Shift-complex sequences

2013 ◽  
Vol 19 (2) ◽  
pp. 199-215 ◽  
Author(s):  
Mushfeq Khan
Keyword(s):  
Kolmogorov Complexity ◽  
Initial Segment ◽  
Random Sequence ◽  
Binary Sequence ◽  
Low Complexity ◽  
Constant Number ◽  
Random Sequences ◽  
Halting Problem ◽  
Complex Sequences ◽  
Do So

AbstractA Martin-Löf random sequence is an infinite binary sequence with the property that every initial segment σ has prefix-free Kolmogorov complexity K(σ) at least ∣σ∣ − c, for some constant c ϵ ω. Informally, initial segments of Martin-Löf randoms are highly complex in the sense that they are not compressible by more than a constant number of bits. However, all Martin-Löf randoms necessarily have contiguous substrings of arbitrarily low complexity. If we demand that all substrings of a sequence be uniformly complex, then we arrive at the notion of shift-complex sequences. In this paper, we collect some of the existing results on these sequences and contribute two new ones. Rumyantsev showed that the measure of oracles that compute shift-complex sequences is zero. We strengthen this result by proving that the Martin-Löf random sequences that do not compute shift-complex sequences are exactly the incomplete ones, in other words, the ones that do not compute the halting problem. In order to do so, we make use of the characterization by Franklin and Ng of the class of incomplete Martin-Löf randoms via a notion of randomness called difference randomness. Turning to the power of shift-complex sequences as oracles, we show that there are shift-complex sequences that do not compute Martin-Löf random (or even Kurtz random) sequences.

On Producing Random Responses

1964 ◽  
Vol 14 (3) ◽  
pp. 931-941 ◽  
Author(s):  
Robert L. Weiss
Keyword(s):  
Individual Differences ◽  
Psychiatric Patients ◽  
Random Sequence ◽  
New Method ◽  
Random Sequences ◽  
Marked Individual ◽  
Binary Choices ◽  
Response Patterning ◽  
Random Responses ◽  
Do So

This paper has surveyed studies of response patterning in an attempt to illustrate the various ways in which this fact of behavior has been used in psychology. The view is expressed that response biasing takes on significance as an important dependent variable when we note that human Ss are unable to generate binary choices in a random sequence even when instructed to do so. Production of random sequences may be intimately related to processes of set and attention; success in generating responses randomly indicates ability to maintain an appropriate set for randomness. Techniques for measuring response patterning, or variability, were reviewed, and a new method was described. Data were presented to illustrate the difficulties encountered by students and psychiatric patients when instructed to generate a random sequence of binary choices. Almost nothing is known about the correlates of ability to maintain a set for randomness, yet there are marked individual differences in this ability. Generating random choices is basically a very non-stimulating task, so that differences in arousal may be of particular importance, when studying this particular kind of set.

scholarly journals Π10 classes and strong degree spectra of relations

2007 ◽  
Vol 72 (3) ◽  
pp. 1003-1018 ◽  
Author(s):  
John Chisholm ◽  
Jennifer Chubb ◽  
Valentina S. Harizanov ◽  
Denis R. Hirschfeldt ◽  
Carl G. Jockusch ◽  
...  
Keyword(s):  
Kolmogorov Complexity ◽  
Initial Segment ◽  
Truth Table ◽  
Linear Ordering ◽  
Computable Structures ◽  
Degree Spectra

AbstractWe study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable subsets of 2ω and Kolmogorov complexity play a major role in the proof.

scholarly journals Continuous higher randomness

2017 ◽  
Vol 17 (01) ◽  
pp. 1750004 ◽  
Author(s):  
Laurent Bienvenu ◽  
Noam Greenberg ◽  
Benoit Monin
Keyword(s):  
Random Sequences ◽  
Turing Reducibility ◽  
Do So

We investigate the role of continuous reductions and continuous relativization in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with respect to van Lambalgen’s theorem and the Miller–Yu/Levin theorem. We study lowness for continuous relativization of randomness, and show the equivalence of the higher analogues of the different characterizations of lowness for Martin-Löf randomness. We also characterize computing higher [Formula: see text]-trivial sets by higher random sequences. We give a separation between higher notions of randomness, in particular between higher weak 2-randomness and [Formula: see text]-randomness. To do so we investigate classes of functions computable from Kleene’s [Formula: see text] based on strong forms of the higher limit lemma.

scholarly journals Von Mises' definition of random sequences reconsidered

1987 ◽  
Vol 52 (3) ◽  
pp. 725-755 ◽  
Author(s):  
Michiel van Lambalgen
Keyword(s):  
Statistical Tests ◽  
Random Sequence ◽  
The Other ◽  
Selection Procedures ◽  
Random Sequences ◽  
Definition Of ◽  
Von Mises ◽  
The Relationship

AbstractWe review briefly the attempts to define random sequences (§0). These attempts suggest two theorems: one concerning the number of subsequence selection procedures that transform a random sequence into a random sequence (§§1–3 and 5); the other concerning the relationship between definitions of randomness based on subsequence selection and those based on statistical tests (§4).

scholarly journals RECYCLE OF RANDOM SEQUENCES

1993 ◽  
Vol 04 (03) ◽  
pp. 569-590 ◽  
Author(s):  
NOBUYASU ITO ◽  
MACOTO KIKUCHI ◽  
YUTAKA OKABE
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Random Sequence ◽  
Random Sequences

The correlation between a random sequence and its transformed sequences is studied. In the case of a permutation operation or, in other words, the shuffling operation, it is shown that the correlation can be so small that the sequences can be regarded as independent random sequences. The applications to the Monte Carlo simulations are also given. This method is especially useful in the Ising Monte Carlo simulation.

scholarly journals Π10 classes with complex elements

2008 ◽  
Vol 73 (4) ◽  
pp. 1341-1353 ◽  
Author(s):  
Stephen Binns
Keyword(s):  
Kolmogorov Complexity ◽  
Binary Sequence ◽  
Computable Function ◽  
Cantor Set ◽  
Linear Orders ◽  
Complex Element ◽  
Bounded Below

AbstractAn infinite binary sequence is complex if the Kolmogorov complexity of its initial segments is bounded below by a computable function. We prove that a class P contains a complex element if and only if it contains a wtt-cover for the Cantor set. That is, if and only if for every Y ⊆ ω there is an X in P such that X ≥wttY. We show that this is also equivalent to the class's being large in some sense. We give an example of how this result can be used in the study of scattered linear orders.

scholarly journals Lowness for effective Hausdorff dimension

2014 ◽  
Vol 14 (02) ◽  
pp. 1450011 ◽  
Author(s):  
Steffen Lempp ◽  
Joseph S. Miller ◽  
Keng Meng Ng ◽  
Daniel D. Turetsky ◽  
Rebecca Weber
Keyword(s):  
Hausdorff Dimension ◽  
Random Sequence ◽  
Effective Dimension ◽  
Random Sequences ◽  
Central Notion

We examine the sequences A that are low for dimension, i.e. those for which the effective (Hausdorff) dimension relative to A is the same as the unrelativized effective dimension. Lowness for dimension is a weakening of lowness for randomness, a central notion in effective randomness. By considering analogues of characterizations of lowness for randomness, we show that lowness for dimension can be characterized in several ways. It is equivalent to lowishness for randomness, namely, that every Martin-Löf random sequence has effective dimension 1 relative to A, and lowishness for K, namely, that the limit of KA(n)/K(n) is 1. We show that there is a perfect [Formula: see text]-class of low for dimension sequences. Since there are only countably many low for random sequences, many more sequences are low for dimension. Finally, we prove that every low for dimension is jump-traceable in order nε, for any ε > 0.

scholarly journals A Model for Quantifying the Agent of a Complex Network in Conditions of Incomplete Awareness

2021 ◽  
pp. 26-35
Author(s):  
Andrey Kalashnikov ◽  
◽  
Konstantin Bugajskij ◽  
Keyword(s):  
Information Systems ◽  
Complex Network ◽  
Kolmogorov Complexity ◽  
Binary Sequence ◽  
Data Transformation ◽  
Point Of View ◽  
Negative Consequences ◽  
Complex Information ◽  
Uncertainty Function ◽  
Qualitative Level

Purpose of the article: development of a mechanism for quantitative evaluation of elements of complex information systems in conditions of insufficient information about the presence of vulnerabilities. Research method: mathematical modeling of uncertainty estimation based on binary convolution and Kolmogorov complexity. Data banks on vulnerabilities and weaknesses are used as initial data for modeling. The result: it is shown that the operation of an element of a complex network can be represented by data transformation procedures, which consist of a sequence of operations in time, described by weaknesses and related vulnerabilities. Each operation can be evaluated at a qualitative level in terms of the severity of the consequences in the event of the implementation of potential weaknesses. The use of binary convolution and universal coding makes it possible to translate qualitative estimates into a binary sequence – a word in the alphabet {0,1}. The sequence of such words — as the uncertainty function — describes the possible negative consequences of implementing data transformation procedures due to the presence of weaknesses in an element of a complex system. It is proposed to use the Kolmogorov complexity to quantify the uncertainty function. The use of a Turing machine for calculating the uncertainty function provides a universal mechanism for evaluating elements of complex information systems from the point of view of information security, regardless of their software and hardware implementation.

scholarly journals Probability and Randomness

2017 ◽  
Author(s):  
Antony Eagle
Keyword(s):  
Twentieth Century ◽  
Kolmogorov Complexity ◽  
Random Sequence ◽  
Frequency Theory ◽  
Good Evidence ◽  
The Other ◽  
Algorithmic Randomness ◽  
Philosophical Literature ◽  
The Relationship ◽  
Mathematical Literature

Early work on the frequency theory of probability made extensive use of the notion of randomness, conceived of as a property possessed by disorderly collections of outcomes. Growing out of this work, a rich mathematical literature on algorithmic randomness and Kolmogorov complexity developed through the twentieth century, but largely lost contact with the philosophical literature on physical probability. The present chapter begins with a clarification of the notions of randomness and probability, conceiving of the former as a property of a sequence of outcomes, and the latter as a property of the process generating those outcomes. A discussion follows of the nature and limits of the relationship between the two notions, with largely negative verdicts on the prospects for any reduction of one to the other, although the existence of an apparently random sequence of outcomes is good evidence for the involvement of a genuinely chancy process.

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