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Page 1701 of Mathematical Reviews Vol. , Issue 87d [page]

<span title="">1987</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The author gives the following theorem: Suppose that B is a non- recursive r.e. set.  ...  The Robinson splitting theorem states that every r.e. degree splits over every lesser low degree.  ... 
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Definable Encodings in the Computably Enumerable Sets

Peter A. Cholak, Leo A. Harrington
<span title="">2000</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nxnim3i24fhcvkvgruk2dej7yu" style="color: black;">Bulletin of Symbolic Logic</a> </i> &nbsp;
What these results have in common is that the guts of the proofs of these theorems uses a new form of definable coding for the computably enumerable sets.  ...  All sets will be computably enumerable non-computable sets and all degrees will be computably enumerable and non-computable, unless otherwise noted.  ...  All sets will be computably enumerable non-computable sets and all degrees will computably enumerable and non-computable, unless otherwise noted.  ... 
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Analogues of Chaitin's Omega in the computably enumerable sets

G. Barmpalias, R. Hölzl, A.E.M. Lewis, W. Merkle
<span title="">2013</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5xfy4fcdxvcvdiybp6iqbffqc4" style="color: black;">Information Processing Letters</a> </i> &nbsp;
These c.e. sets belong to the weak truth table degree of the halting problem, but not every weak truth table complete set has maximum initial segment Kolmogorov complexity.  ...  We show that there are computably enumerable (c.e.) sets with maximum initial segment Kolmogorov complexity amongst all c.e. sets (with respect to both the plain and the prefix-free version of Kolmogorov  ...  Theorem 3 . 1 ( 31 Non-splitting for ≤ C ).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ipl.2013.01.007">doi:10.1016/j.ipl.2013.01.007</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pt6cltcd4zfzxgaovvi35smxt4">fatcat:pt6cltcd4zfzxgaovvi35smxt4</a> </span>
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Splitting and jump inversion in the Turing degrees

S. Barry Cooper, Angsheng Li, Mingji Xia, Benedikt Löwe
<span title="2018-06-07">2018</span> <i title="IOS Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/rtyr26r4evfy3hccqovstp6k5m" style="color: black;">Computability - The Journal of the Assosiation</a> </i> &nbsp;
It is shown that for any computably enumerable degree a = 0, any degree c = 0, and any Turing degree s, if s ≥ 0 ′ , and c.e. in a, then there exists a c.e. degree x with the following properties, (1)  ...  This implies that the Sacks' splitting theorem and the Sacks' jump theorem can be uniformly combined.  ...  The difficulty in splitting computably enumerable Turing degrees is due to a very strong non-extension feature (see for instance, Cooper and Li [2002] ).  ... 
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The search for natural definability in the Turing degrees

Andrew E.M. Lewis-Pye, Benedikt Löwe
<span title="2018-06-07">2018</span> <i title="IOS Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/rtyr26r4evfy3hccqovstp6k5m" style="color: black;">Computability - The Journal of the Assosiation</a> </i> &nbsp;
First consider the non-GL 2 degrees: Theorem 13.1 ( [DGLM] ) All non-GL 2 degrees satisfy the join property. Proof. The proof follows the format for non-GL 2 proofs that we described in Section 9.  ...  Theorem 8.4 (Solovay, extending Scott's basis theorem [DS] ) A degree a is PA iff every non-empty Π 0 1 class contains a member of degree below a. The following theorem is very often useful.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3233/com-170068">doi:10.3233/com-170068</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/t6qdc3cd6zbadg5ixwoy6jxk7e">fatcat:t6qdc3cd6zbadg5ixwoy6jxk7e</a> </span>
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Page 5456 of Mathematical Reviews Vol. , Issue 2002H [page]

<span title="">2002</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
In this paper a splitting theorem about the n-REA degrees is proven.  ...  Peter Cholak (1-NDM; Notre Dame, IN) 2002h:03094 03D25 03D30 Shore, Richard A. (1-CRNL; Ithaca, NY); Slaman, Theodore A. (1-CA; Berkeley, CA) A splitting theorem for n-REA degrees.  ... 
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Page 818 of Mathematical Reviews Vol. , Issue 81C [page]

<span title="">1981</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Then he proves the characteri- | zation theorem: A regular, non-a-recursive set G lying below 0 is of minimal a-degree if and only if it is generic.  ...  Friedberg proved the first splitting theorem [J. Symbolic Logic 23 (1958), 309-316; MR 22 #13; H.  ... 
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Page 4 of Mathematical Reviews Vol. 53, Issue 2 [page]

<span title="">1977</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The author proves that there exists a non-recursive, recursively enumerable degree d such that every recursively enumerable set of degree d is mitotic.  ...  A recursively enumerable set A is said to be mitotic if it is the disjoint union of two recursively enumerable sets of the same degree as A.  ... 
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Splitting and non-splitting in the difference hierarchy

MARAT ARSLANOV
<span title="2016-06-20">2016</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jdpukxfslnfmradqb65llq43na" style="color: black;">Mathematical Structures in Computer Science</a> </i> &nbsp;
In the second part of the paper, we consider a splitting of 0′ with some additional properties.  ...  In this paper, we investigate splitting and non-splitting properties in the Ershov difference hierarchy, in which area major contributions have been made by Barry Cooper with his students and colleagues  ...  Later Cooper and Li [2002b] showed that this non-uniformity leads to the following non-splitting theorem: given n > 1, there exist an n-c.e. degree a, and a c.e. degree b with 0 < b < a such that any  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0960129516000244">doi:10.1017/s0960129516000244</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rxx3bhj4wba33kuftrjblhjezu">fatcat:rxx3bhj4wba33kuftrjblhjezu</a> </span>
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Two splitting theorems for beta-recursion theory

Steven Homer
<span title="">1980</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/56d6znqmijdrbkwh46vzwf5hpq" style="color: black;">Annals of Mathematical Logic</a> </i> &nbsp;
The regular sets theorem of Sacks [12] then enabled Shore to cJnclude that he could split sets in any non-a-recursive a-r,e, degree.  ...  In Section 4 we give a deficiency set construction to show in which /3-degrees the sets split by this theorem lie.  ...  A, /3 and C of the splitting theorem are t.r.e, and regular, the usual corollaries of the splitting theorem follow, Corollary 4 . 3 . 43 If c is a non-zero [3-degree which comains a tame set then there  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0003-4843(80)90015-7">doi:10.1016/0003-4843(80)90015-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/u4ralkm7cvgivlenvz5xbjyvzi">fatcat:u4ralkm7cvgivlenvz5xbjyvzi</a> </span>
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On the degree spectrum of a $\Pi ^0_1$ class

Thomas Kent, Andrew E. M. Lewis
<span title="2010-10-01">2010</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
In section 7 we prove two anti-basis theorems for Π 0 1 classes. We show first that any Π 0 1 class that contains a member of every low degree contains a member of every degree.  ...  For any P ⊆ 2 ω , define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists A ∈ P of degree a.  ...  Acknowledgements The authors would like to thank Doug Cenzer, Steve Simpson, Antonin Kucera and especially Carl Jockusch for their interest and for many very useful comments and suggestions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9947-10-05037-3">doi:10.1090/s0002-9947-10-05037-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4auogboo2vcjbazeuqelvmqjde">fatcat:4auogboo2vcjbazeuqelvmqjde</a> </span>
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Page 1804 of Mathematical Reviews Vol. 53, Issue 6 [page]

<span title="">1977</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
For example, Denisov proved that for each non-recursive, noncreative r.e. m-degree b there is a r.e. m-degree a such that a|b.  ...  It is a simple exercise to verify that conditions (1) and (2) on the degrees of A and C serve, once realized, to dismantle the following large conjecture about “simultaneous splitting and density” in the  ... 
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Π^0_1 classes, strong minimal covers and hyperimmune-free degrees [article]

Andrew E.M. Lewis
<span title="2007-11-02">2007</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We investigate issues surrounding an old question of Yates' as to the existence of a minimal Turing degree with no strong minimal cover, specifically with respect to the hyperimmune-free degrees.  ...  In an unpublished paper [MK] Kumabe has constructed a FPF minimal degree. Theorem 4.4. If A satisfies the splitting-tree property then A is not of FPF degree. Proof.  ...  Theorem 3 . 1 ( 31 Jockusch and Soare). The 0-dominated basis theorem: every non-empty Π 0 1 class contains a member of 0-dominated degree. Definition 3.3.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0711.0287v1">arXiv:0711.0287v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/h4nsngwuzzcntiubw2h3djfseq">fatcat:h4nsngwuzzcntiubw2h3djfseq</a> </span>
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Split-Decomposition Trees with Prime Nodes: Enumeration and Random Generation of Cactus Graphs [article]

Maryam Bahrani, Jérémie Lumbroso
<span title="2017-11-29">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Our goal was to study a simple family of graphs, of which the split-decomposition trees have prime nodes drawn from an enumerable (and manageable!) set of graphs.  ...  produce new enumerative results on graphs---in particular the enumeration of several subclasses of perfect graphs (distance-hereditary, 3-leaf power, ptolemaic).  ...  Acknowledgements We would like to thank the anonymous reviewers for critically reading the manuscript and making suggestions that led to significant improvements to the content and clarity of the paper  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1711.10647v1">arXiv:1711.10647v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zguxw3jnuvdu7gdhyw5cugd2aa">fatcat:zguxw3jnuvdu7gdhyw5cugd2aa</a> </span>
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Split-Decomposition Trees with Prime Nodes: Enumeration and Random Generation of Cactus Graphs [chapter]

Maryam Bahrani, Jérémie Lumbroso
<span title="">2018</span> <i title="Society for Industrial and Applied Mathematics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/orsfoxbevzfrnhxrp6ku3nxqge" style="color: black;">2018 Proceedings of the Fifteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)</a> </i> &nbsp;
Our goal was to study a simple family of graphs, of which the split-decomposition trees have prime nodes drawn from an enumerable (and manageable!) set of graphs.  ...  results on graphs-in particular the enumeration of several subclasses of perfect graphs (distance-hereditary, 3-leaf power, ptolemaic).  ...  Acknowledgements We would like to thank the anonymous reviewers for critically reading the manuscript and making suggestions that led to significant improvements to the content and clarity of the paper  ... 
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