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Random strings and tt-degrees of Turing complete C.E. sets

2014
*
Logical Methods in Computer Science
*

We investigate the truth-table

doi:10.2168/lmcs-10(3:15)2014
fatcat:oepxnqtnvnbqvh4ojk7yuh2spu
*degrees**of*(co-)*c.e*.\*sets*, in particular,*sets**of**random**strings*. ... The latter result proves a conjecture*of*Allender, Friedman*and*Gasarch. We also show that there are two*Turing**complete**c.e*.*sets*whose truth-table*degrees*form a minimal pair. ...*Turing**complete**set*be half*of*a minimal pair in the*tt*-*degrees*, but the other half*of*the minimal pair may also be a*c.e*.*Turing**complete**set*. ...##
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Superhighness

2009
*
Notre Dame Journal of Formal Logic
*

We prove that superhigh

doi:10.1215/00294527-2009-020
fatcat:bjnrgchrvrh2nnq3f5hoqdz7gu
*sets*can be jump traceable, answering a question*of*Cole*and*Simpson. On the other hand, we show that such*sets*cannot be weakly 2-*random*. ... We also study the class superhigh^,*and*show that it contains some, but not all,*of*the noncomputable K-trivial*sets*. ... Superhighness for computably enumerable (*c.e*.)*sets*was introduced by Mohrherr [M] . She proved that the superhigh*c.e*.*degrees*sit properly between the high*and**Turing**complete*(A ≥ T ∅ ′ ) ones. ...##
###
Deep Π^0_1 Classes
[article]

2017
*
arXiv
*
pre-print

We prove a number

arXiv:1403.0450v3
fatcat:i63bvw7llnhodbdw6tn5sjcswq
*of*basic results about depth, negligibility,*and*a variant*of*negligibility that we call*tt*-negligibility. ... We also provide a number*of*examples*of*deep Π^0_1 classes that occur naturally in computability theory*and*algorithmic*randomness*. ... We would also like to thank the anonymous referees for a number*of*helpful suggestions. ...##
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Effective randomness, strong reductions and Demuth's theorem
[article]

2011
*
arXiv
*
pre-print

We also provide some additional results about the

arXiv:1110.1860v2
fatcat:q3rwibndfjcblgsemzw35vfxxe
*Turing**and**tt*-*degrees**of*reals that are*random*with respect to some computable measure. ... We study generalizations*of*Demuth's Theorem, which states that the image*of*a Martin-L\"*of**random*real under a*tt*-reduction is either computable or*Turing*equivalent to a Martin-L\"*of**random*real. ... The family*of**c.e*.*sets*that are*random*for some computable probability measure is therefore not downwards closed in the*Turing**degrees*. ...##
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Randomness and Computability: Open Questions

2006
*
Bulletin of Symbolic Logic
*

All

doi:10.2178/bsl/1154698740
fatcat:int7vbtqbnggtfkbbd3wodxo2q
*sets*will be*sets**of*natural numbers, unless otherwise stated. These*sets*are identified with infinite*strings*over {0, 1}. Other terms used in the literature are sequence*and*real. ... It is time for a new paper about open questions in the currently very active area*of**randomness**and*computability. Ambos-Spies*and*Kučera presented such a paper in 1999 [1]. ... Demuth [6] proved that if A is ML-*random**and*B ≤*tt*A is noncomputable, then the*Turing**degree**of*B also contains a ML-*random**set*. ...##
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Strong reductions in effective randomness

2012
*
Theoretical Computer Science
*

We also provide some additional results about the

doi:10.1016/j.tcs.2012.06.031
fatcat:xvctlday7zaubplrx2pmklgzwu
*Turing**and**tt*-*degrees**of*reals that are*random*with respect to some computable measure. ... We study generalizations*of*Demuth's Theorem, which states that the image*of*a Martin-Löf*random*real under a*tt*-reduction is either computable or*Turing*equivalent to a Martin-Löf*random*real. ... Acknowledgments The authors would like to thank André Nies, Jason Rute,*and*two anonymous referees for very helpful comments*and*suggestions. ...##
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Kobayashi compressibility

2017
*
Theoretical Computer Science
*

We prove that Kobayashi compressibility can be used in order to define Martin-Löf

doi:10.1016/j.tcs.2017.02.029
fatcat:63iwdmnqjrcutctsg5oiikbnt4
*randomness*, a strong version*of*finite*randomness**and*Kurtz*randomness*, strictly in terms*of**Turing*reductions. ... Moreover these*randomness*notions naturally correspond to*Turing*reducibility, weak truth-table reducibility*and*truth-table reducibility respectively. ... Kummer [Kum96] showed that array non-computable*c.e*.*degrees*contain*c.e*.*sets*A such that C(A n ) > 2 log n − c for some constant c*and*infinitely many n, while the plain complexity*of*the*c.e*.*sets*...##
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A Bounded Jump for the Bounded Turing Degrees

2014
*
Notre Dame Journal of Formal Logic
*

We demonstrate several properties

doi:10.1215/00294527-2420660
fatcat:fbv46wmssbgqzmgyhb5lh6hfr4
*of*the bounded jump, including that it is strictly increasing*and*order preserving on the bounded*Turing*(bT)*degrees*(also known as the weak truth-table*degrees*). ... Finally, we prove that the analogue*of*Shoenfield inversion holds for the bounded jump on the bounded*Turing**degrees*. ... We do not know if Theorem 6.2 holds if we add the requirement that Y is*c.e*. We can also look at concepts related to the*Turing*jump. ...##
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Key developments in algorithmic randomness
[article]

2020
*
arXiv
*
pre-print

The goal

arXiv:2004.02851v1
fatcat:hacuzbqogfdohim27pmtsqzsti
*of*this introductory survey is to present the major developments*of*algorithmic*randomness*with an eye toward its historical development. ... who want to develop a sense*of*the field quickly*and*interesting for researchers already in the field who would like to see these results presented in chronological order. ... Acknowledgments The authors would like to thank the referees who provided extraordinarily perceptive*and*useful comments on this survey. ...##
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Kobayashi compressibility
[article]

2017
*
arXiv
*
pre-print

We prove that Kobayashi compressibility can be used in order to define Martin-Loef

arXiv:1608.00692v2
fatcat:pwaducmly5h4nfa5yc4nz4whji
*randomness*, a strong version*of*finite*randomness**and*Kurtz*randomness*, strictly in terms*of**Turing*reductions. ... Moreover these*randomness*notions naturally correspond to*Turing*reducibility, weak truth-table reducibility*and*truth-table reducibility respectively. ... Kummer [24] showed that array non-computable*c.e*.*degrees*contain*c.e*.*sets*A such that C(A ↾ n ) > 2 log n − c for some constant c*and*infinitely many n, while the plain complexity*of*the*c.e*.*sets*...##
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Characterizing the strongly jump-traceable sets via randomness
[article]

2011
*
arXiv
*
pre-print

To do so, we connect cost function strength

arXiv:1109.6749v1
fatcat:2i3bgtix3rctnmvaxz6zte62ai
*and*the strength*of**randomness*notions. ... This result gives a full correspondence between obedience*of*cost functions*and*being computable from Δ^0_2 1-*random**sets*. ... Now, the class*of*PAcomplete*sets*is upward closed in the*Turing**degrees*, hence A ⊕ Z is PA-*complete*, is superlow,*and*computes A. ...##
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Aspects of Chaitin's Omega
[article]

2018
*
arXiv
*
pre-print

The halting probability

arXiv:1707.08109v5
fatcat:hpjvbxxcdfbdpbxcuxkttsde3i
*of*a*Turing*machine,also known as Chaitin's Omega, is an algorithmically*random*number with many interesting properties. ... The purpose*of*this survey is to expose these developments*and*tell a story about Omega, which outlines its multifaceted mathematical properties*and*roles in algorithmic*randomness*. ... Unfortunately, it was discovered in [1] that there is no*complete**c.e*.*set*in the Solovay*degrees**and*, even worse, for each*c.e*.*set*A there exists a*c.e*.*set*B*of*strictly larger Solovay*degree*than ...##
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Superhighness and Strong Jump Traceability
[chapter]

2009
*
Lecture Notes in Computer Science
*

Let A be a

doi:10.1007/978-3-642-02927-1_60
fatcat:quzpuqeaefhbdpndwbar4puq2a
*c.e*.*set*. Then A is strongly jump traceable if*and*only if A is*Turing*below each superhigh Martin-Löf*random**set*. The proof combines priority with measure theoretic arguments. ... Thus our main result is that a*c.e*.*set*A is strongly jump traceable if*and*only if A is*Turing*below each superhigh Martin-Löf*random**set*. ...*Turing**degrees*. By a result*of*Hirschfeldt*and*Miller (see [7, 5.3 .15]), for each null Σ 0 3 class, the corresponding diamond class contains a promptly simple*set*A. ...##
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Characterizing the strongly jump-traceable sets via randomness

2012
*
Advances in Mathematics
*

Together with a theorem

doi:10.1016/j.aim.2012.06.005
fatcat:62prazncfjhllisvoxkqvwmaoi
*of*Greenberg*and*Nies (ibd.), this result gives a full correspondence between obedience*of*cost functions*and*being computable from 0 2 1-*random**sets*. ... To do so, we connect cost function strength*and*the strength*of**randomness*notions. ... Acknowledgments The first*and*third authors were partially supported by the Marsden Fund*of*New Zealand. ...##
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Extending and interpreting Post's programme

2010
*
Annals of Pure and Applied Logic
*

*of*

*sets*. ... The second looks at how new types

*of*information coming from the recent growth

*of*research into

*randomness*,

*and*the revealing

*of*unexpected new computability-theoretic infrastructure. ...

*And*in relation to other reducibilities, it is worth mentioning that Arslanov's

*completeness*criterion characterizes the

*Turing*

*complete*

*and*wtt-

*complete*but not the

*tt*-

*complete*

*c.e*.

*sets*. ...

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