The Computational Content of Intrinsic Density

Astor, Eric P.

doi:10.1017/jsl.2018.4

Cited by 4 publications

(14 citation statements)

References 22 publications

(52 reference statements)

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“…Astor [3] proved that the Turing degrees which contain an infinite intrinsically small set are those which are not weakly computably traceable. Kjos-Hanssen et al [10] characterized these degrees as those which are High or DNC.…”

Section: Functions and Intrinsic Densitymentioning

confidence: 99%

“…Recall that hyperimmune sets are infinite by definition, so hypersimple sets are co-infinite.) One question left open in [2] (later answered by Astor in [3] using a degree argument) was whether or not a hypersimple set could have lower density 0, or at least non-1 lower density. The answer is yes, showing that hypersimple sets need not have defined density.…”

Section: Hyperimmunity and Intrinsic Smallnessmentioning

confidence: 99%

“…While Theorem 3.2 shows that hyperimmunity and intrinsic smallness are unrelated notions of smallness, we would like to know whether it is possible to provide a simple characterization of intrinsic smallness using principal functions. Perhaps the most natural candidate is that of weak computable traceability from [3], which does provide us with a useful test for intrinsic smallness:…”

Section: Hyperimmunity and Intrinsic Smallnessmentioning

confidence: 99%

“…Furthermore, is enough to ensure A has intrinsic density zero.) Of special interest is the property of having intrinsic density , which has been studied extensively by Astor [2 , 3 ] in relation with other notions of smallness such as immunity. We will refer to sets that have intrinsic density as intrinsically small to ease notation slightly.…”

Section: Introductionmentioning

confidence: 99%

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Intrinsic Smallness

Miller¹

2020

J. symb. log.

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Recent work in computability theory has focused on various notions of asymptotic computability, which capture the idea of a set being “almost computable.” One potentially upsetting result is that all four notions of asymptotic computability admit “almost computable” sets in every Turing degree via coding tricks, contradicting the notion that “almost computable” sets should be computationally close to the computable sets. In response, Astor introduced the notion of intrinsic density: a set has defined intrinsic density if its image under any computable permutation has the same asymptotic density. Furthermore, introduced various notions of intrinsic computation in which the standard coding tricks cannot be used to embed intrinsically computable sets in every Turing degree. Our goal is to study the sets which are intrinsically small, i.e. those that have intrinsic density zero. We begin by studying which computable functions preserve intrinsic smallness. We also show 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. Our discussion concludes by relativizing the notion of intrinsic smallness and discussing intrinsic computability as it relates to our study of intrinsic smallness.

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Section: Functions and Intrinsic Densitymentioning

confidence: 99%

Section: Hyperimmunity and Intrinsic Smallnessmentioning

confidence: 99%

Section: Hyperimmunity and Intrinsic Smallnessmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Intrinsic Smallness

Miller¹

2020

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“…DNC and FPF functions play an important part in computability theory, for example in the work of Kučera [17]. See Astor [3] for a recent example of their use, or Downey and Hirschfeldt [7] for many more. They are also closely related to the study of complete extensions of Peano Arithmetic, see e.g.…”

Section: Gödel Rossermentioning

confidence: 99%

Fixed point theorems for precomplete numberings

Barendregt

Terwijn

2019

Annals of Pure and Applied Logic

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In the context of his theory of numberings, Ershov showed that Kleene's recursion theorem holds for any precomplete numbering. We discuss various generalizations of this result. Among other things, we show that Arslanov's completeness criterion also holds for every precomplete numbering, and we discuss the relation with Visser's ADN theorem, as well as the uniformity or nonuniformity of the various fixed point theorems. Finally, we base numberings on partial combinatory algebras and prove a generalization of Ershov's theorem in this context.

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Intrinsic density, asymptotic computability, and stochasticity

Miller

2021

Bull. symb. log

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There are many computational problems which are generally “easy” to solve but have certain rare examples which are much more difficult to solve. One approach to studying these problems is to ignore the difficult edge cases. Asymptotic computability is one of the formal tools that uses this approach to study these problems. Asymptotically computable sets can be thought of as almost computable sets, however every set is computationally equivalent to an almost computable set. Intrinsic density was introduced as a way to get around this unsettling fact, and which will be our main focus.Of particular interest for the first half of this dissertation are the intrinsically small sets, the sets of intrinsic density $0$ . While the bulk of the existing work concerning intrinsic density was focused on these sets, there were still many questions left unanswered. The first half of this dissertation answers some of these questions. We proved some useful closure properties for the intrinsically small sets and applied them to prove separations for the intrinsic variants of asymptotic computability. We also completely separated hyperimmunity and intrinsic smallness in the Turing degrees and resolved some open questions regarding the relativization of intrinsic density.For the second half of this dissertation, we turned our attention to the study of intermediate intrinsic density. We developed a calculus using noncomputable coding operations to construct examples of sets with intermediate intrinsic density. For almost all $r\in (0,1)$ , this construction yielded the first known example of a set with intrinsic density r which cannot compute a set random with respect to the r-Bernoulli measure. Motivated by the fact that intrinsic density coincides with the notion of injection stochasticity, we applied these techniques to study the structure of the more well-known notion of MWC-stochasticity.Abstract prepared by Justin Miller.E-mail: jmille74@nd.eduURL: https://curate.nd.edu/show/6t053f4938w

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The Computational Content of Intrinsic Density

Cited by 4 publications

References 22 publications

Intrinsic Smallness

Intrinsic Smallness

Fixed point theorems for precomplete numberings

Intrinsic density, asymptotic computability, and stochasticity

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