DEEP CLASSES

2016 ◽  
Vol 22 (2) ◽  
pp. 249-286 ◽  
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.

scholarly journals Topics in Algorithmic Randomness and Computability Theory

2021 ◽  
Author(s):  
◽  
Michael McInerney
Keyword(s):  
Binary Sequence ◽  
Computability Theory ◽  
Binary Sequences ◽  
Random Sets ◽  
Turing Degrees ◽  
Algorithmic Randomness ◽  
Computational Power ◽  
Definition Of ◽  
Compare And Contrast

<p>This thesis establishes results in several different areas of computability theory.  The first chapter is concerned with algorithmic randomness. A well-known approach to the definition of a random infinite binary sequence is via effective betting strategies. A betting strategy is called integer-valued if it can bet only in integer amounts. We consider integer-valued random sets, which are infinite binary sequences such that no effective integer-valued betting strategy wins arbitrarily much money betting on the bits of the sequence. This is a notion that is much weaker than those normally considered in algorithmic randomness. It is sufficiently weak to allow interesting interactions with topics from classical computability theory, such as genericity and the computably enumerable degrees. We investigate the computational power of the integer-valued random sets in terms of standard notions from computability theory.  In the second chapter we extend the technique of forcing with bushy trees. We use this to construct an increasing ѡ-sequence 〈an〉 of Turing degrees which forms an initial segment of the Turing degrees, and such that each an₊₁ is diagonally noncomputable relative to an. This shows that the DNR₀ principle of reverse mathematics does not imply the existence of Turing incomparable degrees.   In the final chapter, we introduce a new notion of genericity which we call ѡ-change genericity. This lies in between the well-studied notions of 1- and 2-genericity. We give several results about the computational power required to compute these generics, as well as other results which compare and contrast their behaviour with that of 1-generics.</p>

Algorithmic Randomness and Measures of Complexity

2013 ◽  
Vol 19 (3) ◽  
pp. 318-350 ◽  
Author(s):  
George Barmpalias
Keyword(s):  
Initial Segment ◽  
Computability Theory ◽  
Algorithmic Randomness ◽  
Relative Complexity ◽  
Recent Advances

AbstractWe survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.

scholarly journals Topics in Algorithmic Randomness and Computability Theory

2021 ◽  
Author(s):  
◽  
Michael McInerney
Keyword(s):  
Binary Sequence ◽  
Computability Theory ◽  
Binary Sequences ◽  
Random Sets ◽  
Turing Degrees ◽  
Algorithmic Randomness ◽  
Computational Power ◽  
Definition Of ◽  
Compare And Contrast

<p>This thesis establishes results in several different areas of computability theory.  The first chapter is concerned with algorithmic randomness. A well-known approach to the definition of a random infinite binary sequence is via effective betting strategies. A betting strategy is called integer-valued if it can bet only in integer amounts. We consider integer-valued random sets, which are infinite binary sequences such that no effective integer-valued betting strategy wins arbitrarily much money betting on the bits of the sequence. This is a notion that is much weaker than those normally considered in algorithmic randomness. It is sufficiently weak to allow interesting interactions with topics from classical computability theory, such as genericity and the computably enumerable degrees. We investigate the computational power of the integer-valued random sets in terms of standard notions from computability theory.  In the second chapter we extend the technique of forcing with bushy trees. We use this to construct an increasing ѡ-sequence 〈an〉 of Turing degrees which forms an initial segment of the Turing degrees, and such that each an₊₁ is diagonally noncomputable relative to an. This shows that the DNR₀ principle of reverse mathematics does not imply the existence of Turing incomparable degrees.   In the final chapter, we introduce a new notion of genericity which we call ѡ-change genericity. This lies in between the well-studied notions of 1- and 2-genericity. We give several results about the computational power required to compute these generics, as well as other results which compare and contrast their behaviour with that of 1-generics.</p>

Classifying model-theoretic properties

2008 ◽  
Vol 73 (3) ◽  
pp. 885-905 ◽  
Author(s):  
Chris J. Conidis
Keyword(s):  
Model Theory ◽  
Computability Theory ◽  
Prime Model ◽  
Decidable Theory ◽  
Dense Sets ◽  
Equivalent Properties ◽  
The Relationship

AbstractIn 2004 Csima, Hirschfeldt, Knight, and Soare [1] showed that a set A ≤T 0′ is nonlow2 if and only if A is prime bounding, i.e., for every complete atomic decidable theory T, there is a prime model computable in A. The authors presented nine seemingly unrelated predicates of a set A, and showed that they are equivalent for sets. Some of these predicates, such as prime bounding, and others involving equivalence structures and abelian p-groups come from model theory, while others involving meeting dense sets in trees and escaping a given function come from pure computability theory.As predicates of A, the original nine properties are equivalent for sets; however, they are not equivalent in general. This article examines the (degree-theoretic) relationship between the nine properties. We show that the nine properties fall into three classes, each of which consists of several equivalent properties. We also investigate the relationship between the three classes, by determining whether or not any of the predicates in one class implies a predicate in another class.

scholarly journals Slaves to Rome: The Rhetoric of Mastery in Titus' Speech to the Jews (Bellum Judaicum 6.328-50)

Ramus ◽  
2007 ◽  
Vol 36 (1) ◽  
pp. 25-38 ◽  
Author(s):  
Myles Lavan
Keyword(s):  
Historical Account ◽  
Self Presentation ◽  
Image Position ◽  
Relationship Of ◽  
The Relationship ◽  
Made In

(BJ6.350)Those who discard their weapons and surrender their persons, I will let live. Like a lenient master in a household, I will punish the incorrigible but preserve the rest for myself.So ends Titus' address to the embattled defenders of Jerusalem in the sixth book of Josephus'Jewish War(6.328-50). It is the most substantial instance of communication between Romans and Jews in the work. Titus compares himself to the master of a household and the Jewish rebels to his slaves. Is this how we expect a Roman to describe empire? If not, what does it mean for our understanding of the politics of Josephus' history? The question is particularly acute given that this is not just any Roman but Titus himself: heir apparent and, if we believe Josephus, the man who read and approved this historical account. It is thus surprising that, while the speeches of Jewish advocates of submission to Rome such as Agrippa II (2.345-401) and Josephus himself (5.362-419) have long fascinated readers, Titus' speech has received little or no attention. Remarkably, it is not mentioned in any of three recent collections of essays on Josephus. This paper aims to highlight the rhetorical choices that Josephus has made in constructing this voice for Titus—particularly his self-presentation as master—and the interpretive questions these raise for his readers. It should go without saying that the relationship of this text to anything that Titus may have said during the siege is highly problematic. (Potentially more significant, but unfortunately no less speculative, is the question of how it might relate to any speech recorded in the commentaries of Vespasian and Titus that Josephus appears to have used as a source.) What we have is a Josephan composition that is embedded in the broader narrative of theJewish War.

A note on the relationship between live weight and the incidence of oestrus in Hereford heifers

1980 ◽  
Vol 31 (2) ◽  
pp. 221-222 ◽  
Author(s):  
R. D. H. Cohen ◽  
D. L. Garden ◽  
J. P. Langlands
Keyword(s):  
Live Weight ◽  
Weight Range ◽  
Image Position ◽  
The Relationship

ABSTRACTThe incidence of oestrus in 603 Hereford heifers with a mean age of 545 days (range 515 to 575) was related to live weight (range 130 to 376 kg). Observations were ranked in order of ascending live weight and the percentage of heifers showing oestrus was calculated for consecutive 10 kg increments in live weight from 130 to 380 kg. The percentage showing oestrus was subjected to the probit transformation (Y, probit units) and was then related to live weight (X, kg):It was predicted that 50% of the heifers showed oestrus at 231 ± 1 kg, and that 5% and 95% showed oestrus at 187 and 280 kg respectively.

Sublattices and Initial Segments of the Degrees of Unsolvability

1970 ◽  
Vol 22 (3) ◽  
pp. 569-581 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Lattice Theory ◽  
Initial Segment ◽  
Finite Lattice ◽  
Equivalence Relations ◽  
Finite Lattices ◽  
Image Position ◽  
Degrees Of Unsolvability

In this paper we shall prove that every finite lattice is isomorphic to a sublattice of the degrees of unsolvability, and that every one of a certain class of finite lattices is isomorphic to an initial segment of degrees.Acknowledgment. I am grateful to Ralph McKenzie for his assistance in matters of lattice theory.1. Representation of lattices. The equivalence lattice of the set S consists of all equivalence relations on S, ordered by setting θ ≦ θ’ if for all a and b in S, a θ b ⇒ a θ’ b. The least upper bound and greatest lower bound in are given by the ⋃ and ⋂ operations:

INITIAL SEGMENTS OF THE ENUMERATION DEGREES

2016 ◽  
Vol 81 (1) ◽  
pp. 316-325 ◽  
Author(s):  
HRISTO GANCHEV ◽  
ANDREA SORBI
Keyword(s):  
Local Structure ◽  
Initial Segment ◽  
Nonzero Element ◽  
Enumeration Degrees ◽  
Enumeration Degree ◽  
Image Position

AbstractUsing properties of${\cal K}$-pairs of sets, we show that every nonzero enumeration degreeabounds a nontrivial initial segment of enumeration degrees whose nonzero elements have all the same jump asa. Some consequences of this fact are derived, that hold in the local structure of the enumeration degrees, including: There is an initial segment of enumeration degrees, whose nonzero elements are all high; there is a nonsplitting high enumeration degree; every noncappable enumeration degree is high; every nonzero low enumeration degree can be capped by degrees of any possible local jump (i.e., any jump that can be realized by enumeration degrees of the local structure); every enumeration degree that bounds a nonzero element of strictly smaller jump, is bounding; every low enumeration degree below a non low enumeration degreeacan be capped belowa.

scholarly journals Correlated activity favors synergistic processing in local cortical networks in vitro at synaptically relevant timescales

2020 ◽  
Vol 4 (3) ◽  
pp. 678-697
Author(s):  
Samantha P. Sherrill ◽  
Nicholas M. Timme ◽  
John M. Beggs ◽  
Ehren L. Newman
Keyword(s):  
Information Processing ◽  
Mutual Information ◽  
Information Transmission ◽  
Cortical Circuits ◽  
Correlated Activity ◽  
Neural Information ◽  
Neural Information Processing ◽  
The Relationship ◽  
Information Values

Neural information processing is widely understood to depend on correlations in neuronal activity. However, whether correlation is favorable or not is contentious. Here, we sought to determine how correlated activity and information processing are related in cortical circuits. Using recordings of hundreds of spiking neurons in organotypic cultures of mouse neocortex, we asked whether mutual information between neurons that feed into a common third neuron increased synergistic information processing by the receiving neuron. We found that mutual information and synergistic processing were positively related at synaptic timescales (0.05–14 ms), where mutual information values were low. This effect was mediated by the increase in information transmission—of which synergistic processing is a component—that resulted as mutual information grew. However, at extrasynaptic windows (up to 3,000 ms), where mutual information values were high, the relationship between mutual information and synergistic processing became negative. In this regime, greater mutual information resulted in a disproportionate increase in redundancy relative to information transmission. These results indicate that the emergence of synergistic processing from correlated activity differs according to timescale and correlation regime. In a low-correlation regime, synergistic processing increases with greater correlation, and in a high-correlation regime, synergistic processing decreases with greater correlation.

Divergent population processes and mammal outbreaks

1967 ◽  
Vol 4 (01) ◽  
pp. 1-8 ◽  
Author(s):  
P. Holgate
Keyword(s):  
Infinite System ◽  
Transition Probabilities ◽  
Positive Probability ◽  
System Of Differential Equations ◽  
Mathematical Ecology ◽  
Birth And Death Processes ◽  
Sample Paths ◽  
Image Position ◽  
Birth Processes ◽  
Necessary And Sufficient

Birth and death processes, i.e. Markov chains {X(t)}, 0 ≦ t ≦ ∞, taking values in the set I of non-negative integers, for which the infinitesimal transition probabilities are have found wide application in mathematical ecology . In order that the infinite system of differential Equations (1) should have solutions such that Pij (t) ≦ 0, all t and it is necessary and sufficient that . The conditions ensure that with probability one, the sample paths of X(t) have no discontinuities worse than jumps, and in this case the solution is unique. If (3) does not hold then for some t 1 at least, there is a set of sample paths of positive probability, each of which has t 1 as a limit point of jumps. For pure birth processes, (in which μj = 0) this clearly implies X(t) → ∞ as t → t 1.

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