# Asymptotic density, immunity and randomness

@article{Astor2015AsymptoticDI, title={Asymptotic density, immunity and randomness}, author={Eric P. Astor}, journal={Comput.}, year={2015}, volume={4}, pages={141-158} }

In 2012, inspired by developments in group theory and complexity, Jockusch and Schupp introduced generic com- putability, capturing the idea that an algorithm might work correctly except for a vanishing fraction of cases. However, we observe that their definition of a negligible set is not computably invariant (and thus not well-defined on the 1-degrees), resulting in some failures of intuition and a break with standard expectations in computability theory. To strengthen their approach, we…

## 10 Citations

### On New Notions of Algorithmic Dimension, Immunity, and Medvedev Degree

- Mathematics, Computer Science
- 2022

A new notion of algorithmic dimension, the inescapable dimension, which lies between the effective Hausdorff and packing dimensions, is investigated, and an embedding of the Turing degrees into notions of dimension is obtained.

### Stochasticity as Density

- Mathematics
- 2020

Intrinsic density was introduced by Astor to study asymptotic computability. Intrinsically small sets, those of intrinsic density zero, serve as the basis for generalizing classical asymptotic…

### INTRINSIC SMALLNESS

- Computer Science, PhilosophyThe Journal of Symbolic Logic
- 2020

It is shown that intrinsic smallness and hyperimmunity are computationally independent notions of smallness, i.e. any hyperimmune degree contains a Turing-equivalent hyperimmune set which is “as large as possible” and therefore not intrinsically small.

### L O ] 3 0 A ug 2 01 9 Intrinsic Smallness

- Computer Science, Philosophy
- 2019

It is shown that intrinsic smallness and hyperimmunity are computationally independent notions of smallness, i.e. any hyperimmune degree contains a Turing-equivalent hyperimmune set which is “as large as possible” and therefore not intrinsically small.

### THE COMPUTATIONAL CONTENT OF INTRINSIC DENSITY

- MathematicsThe Journal of Symbolic Logic
- 2018

It is proved that sets with well-defined intrinsic density (and particularly intrinsic density 0) exist only in Turing degrees that are either high (${\bf{a}}\prime { \ge _{\rm{T}}}\emptyset \prime \prime$) or compute a diagonally noncomputable function.

### COARSE REDUCIBILITY AND ALGORITHMIC RANDOMNESS

- Computer Science, MathematicsThe Journal of Symbolic Logic
- 2016

This paper studies the extent to which noncomputable information can be effectively recovered from all coarse descriptions of a given set A, especially when A is effectively random in some sense, and shows that if A is 1-random and B is computable from every coarse description D of A, then B is K-trivial, which implies thatif A is in fact weakly 2-random then B has to be computable.

### Intermediate intrinsic density and randomness

- Computer Science, MathematicsComput.
- 2021

This set will be the first known example of an intrinsic density r set which cannot compute any r-Bernoulli random set, and will formalize the into and within noncomputable coding methods which work well with intrinsic density.

### Asymptotic Density and the Theory of Computability: A Partial Survey

- MathematicsComputability and Complexity
- 2017

This paper surveys recent work on how classical asymptotic density interacts with the theory of computability and includes a few easy proofs to illustrate the flavor of the subject.

### Dense computability, upper cones, and minimal pairs

- Mathematics, Computer ScienceComput.
- 2019

It is shown that nontrivial upper cones in the generic, dense, and effective dense degrees are of measure $0$ and use this fact to show that there are minimal pairs in the dense degrees.

### Degrees of sets having no subsets of higher m- and tt-degree

- MathematicsComput.
- 2021

It is established that each computably enumerable weak truth-table degree contains m-introimmune Π 1 0 -sets; each hyperimmune degree contains bi-m- introimmune sets.

## References

SHOWING 1-10 OF 31 REFERENCES

### Computability and randomness

- Computer Science
- 2009

This book provides a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory.

### Asymptotic density and computably Enumerable Sets

- Mathematics, Computer ScienceJ. Math. Log.
- 2013

The lower densities, upper densities and densities of both computable and computably enumerable sets are characterized and it is proved that every nonzero c.e. degree contains a set which is generically computable but not coarsely computable.

### Algorithmic Randomness and Complexity

- Computer Science, MathematicsTheory and Applications of Computability
- 2010

This chapter discusses Randomness-Theoretic Weakness, Omega as an Operator, Complexity of C.E. Sets, and other Notions of Effective Randomness.

### From Bi-Immunity to Absolute Undecidability

- MathematicsThe Journal of Symbolic Logic
- 2009

It is shown how to use Walsh–Hadamard codes to build a truth-table functional which maps any sequence A to a sequence B, such that given any restriction of B to a set of positive upper density, one can recover A, implying that if A is non-computable, then B is absolutely undecidable.

### Recursively enumerable sets of positive integers and their decision problems

- Mathematics
- 1944

Introduction. Recent developments of symbolic logic have considerable importance for mathematics both with respect to its philosophy and practice. That mathematicians generally are oblivious to the…

### Uniformity in Computable Structure Theory

- Computer Science, Mathematics
- 2001

This work considers and compares two different notions of uniformity, previously studied by Kudinov and by Ventsov, and investigates the effects of adding uniformity requirements to concepts in computable structure theory such as computable categoricity and intrinsic computability.

### Nonexistence of minimal pairs for generic computability

- Mathematics, Computer ScienceThe Journal of Symbolic Logic
- 2013

The primary result of this paper is that there are no minimal pairs for generic computability, answering a question of Jockusch and Schupp.

### Classes of recursively enumerable sets and their decision problems

- Mathematics, Computer Science
- 1953

This paper considers classes whose elements are re-cursively enumerable sets of non-negative integers whose properties are complete recursive enumerability and complete recursiveness.

### Algorithmic randomness and complexity. Theory and Applications of Computability

- Computer Science
- 2012

This book is very referred for you because it gives not only the experience but also lesson, it is about this book that will give wellness for all people from many societies.

### Recursively enumerable sets and degrees - a study of computable functions and computability generated sets

- Computer SciencePerspectives in mathematical logic
- 1987

The author has managed to give a coherent exposition of a rather complex and messy area of logic, and with this book degree-theory is far more accessible to students and logicians in other fields than it used to be.