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Relevant bibliographies by topics / Borel hierarchy

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Academic literature on the topic 'Borel hierarchy'

Author: Grafiati

Published: 4 June 2021

Last updated: 27 April 2022

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Journal articles on the topic "Borel hierarchy"

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Boom, M. Vanden. "The effective Borel hierarchy." Fundamenta Mathematicae 195, no. 3 (2007): 269–89. http://dx.doi.org/10.4064/fm195-3-4.

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Duparc, J. "Wadge hierarchy and Veblen hierarchy Part I: Borel sets of finite rank." Journal of Symbolic Logic 66, no. 1 (2001): 56–86. http://dx.doi.org/10.2307/2694911.

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AbstractWe consider Borel sets of finite rank A ⊆ ∧ω where cardinality of Λ is less than some uncountable regular cardinal . We obtain a “normal form” of A, by finding a Borel set Ω, such that A and Ω continuously reduce to each other. In more technical terms: we define simple Borel operations which are homomorphic to ordinal sum, to multiplication by a countable ordinal, and to ordinal exponentiation of base , under the map which sends every Borel set A of finite rank to its Wadge degree.

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Becher, Verónica, Pablo Ariel Heiber, and Theodore A. Slaman. "Normal numbers and the Borel hierarchy." Fundamenta Mathematicae 226, no. 1 (2014): 63–77. http://dx.doi.org/10.4064/fm226-1-4.

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Gartside, Paul M., and Joseph T. H. Lo. "The hierarchy of Borel universal sets." Topology and its Applications 119, no. 2 (2002): 117–29. http://dx.doi.org/10.1016/s0166-8641(01)00070-0.

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Finkel, Olivier. "Borel hierarchy and omega context free languages." Theoretical Computer Science 290, no. 3 (2003): 1385–405. http://dx.doi.org/10.1016/s0304-3975(02)00042-7.

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VELDMAN, WIM. "THE FINE STRUCTURE OF THE INTUITIONISTIC BOREL HIERARCHY." Review of Symbolic Logic 2, no. 1 (2009): 30–101. http://dx.doi.org/10.1017/s1755020309090121.

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In intuitionistic analysis, a subset of a Polish space like ℝ or ${\cal N}$ is called positively Borel if and only if it is an open subset of the space or a closed subset of the space or the result of forming either the countable union or the countable intersection of an infinite sequence of (earlier constructed) positively Borel subsets of the space. The operation of taking the complement is absent from this inductive definition, and, in fact, the complement of a positively Borel set is not always positively Borel itself (see Veldman, 2008a). The main result of Veldman (2008a) is that, assumi

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Ros, Luca Motto. "Borel-amenable reducibilities for sets of reals." Journal of Symbolic Logic 74, no. 1 (2009): 27–49. http://dx.doi.org/10.2178/jsl/1231082301.

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AbstractWe show that if ℱ is any “well-behaved” subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on (ωω) induced by ℱ turns out to look like the Wadge hierarchy (which is the special case where ℱ is the set of continuous functions).

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Veldman, Wim. "The Borel Hierarchy Theorem from Brouwer's intuitionistic perspective." Journal of Symbolic Logic 73, no. 1 (2008): 1–64. http://dx.doi.org/10.2178/jsl/1208358742.

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AbstractIn intuitionistic analysis, Brouwer's Continuity Principle implies, together with an Axiom of Countable Choice, that the positively Borel sets form a genuinely growing hierarchy: every level of the hierarchy contains sets that do not occur at any lower level.

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Duparc, J. "The Steel hierarchy of ordinal valued Borel mappings." Journal of Symbolic Logic 68, no. 1 (2003): 187–234. http://dx.doi.org/10.2178/jsl/1045861511.

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AbstractGiven well ordered countable sets of the form Λϕ, we consider Borel mappings from Λϕω with countable image inside the ordinals. The ordinals and Λϕω are respectively equipped with the discrete topology and the product of the discrete topology on Λϕ. The Steel well-ordering on such mappings is denned by ϕ ≤ ψ iff there exists a continuous function f such that ϕ(x) ≤ ψof(x) holds for any x ϵ Λϕω. It induces a hierarchy of mappings which we give a complete description of. We provide, for each ordinal α, a mapping whose rank is precisely α in this hierarchy and we also compute the height o

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Finkel, Olivier, and Pierre Simonnet. "On Recognizable Tree Languages Beyond the Borel Hierarchy." Fundamenta Informaticae 95, no. 2-3 (2009): 287–303. http://dx.doi.org/10.3233/fi-2009-151.

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Dissertations / Theses on the topic "Borel hierarchy"

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Fournier, Kevin. "The Wadge Hierarchy : Beyond Borel Sets." Sorbonne Paris Cité, 2016. http://www.theses.fr/2016USPCC005.

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Cette thèse est dévolue à l'étude des sous-ensembles Ale non-boréliens de l'espace de Baire. Dans une première partie, nous généralisons les résultats obtenus par Duparc et Louveau pour obtenir une description complète de la hiérarchie de Wadge des différences croissantes d'ensembles coanalytiques, sous l'hypothèse que tous les ensembles analytiques sont déterminés. Ensuite, nous étudions certaines classes incluses dans la classe A1/2, les classes de Selivanovski et celles de Kolmogorov, et donnons un fragment de leur hiérarchie de Wadge. Finalement, nous appliquons les techniques et résultats

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Cotton, Michael R. "Determinacy in the Low Levels of the Projective Hierarchy." Miami University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=miami1343245802.

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Hummel, Szczepan. "Topological Complexity of Sets Defined by Automata and Formulas." Doctoral thesis, 2017. https://depotuw.ceon.pl/handle/item/2337.

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In this thesis we consider languages of infinite words or trees defined by automata of various types or formulas of various logics. We ask about the highest possible position in the Borel or the projective hierarchy inhabited by sets defined in a given formalism. The answer to this question is called the topological complexity of the formalism.It is shown that the topological complexity of Monadic Second Order Logic extended with the unbounding quantifier (introduced by Bojańczyk to express some asymptotic properties) over ω-words is the whole projective hierarchy. We also give the exact topo

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Book chapters on the topic "Borel hierarchy"

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Miller, Arnold W. "Borel Hierarchy." In Descriptive Set Theory and Forcing. Springer Berlin Heidelberg, 1995. http://dx.doi.org/10.1007/978-3-662-21773-3_2.

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Kechris, Alexander S. "The Borel Hierarchy." In Graduate Texts in Mathematics. Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4612-4190-4_22.

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Formenti, Enrico, Markus Holzer, Martin Kutrib, and Julien Provillard. "ω-rational Languages: High Complexity Classes vs. Borel Hierarchy." In Language and Automata Theory and Applications. Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-04921-2_30.

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Skurczyński, Jerzy. "The Borel hierarchy is infinite in the class of regular sets of trees." In Fundamentals of Computation Theory. Springer Berlin Heidelberg, 1989. http://dx.doi.org/10.1007/3-540-51498-8_40.

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Downey, Rod, and Noam Greenberg. "Introduction." In A Hierarchy of Turing Degrees. Princeton University Press, 2020. http://dx.doi.org/10.23943/princeton/9780691199665.003.0001.

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This introductory chapter provides an overview of computability theory. The roots of computability theory go back to the work of Borel, Dedekind, Hermann, Dehn, and others in the late nineteenth and early twentieth centuries. From a modern point of view, these authors were highly interested in algorithmic procedures in algebra. What does it take to perform a certain construction? In computability theory, this question is the basis of a long-term programme which seeks to understand the relationship between dynamic properties of sets and their algorithmic complexity. The main thesis of this book is that where the computably enumerable (c.e.) Turing degrees are concerned, a degree can compute complicated objects if and only if some functions in the degree are difficult to approximate. Computability-theoretic tools allow one to quantify precisely what is meant by “difficult to approximate,”

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Downey, Rod, and Noam Greenberg. "Presentations of left-c.e. reals." In A Hierarchy of Turing Degrees. Princeton University Press, 2020. http://dx.doi.org/10.23943/princeton/9780691199665.003.0005.

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This chapter examines presentations of left–c.e. reals, proving Theorem 1.4. One of the main ideas of this book is unifying the combinatorics of constructions in various subareas of computability theory. The chapter looks at one such subarea: algorithmic randomness. It provides a brief account of the basics of algorithmic randomness, and includes the basic definitions required in the chapter. While algorithmic randomness has a history going back to the early work of Borel on normal numbers, von Mises, and even Turing, the key concept in the modern incarnation of algorithmic information theory is Martin-Löf randomness. A notion of randomness is determined by a countable collection of null sets, with each null set considered a statistical test. Elements of the null sets are those which have failed the test; they are atypical, in the sense of measure. One of the reasons the notion of ML-randomness is central is that it is robust.

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Velji, Jamel A. "Taʾwīl of an Apocalyptic Transcript II: The Book of Righteousness and True Guidance." In An Apocalyptic History of the Early Fatimid Empire. Edinburgh University Press, 2016. http://dx.doi.org/10.3366/edinburgh/9780748690886.003.0005.

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Chapter four examines another text of taʾwīl dated to the Fatimid revolution, the Kitāb al-rushd wa-l-hidāya, or Book of righteousness and true guidance. It discusses how numerical correspondences between Quranic chapters and members of the Fatimid hierarchy also became a hermeneutic tool pointing to the imminent advent of the mahdi. The Kitāb al-rushd wa-l-hidāya also reflects the Kitāb al-kashf’s deployment of symbolism that equated elements of the Quran’s eschatological or apocalyptic imagery with the mahdi and his advent. The chapter ends with a very brief examination of numismatic materials, illustrating how some of the earliest Fatimid coins bore Quranic inscriptions that could be read as a reflection of the Fatimids’ emergence as having inaugurating the awaited earthly utopia.

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