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

Mingzhong Cai, Rodney Downey, Rachel Epstein, Steffen Lempp, Joseph Miller, S. Barry Cooper
2014 Logical Methods in Computer Science  
We investigate the truth-table 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.  ... 
doi:10.2168/lmcs-10(3:15)2014 fatcat:oepxnqtnvnbqvh4ojk7yuh2spu

Superhighness

Bjørn Kjos-Hanssen, Andrée Nies
2009 Notre Dame Journal of Formal Logic  
We prove that superhigh 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.  ... 
doi:10.1215/00294527-2009-020 fatcat:bjnrgchrvrh2nnq3f5hoqdz7gu

Deep Π^0_1 Classes [article]

Laurent Bienvenu, Christopher P. Porter
2017 arXiv   pre-print
We prove a number 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.  ... 
arXiv:1403.0450v3 fatcat:i63bvw7llnhodbdw6tn5sjcswq

Effective randomness, strong reductions and Demuth's theorem [article]

Laurent Bienvenu, Christopher Porter
2011 arXiv   pre-print
We also provide some additional results about the 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.  ... 
arXiv:1110.1860v2 fatcat:q3rwibndfjcblgsemzw35vfxxe

Randomness and Computability: Open Questions

Joseph S. Miller, André Nies
2006 Bulletin of Symbolic Logic  
All 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.  ... 
doi:10.2178/bsl/1154698740 fatcat:int7vbtqbnggtfkbbd3wodxo2q

Strong reductions in effective randomness

Laurent Bienvenu, Christopher Porter
2012 Theoretical Computer Science  
We also provide some additional results about the 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.  ... 
doi:10.1016/j.tcs.2012.06.031 fatcat:xvctlday7zaubplrx2pmklgzwu

Kobayashi compressibility

George Barmpalias, Rodney G. Downey
2017 Theoretical Computer Science  
We prove that Kobayashi compressibility can be used in order to define Martin-Löf 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  ... 
doi:10.1016/j.tcs.2017.02.029 fatcat:63iwdmnqjrcutctsg5oiikbnt4

A Bounded Jump for the Bounded Turing Degrees

Bernard Anderson, Barbara Csima
2014 Notre Dame Journal of Formal Logic  
We demonstrate several properties 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.  ... 
doi:10.1215/00294527-2420660 fatcat:fbv46wmssbgqzmgyhb5lh6hfr4

Key developments in algorithmic randomness [article]

Johanna N.Y. Franklin, Christopher P. Porter
2020 arXiv   pre-print
The goal 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.  ... 
arXiv:2004.02851v1 fatcat:hacuzbqogfdohim27pmtsqzsti

Kobayashi compressibility [article]

George Barmpalias, Rodney G. Downey
2017 arXiv   pre-print
We prove that Kobayashi compressibility can be used in order to define Martin-Loef 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  ... 
arXiv:1608.00692v2 fatcat:pwaducmly5h4nfa5yc4nz4whji

Characterizing the strongly jump-traceable sets via randomness [article]

Noam Greenberg, Denis Hirschfeldt, Andre Nies
2011 arXiv   pre-print
To do so, we connect cost function strength 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.  ... 
arXiv:1109.6749v1 fatcat:2i3bgtix3rctnmvaxz6zte62ai

Aspects of Chaitin's Omega [article]

George Barmpalias
2018 arXiv   pre-print
The halting probability 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  ... 
arXiv:1707.08109v5 fatcat:hpjvbxxcdfbdpbxcuxkttsde3i

Superhighness and Strong Jump Traceability [chapter]

André Nies
2009 Lecture Notes in Computer Science  
Let A be a 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.  ... 
doi:10.1007/978-3-642-02927-1_60 fatcat:quzpuqeaefhbdpndwbar4puq2a

Characterizing the strongly jump-traceable sets via randomness

Noam Greenberg, Denis R. Hirschfeldt, André Nies
2012 Advances in Mathematics  
Together with a theorem 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.  ... 
doi:10.1016/j.aim.2012.06.005 fatcat:62prazncfjhllisvoxkqvwmaoi

Extending and interpreting Post's programme

S. Barry Cooper
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.  ... 
doi:10.1016/j.apal.2009.06.007 fatcat:dkavrxgjaveurbege66nxmqnui
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