson proved the following result: \emph{Let $f$ be $AC^*G$ on an interval $[a,b]$. Then the total variation measure $\mu = \mu_f$ associated with $f$ has the following properties: a) $\mu$ is a $\sigma$-finite Borel measure on $[a,b]$; b) $\mu$ is absolutely continuous with respect to Lebesgue measure; \linebreak c) There is a sequence of closed sets $F_n$ whose union is all of $[a,b]$ such that $\mu(F_n) < \infty$ for each $n$; d) $\mu(B) = \mu_f(B) = \int_B|f^\prime(x)|\, dx$ for every Borel set $B \subset [a,b]$. Conversely, if a measure $\mu$ satisfies conditions \linebreak a)--c) then there exists an $AC^*G$ function $f$ for which the representation d) is valid.} In this paper we improve Thomson's theorem as follows: in the first part we ask $f$ to be only $VB^*G \cap (N)$ on a Lebesgue measurable subset $P$ of $[a,b]$ and continuous at each point of $P$; the converse is also true even for $\mu$ defined on the Lebesgue measurable subsets of $P$ (see Theorem \ref{T2} and the two examples in Remark~\ref{R1}).

We prove the following two results. 1) There exists a non-separable complete metric space whose Wijsman hypertopology is not Čech-complete. 2) There exist a non-separable metrizable space and two compatible metrics on it, such that the collections of the Borel sets generated by the relative Wijsman hypertopologies do not coincide.

The Banach-Zarecki Theorem states that \(VB \cap (N) = AC\) for continuous functions on a closed set. Hence it is a linear space. In this article we show that \(VB \cap (N)\) is a linear space on any real Borel set. Hence \(VBG \cap (N)\) will also be a real linear space for Borel measurable functions defined on an interval. As a consequence of this result, we show that the \(AK_N\) integral of Gordon (\cite{G14}) is well defined. We also give answers to Gordon’s questions in \cite{G14}.

We consider two-person zero-sum stochastic games with limit superior payoff function and Borel measurable state and action spaces. The games are shown to have a value and the value function is calculated by transfinite iteration of an operator and proved to be upper analytic. The paper extends results of our earlier article [17] in which the same class of games was considered for countable state spaces and finite action sets.

We investigate the algebraic sums of sets for a large class of invari-ant ˙-ideals and ˙- elds of subsets of the real line. We give a simpleexample of two Borel subsets of the real line such that its algebraicsum is not a Borel set. Next we show a similar result to Proposition 2from A. Kharazishvili paper [4]. Our results are obtained for ideals withcoanalytical bases. 1 Introduction We shall work in ZFC set theory. By !we denote natural numbers. By 4wedenote the symmetric di erence of sets. The cardinality of a set Xwe denoteby jXj. By R we denote the real line and by Q we denote rational numbers. IfAand Bare subsets of R n and b2R , then A+B= fa+b: a2A^b2Bgand A+ b= A+ fbg. Similarly, if A R, B R n and a2R, then AB=fab: a2A^b2Bgand aB= fagB.We say that a family Fof subsets of R is invariant if for each A2F, a2Qand b2R we have aA+ b2F(see [3]).Let Ebe a polish space. If x2Eand ">0, then by B(x;") we denote theball with center xand radius ". The family of Borel subsets of Ewe denoteby Bor(E). For each

Let $I$ and $J$ be \s-ideals on Polish spaces $X$ and $Y$, respectively. We say that the pair $\langle I,J\rangle$ has the Fubini Property (FP) if for every Borel subset $B$ of $X\times Y$, if all its sections $B_x= \{y\: \langle x,y\rangle\in B\}$ are in $J$, then its sections $B^y=\{x\: \langle x,y\rangle\in B\}$ are in $I$, for every $y$ outside a set from $J$. We study the question, which pairs of $\sigma$-ideals have the Fubini Property. We show, in particular, that:-- $\langle$ MGR$(X), J\rangle$ satisfies FP, for every $J$ generated by any family of closed subsets of $Y$ (MGR$(X)$ is the $\sigma$-ideal of all meager subsets of $X$),-- $\langle$ NULL$_\mu, J \rangle$ satisfies FP, whenever $J$ is generated by any of the following families of closed subsets of $Y$ (NULL$_mu$ is the $\sigma$-ideal of all subsets of $X$, having outer measure zero with respect to a Borel $\sigma$-finite continuous measure $\mu$ on $X$):(i) all closed sets of cardinality $\leq 1$, (ii) all compact sets, (iii) all closed sets in NULL$_\nu$ for a Borel \s-finite continuous measure $\nu$ on $Y$, (iv) all closed subsets of a $\hbox { $\mathbf{\Pi}^1_1$}$ set $A\subseteq Y$. We also prove that $\langle$MGR$(X)$, MGR$(Y)\rangle$ and $\langle$ NULL$_\mu$, NULL$_\nu\rangle$ are essentially the only cases of FP in the class of \s-ideals obtained from MGR$(X)$ and NULL$_\mu$ by the operations of Borel isomorphism, product, extension and countable intersection.

The subset of Riemann integrable functions in L1[0,1] is a Borel set which is Π30-complete, i.e. Fσδ but not Gδσ.

The purpose of this note is to study s -covering maps f :X→Y with compact or more generally with complete (in the given metric on X ) fibers f −1 (y) . The question of whether f is inductively perfect has been recently negatively solved by Debs and Saint Raymond. In the first part of this paper we show that the answer is “yes” if f −1 (S) is complete in the given metric on X for every convergent including its limit sequence S⊂Y . In the second part point-harmonious maps are used in considering the properties of f . We prove that if X is a Borel set of the multiplicative class α then Y is a Borel set of the multiplicative class 3+α .

A family of games $G = G(\sigma, u)$ is defined such that (a) for each $\sigma$ the set of all $u$ for which Player I can force a win in $G(\sigma, u)$ is a Borel set $B(u)$ and (b) every Borel set is a $B(u)$ for some $u$.

In this paper, the essential properties of general Lebesgue outer measure are discussed. The complete measure space, consisting of the general Lebesgue outer measure restricted to the measurable sets, is developed and this measure is shown to be unique. Two characterizations of measurable sets are discussed. The Borel sets are inves- tigated in the plane and more generally, in n-space, and it is shown that the a-algebra of Borel sets is equal to the product a-algebra of Borel sets on the line. Finally, the interrelationships between Lebesgue measure in the plane and the product measure of Lebesgue measures on the line are investigated. It is shown that the a-algebra of Lebesgue measurable sets properly contains the product a-algebra and that these two measures agree on the product a-algebra. It is also proven that the a-algebra of Lebesgue measurable sets is the completion of the product a-algebra. Examples are provided to illustrate that the product measure spaces discussed are not complete as well as an example of a subset of the plane which is not Lebesgue measurable. http://archive.org/details/outermeasurebore1094515122 Ensign, United States Navy Approved for public release; distribution is unlimited.