Mass problems associated with effectively closed sets

Simpson, Stephen G.

doi:10.2748/tmj/1325886278

Cited by 27 publications

(8 citation statements)

References 92 publications

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“…In the alternative context of the Muchnik and Medvedev degrees (see [22,39,41] for background), related work has recently been done by Miyabe [32]. He views randomness notions as mass problems (so there is no relativization).…”

Section: The Viewpoint Of Reverse Mathematicsmentioning

confidence: 99%

Randomness Notions and Reverse Mathematics

Nies¹,

Shafer²

2019

J. symb. log.

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We investigate the strength of a randomness notion R as a set-existence principle in second-order arithmetic: for each Z there is X that is R-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in RCA0. We verify that RCA0 proves the basic implications among randomness notions: 2-random ⇒ weakly 2-random ⇒ Martin-Löf random ⇒ computably random ⇒ Schnorr random. Also, over RCA0 the existence of computable randoms is equivalent to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-Löf randoms, and we describe a sense in which this result is nearly optimal.

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Section: The Viewpoint Of Reverse Mathematicsmentioning

confidence: 99%

Randomness Notions and Reverse Mathematics

Nies¹,

Shafer²

2019

J. symb. log.

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“…Medvedev equivalence is usually introduced in the context of mass problems [Sim11]: S ≤ M S if it is easier to find an element of S than to find an element of S , in the sense that, if we find an element y of S , then we also obtain in the same way an element of S (namely Φ(y)).…”

Section: Definition 3 Let S S Two Subsets Ofmentioning

confidence: 99%

Pursuit of the Universal

Beckmann¹,

Bienvenu²,

Jonoska³

2016

Lecture Notes in Computer Science

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Abstract.A general correction grammar for a language L is a program g that, for each (x, t) ∈ N 2 , issues a yes or no (where when t = 0, the answer is always no) which is g's t-th approximation in answering "x∈L?"; moreover, g's sequence of approximations for a given x is required to converge after finitely many mind-changes. The set of correction grammars can be transfinitely stratified based on O, Kleene's system of notation for constructive ordinals. For u ∈ O, a u-correction grammar's mind changes have to fit a count-down process from ordinal notation u; these u-correction grammars capture precisely the Σ −1 u sets in Ershov's hierarchy of sets for Δ 0 2 . Herein we study the relative succinctness between these classes of correction grammars. Example: Given u and v, transfinite elements of O with u H(iv). We also exhibit relative succinctness progressions in these systems of grammars and study the "information-theoretic" underpinnings of relative succinctness. Along the way, we verify and improve slightly a 1972 conjecture of Meyer and Bagchi.

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“…An easy argument [69,Lemma 3.3.5] shows that S w has an alternative characterization as the lattice of Muchnik degrees of nonempty, lightface Σ 0 3 subsets of N N . An easy argument [69,Lemma 3.3.5] shows that S w has an alternative characterization as the lattice of Muchnik degrees of nonempty, lightface Σ 0 3 subsets of N N .…”

Section: The Lattices E W and S Wmentioning

confidence: 99%

Evolving Computability

2015

Lecture Notes in Computer Science

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Mass problems associated with effectively closed sets

Cited by 27 publications

References 92 publications

Randomness Notions and Reverse Mathematics

Randomness Notions and Reverse Mathematics

Pursuit of the Universal

Evolving Computability

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