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Journal ArticleDOI

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

01 Dec 2019-The Bulletin of Symbolic Logic (Cambridge University Press (CUP))-Vol. 25, Iss: 4, pp 451-452
TL;DR: Modifications of Semmes's game characterization of the Borel functions are defined, obtaining game characterizations of the Baire class $\alpha$ functions for each fixed $\alpha < \omega_1$.
Abstract: 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.

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Citations
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01 Jan 2016
TL;DR: The classical descriptive set theory is universally compatible with any devices to read and is available in the book collection an online access to it is set as public so you can download it instantly.
Abstract: Thank you for downloading classical descriptive set theory. Maybe you have knowledge that, people have search numerous times for their chosen novels like this classical descriptive set theory, but end up in infectious downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they juggled with some malicious virus inside their laptop. classical descriptive set theory is available in our book collection an online access to it is set as public so you can download it instantly. Our digital library hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the classical descriptive set theory is universally compatible with any devices to read.

641 citations

Journal ArticleDOI
01 Jun 1978
TL;DR: The motivation for ONAG may have been, and perhaps was-and I would like to think that it was-the attempt to bridge the theory gap between nim-like and chess-like games.
Abstract: Some readers know to play the game of nim well, fewer play a perfect annihilation game, and nobody knows whether there exists an opening move in chess that will guarantee a win for white. These games and many more, belong to the family of combinatorial games, by which we mean the set of all two-player perfect-information games without chance moves and with outcomes lose or win (and sometimes: dynamic tie). The motivation for ONAG may have been, and perhaps was-and I would like to think that it was-the attempt to bridge the theory gap between nim-like and chess-like games. Why is there a gap? Every combinatorial game can be described as a directed graph called game-graph, whose vertices are the game positions, and (u, v) is a directed edge if and only if there is a move from position u to position v. Denote by N the set of all positions from which the Next (first) player can force a win; by P the set of all positions from which the Previous (second) player can force a win; and by T the set of all (dynamic) Tie positions, which are positions from which no player can force a win and therefore both can avoid losing. In an acyclic game-graph there cannot be any tie positions. The N, P, T classification of any game graph R = (V, E) can be determined in 0(\V\ + \E\) steps [8]. For both nim and chess, a finite game-graph can be constructed and the N, P, T classification can be determined. So both games are solvable in principle. If we play nim with n piles, each pile containing at most k tokens, then the game-graph contains (k + \) vertices. Suppose that in (generalized) chess played on an « X « board there are k different pieces. If k is about n/2, then the game-graph of chess contains O (2") vertices. So both game-graphs have exponentially many vertices, and thus both games appear intractable in the usual sense of computational complexity [1, Chapter 10], [14, Chapter 9], namely a computation appears to be required which is asymptotically exponential. From a computational efficiency standpoint, the essential difference between nim and chess is that nim can be viewed as a disjunctive compound (sum) of independent games, namely the individual piles. A disjunctive

306 citations

DOI
12 Dec 2016

184 citations

Book ChapterDOI
01 Jan 2014

144 citations

29 May 2019
TL;DR: A generic quantum algorithmic framework, capable of working with exponentially large matrices, that can apply polynomial transformations to the singular values of a block of a unitary is developed, and unifies a large number of prominent quantum algorithms.
Abstract: In this dissertation we study how efficiently quantum computers can solve various problems, and how large speedups can be achieved compared to classical computers. In particular we develop a generic quantum algorithmic framework that we call "quantum singular value transformation", capable of working with exponentially large matrices, that can apply polynomial transformations to the singular values of a block of a unitary. We show how quantum singular value transformation unifies a large number of prominent quantum algorithms, and show several problems where it leads to new quantum algorithms or improves earlier approaches. We develop an improved version of Jordan's quantum algorithm for gradient computation that can speed up the training of variational quantum optimization, and prove an essentially matching lower bound on quantum gradient computation. We also show that a quantum computer can very efficiently compute an approximate subgradient of a convex Lipschitz function. Combining this with some recent classical results we get improvements for black-box convex optimization problems. Then we take a new perspective on quantum SDP-solvers, introducing several new techniques, and improve on all prior quantum algorithms for SDP-solving. Finally we study the variable version of the Lovasz Local Lemma (LLL) and its quantum generalization. We improve on the previous constructive quantum results by designing an algorithm that works efficiently for non-commuting terms as well, assuming that the system is "uniformly" gapped. For the variable version of the classical LLL we find optimal bounds for the "guaranteed-to-be-feasible" probabilities on cyclic dependency graphs.

22 citations

References
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Book
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3,954 citations

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02 Nov 2011
TL;DR: The Foundations of Set Theory and Infinitary Combinatorics are presented, followed by a discussion of easy Consistency Proofs and Defining Definability.
Abstract: The Foundations of Set Theory. Infinitary Combinatorics. The Well-Founded Sets. Easy Consistency Proofs. Defining Definability. The Constructible Sets. Forcing. Iterated Forcing. Bibliography. Indexes.

1,506 citations

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TL;DR: In this paper, the development of Mathematics within Subsystems of Z2 is discussed, with a focus on recursive comprehension and weak Konig's lemma, and a discussion of models of sub-systems.
Abstract: List of tables Preface Acknowledgements 1. Introduction Part I. Development of Mathematics within Subsystems of Z2: 2. Recursive comprehension 3. Arithmetical comprehension 4. Weak Konig's lemma 5. Arithmetical transfinite recursion 6. pi11 comprehension Part II. Models of Subsystems of Z2: 7. ss-models 8. omega-models 9. Non-omega-models Part III. Appendix: 10. Additional results Bibliography Index.

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TL;DR: Theories of Recursive functions, Hierarchies of recursive functions, and Arithmetical sets: Recursively enumerable sets.
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1,055 citations

Book
01 Jan 1988
TL;DR: If the lesson of the paradoxes of set theory is that a predicate need no more have a set as extension than the name "Santa Claus" need denote someone, then perhaps the lessons of the liar paradox is that nothing answers to a liar sentence.
Abstract: Most of us have learnt to live with the fact that some singular terms do not denote. It would be fun were it otherwise, but no jolly fat man at the North Pole brings presents to good children at Yuletide; the name "Santa Claus" does not denote. Sets and classes may first have presented themselves to us as extensions of predicates, but it seems now the consensus that the paradoxes of set theory show that not all predicates have sets as extensions. Ramsey notwithstanding, many agree with the early Russell, and with Poincare, in seeing an affinity between Russell's paradox and the paradox of the liar. So if the lesson of the paradoxes of set theory is that a predicate need no more have a set as extension than the name "Santa Claus" need denote someone, then perhaps the lesson of the liar paradox is that nothing answers to a liar sentence.i To try this idea out systematically, we want some categories. Some singular terms denote, and "object" is a general term for the sort of thing singular terms succeed in denoting. We may say that there is no object denoted by the name "Santa Claus".

810 citations


"Games for Functions: Baire Classes,..." refers background in this paper

  • ...and later popularized by Aczel [1]—in the style of Aczel [1, Chapter 6], in ZFC− + AFA the set of informal trees can be defined as the greatest fixed point of the class operator Φ defined by letting Φ(X) be the class of all countable sets of elements of the form (n, T ), with n ∈ ω and T a countable subset of X....

    [...]