Investigations of mass problems

Mass Problems and Randomness

Bulletin of Symbolic Logic ◽

10.2178/bsl/1107959497 ◽

2005 ◽

Vol 11 (1) ◽

pp. 1-27 ◽

Cited By ~ 40

Author(s):

Stephen G. Simpson

Keyword(s):

Computational Complexity ◽

Positive Measure ◽

Proof Theory ◽

Reverse Mathematics ◽

Equivalence Classes ◽

Distributive Lattices ◽

Intermediate Degree ◽

Associated Mass ◽

Weakly Reducible ◽

Mass Problems

AbstractA mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if every member of Q Turing computes a member of P. We say that P is strongly reducible to Q if every member of Q Turing computes a member of P via a fixed Turing functional. The weak degrees and strong degrees are the equivalence classes of mass problems under weak and strong reducibility, respectively. We focus on the countable distributive lattices ω and s of weak and strong degrees of mass problems given by nonempty subsets of 2ω. Using an abstract Gödel/Rosser incompleteness property, we characterize the subsets of 2ω whose associated mass problems are of top degree in ω and s, respectively Let R be the set of Turing oracles which are random in the sense of Martin-Löf, and let r be the weak degree of R. We show that r is a natural intermediate degree within ω. Namely, we characterize r as the unique largest weak degree of a subset of 2ω of positive measure. Within ω we show that r is meet irreducible, does not join to 1, and is incomparable with all weak degrees of nonempty thin perfect subsets of 2ω. In addition, we present other natural examples of intermediate degrees in ω. We relate these examples to reverse mathematics, computational complexity, and Gentzen-style proof theory.

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DEEP CLASSES

Bulletin of Symbolic Logic ◽

10.1017/bsl.2016.9 ◽

2016 ◽

Vol 22 (2) ◽

pp. 249-286 ◽

Cited By ~ 3

Author(s):

LAURENT BIENVENU ◽

CHRISTOPHER P. PORTER

Keyword(s):

Mutual Information ◽

Initial Segment ◽

Computability Theory ◽

Binary Sequences ◽

Algorithmic Randomness ◽

Positive Probability ◽

Probabilistic Algorithm ◽

Image Position ◽

The Relationship ◽

Mass Problems

AbstractA set of infinite binary sequences ${\cal C} \subseteq 2$ℕ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of ${\rm{\Pi }}_1^0 $ classes. In this paper, we introduce the notion of depth for ${\rm{\Pi }}_1^0 $ classes, which is a stronger form of negligibility. Whereas a negligible ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute a member of ${\cal C}$ with positive probability, a deep ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute an initial segment of a member of ${\cal C}$ with high probability. That is, the probability of computing a length n initial segment of a deep ${\rm{\Pi }}_1^0 $ class converges to 0 effectively in n.We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call tt-negligibility. We provide a number of examples of deep ${\rm{\Pi }}_1^0 $ classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin’s Ω.

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Solutions To Calendar

Mathematics Teacher ◽

10.5951/mt.89.2.0130 ◽

1996 ◽

Vol 89 (2) ◽

pp. 130-131

Keyword(s):

High School ◽

Mathematics Education ◽

Education Student ◽

University Of Georgia ◽

Ann Arbor ◽

Saint Francis ◽

The University ◽

Student Association ◽

Mass Problems

Problems 1–3 were submitted by Karen Doyle Walton, Allentown College of Saint Francis de Sales, Center Valley, PA 18034, and Zachary Walton, a student at Haryard Uniyersity, Cambridge, Mass. Problems 4–5 were furnished by Doug Wagner, 1995 PineDa Drive, Grayson, GA 30221. Problems 6–14 and 16–20 were sent in by teachers at Adlai E. Stevenson High School, Lincolnshire, IL 60069: 6, 7, and 19 by Dene Hamilton; 8, 9, and 10 by Joe Bettina; 11, 12, and 20 by Kathie Rauch; 13 by Neal Roys; 14 and 16 by Tim Kanold; and 17 and 18 by Scott Oliver. Problems 21–24 and 26–28 were created by the Mathematics Education Student Association at the University of Georgia, Athens, GA 30602: Karen Bell, Denise Spangler Mewborn, Mary Beth Searcy, Barry Shealy, Ron Tzur, and Bryan Whitfield. Problems 15 and 25 were taken from 101 Puzzle Problems by Nathaniel B. Bates and Sanderson M. Smith (Concord, Mass.: Bates Publishing Co., 1980). Problem 29 was submitted by Melvin R. Wilson, 610 E. University, Ann Arbor, MI 48109.

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MASS PROBLEMS AND HYPERARITHMETICITY

Journal of Mathematical Logic ◽

10.1142/s0219061307000652 ◽

2007 ◽

Vol 07 (02) ◽

pp. 125-143 ◽

Cited By ~ 16

Author(s):

JOSHUA A. COLE ◽

STEPHEN G. SIMPSON

Keyword(s):

Equivalence Class ◽

Cantor Space ◽

Mass Problem ◽

Ordinal Numbers ◽

Hyperarithmetical Hierarchy ◽

Index Sets ◽

Weakly Reducible ◽

Mass Problems

A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let [Formula: see text] be the lattice of weak degrees of mass problems associated with nonempty [Formula: see text] subsets of the Cantor space. The lattice [Formula: see text] has been studied in previous publications. The purpose of this paper is to show that [Formula: see text] partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in [Formula: see text] which are indexed by the ordinal numbers less than [Formula: see text] and which correspond to the hyperarithmetical hierarchy. Namely, for each [Formula: see text], let hα be the weak degree of 0(α), the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p* be the weak degree of the mass problem P* = {Y | ∃X (X ∈ P and BLR (X) ⊆ BLR (Y))} where BLR (X) is the set of functions which are boundedly limit recursive in X. Let 1 be the top degree in [Formula: see text]. We prove that all of the weak degrees [Formula: see text], [Formula: see text], are distinct and belong to [Formula: see text]. In addition, we prove that certain index sets associated with [Formula: see text] are [Formula: see text] complete.

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Mass problems and almost everywhere domination

Mathematical Logic Quarterly ◽

10.1002/malq.200710013 ◽

2007 ◽

Vol 53 (4-5) ◽

pp. 483-492 ◽

Cited By ~ 8

Author(s):

Stephen G. Simpson

Keyword(s):

Almost Everywhere ◽

Mass Problems

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The naturalness in the BLMSSM and B-LSSM

The European Physical Journal C ◽

10.1140/epjc/s10052-020-08768-0 ◽

2020 ◽

Vol 80 (12) ◽

Author(s):

Xing-Xing Dong ◽

Tai-Fu Feng ◽

Shu-Min Zhao ◽

Hai-Bin Zhang

Keyword(s):

Experimental Data ◽

Higgs Boson ◽

Neutrino Mass ◽

Experimental Results ◽

Fine Tuning ◽

Boson Mass ◽

Higgs Boson Mass ◽

The Right ◽

Mass Problems

AbstractIn order to interpret the Higgs boson mass and its decays naturally, we hope to examine the BLMSSM and B-LSSM. In the both models, the right-handed neutrino superfields are introduced to better explain the neutrino mass problems. In this paper, we introduce the fine-tuning to acquire the physical Higgs boson mass. Besides, the method of $$\chi ^2$$ χ 2 analyses will be adopted in the BLMSSM and B-LSSM to fit the experimental data. Therefore, we can obtain the reasonable theoretical values of the Higgs decays and muon $$g-2$$ g - 2 that are in accordance with the experimental results respectively in the BLMSSM and B-LSSM.

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