scholarly journals DEMUTH’S PATH TO RANDOMNESS

2015 ◽  
Vol 21 (3) ◽  
pp. 270-305 ◽  
Author(s):  
ANTONÍN KUČERA ◽  
ANDRÉ NIES ◽  
CHRISTOPHER P. PORTER
Keyword(s):  
Computability Theory ◽  
Truth Table ◽  
Algorithmic Randomness ◽  
Constructive Mathematics ◽  
Russian School ◽  
Constructive Analysis ◽  
Innovative Work ◽  
Classical Language ◽  
Computable Functions ◽  
Null Set

AbstractOsvald Demuth (1936–1988) studied constructive analysis from the viewpoint of the Russian school of constructive mathematics. In the course of his work he introduced various notions of effective null set which, when phrased in classical language, yield a number of major algorithmic randomness notions. In addition, he proved several results connecting constructive analysis and randomness that were rediscovered only much later.In this paper, we trace the path that took Demuth from his constructivist roots to his deep and innovative work on the interactions between constructive analysis, algorithmic randomness, and computability theory. We will focus specifically on (i) Demuth’s work on the differentiability of Markov computable functions and his study of constructive versions of the Denjoy alternative, (ii) Demuth’s independent discovery of the main notions of algorithmic randomness, as well as the development of Demuth randomness, and (iii) the interactions of truth-table reducibility, algorithmic randomness, and semigenericity in Demuth’s work.

RELATIVIZING CHAITIN'S HALTING PROBABILITY

2005 ◽  
Vol 05 (02) ◽  
pp. 167-192 ◽  
Author(s):  
ROD DOWNEY ◽  
DENIS R. HIRSCHFELDT ◽  
JOSEPH S. MILLER ◽  
ANDRÉ NIES
Keyword(s):  
Computability Theory ◽  
Algorithmic Randomness ◽  
Jump Operator ◽  
Random Real ◽  
Primary Object ◽  
Object Of Study ◽  
Free Machine ◽  
Computable Functions ◽  
Omega Operator

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let [Formula: see text] be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory and this Omega operator. But unlike the jump, which is invariant (up to computable permutation) under the choice of an effective enumeration of the partial computable functions, [Formula: see text] can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that [Formula: see text] and [Formula: see text] are 1-random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness.

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.

Presentations of left-c.e. reals

2020 ◽  
pp. 106-133
Author(s):  
Rod Downey ◽  
Noam Greenberg
Keyword(s):  
Statistical Test ◽  
Algorithmic Randomness ◽  
Normal Numbers ◽  
Algorithmic Information Theory ◽  
Countable Collection ◽  
Test Elements ◽  
Algorithmic Information ◽  
Von Mises ◽  
Main Ideas ◽  
Null Set

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.

scholarly journals A Constructive Analysis of Convex-Valued Demand Correspondence for Weakly Uniformly Rotund and Monotonic Preference

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
Yasuhito Tanaka ◽  
Atsuhiro Satoh
Keyword(s):  
Demand Function ◽  
Preference Relation ◽  
Constructive Mathematics ◽  
Closed Graph ◽  
Preference Relations ◽  
Constructive Analysis ◽  
Continuous Demand

Bridges (1992) has constructively shown the existence of continuous demand function for consumers with continuous, uniformly rotund preference relations. We extend this result to the case of multivalued demand correspondence. We consider a weakly uniformly rotund and monotonic preference relation and will show the existence of convex-valued demand correspondence with closed graph for consumers with continuous, weakly uniformly rotund and monotonic preference relations. We follow the Bishop style constructive mathematics according to Bishop and Bridges (1985), Bridges and Richman (1987), and Bridges and Vîţă (2006).

scholarly journals Uniform distribution and algorithmic randomness

2013 ◽  
Vol 78 (1) ◽  
pp. 334-344 ◽  
Author(s):  
Jeremy Avigad
Keyword(s):  
Uniform Distribution ◽  
Unit Interval ◽  
The Other ◽  
Weyl’S Theorem ◽  
Algorithmic Randomness ◽  
Random Real ◽  
Other Hand ◽  
Random Reals ◽  
Null Set

AbstractA seminal theorem due to Weyl [14] states that if (an) is any sequence of distinct integers, then, for almost every x ∈ ℝ, the sequence (anx) is uniformly distributed modulo one. In particular, for almost every x in the unit interval, the sequence (anx) is uniformly distributed modulo one for every computable sequence (an) of distinct integers. Call such an x UD random. Here it is shown that every Schnorr random real is UD random, but there are Kurtz random reals that are not UD random. On the other hand, Weyl's theorem still holds relative to a particular effectively closed null set, so there are UD random reals that are not Kurtz random.

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.

A Bigger Picture

2019 ◽  
pp. 154-167
Author(s):  
John Stillwell
Keyword(s):  
Reverse Mathematics ◽  
Computability Theory ◽  
Peano Arithmetic ◽  
Equivalence Classes ◽  
Constructive Mathematics

This chapter aims to pick up some of the ideas dropped from this book and set them in a bigger picture of logic and computability theory. It begins with a sketch of constructive mathematics. Originally developed by a minority of mathematicians opposed to using actual infinities, constructive mathematics contributed some useful techniques for computable mathematics in systems such as RCA0. This is followed by discussions on the completeness of logic and the incompleteness of Peano arithmetic (PA) and related systems. These results reveal mathematics as an arena where theorems cannot always be proved outright, but in which all of their logical equivalents can be found. This creates the possibility of reverse mathematics, where one seeks equivalents that are suitable as axioms. Next, the chapter explains how computability theory helps to distinguish the equivalence classes of theorems, and finally makes a few speculative remarks on the ordering of the equivalence classes, and how this throws light on the concept of mathematical depth.

scholarly journals On the Definitional Embeddability of the Combinatory Logic Theory into the First-Order Predicate Calculus

2015 ◽  
Vol 21 (2) ◽  
pp. 9-14
Author(s):  
В. И. Шалак
Keyword(s):  
Predicate Logic ◽  
Computability Theory ◽  
Predicate Calculus ◽  
Combinatory Logic ◽  
Applied Theory ◽  
First Order ◽  
Computable Functions ◽  
Order Predicate Calculus ◽  
Classical Predicate

In this article we prove a theorem on the definitional embeddability of the combinatory logic into the first-order predicate calculus without equality. Since all efficiently computable functions can be represented in the combinatory logic, it immediately follows that they can be represented in the first-order classical predicate logic. So far mathematicians studied the computability theory as some applied theory. From our theorem it follows that the notion of computability is purely logical. This result will be of interest not only for logicians and mathematicians but also for philosophers who study foundations of logic and its relation to mathematics.

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>

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