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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Exact pairs for the ideal of the $K$-trivial seque
nces\n in the Turing degrees - Barmpalias\, G (Chi
nese Academy of Sciences)
DTSTART;TZID=Europe/London:20120705T110000
DTEND;TZID=Europe/London:20120705T120000
UID:TALK38873AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/38873
DESCRIPTION:The $K$-trivial sets form an ideal in the Turing d
egrees\, which is\ngenerated by its computably enu
merable (c.e.) members and has an exact\npair belo
w the degree of the halting problem. The question
of whether\nit has an exact pair in the c.e. degre
es was first raised in a published\nlist of questi
ons in the Bulletin of Symbolic Logic in 2006 by M
iller and\nNies and later in Nies' book on computa
bility and randomness. Moreover it\nwas featured i
n several conferences in algorithmic randomness\,
since 2005.\n\nWe give a negative answer to this q
uestion. In fact\, we show the\nfollowing stronger
statement in the c.e. degrees. There exists a\n$K
$-trivial degree $mathbf{d}$ such that for all deg
rees $mathbf{a}\,\nmathbf{b}$ which are not $K$-tr
ivial and $mathbf{a}>mathbf{d}\,\nmathbf{b}>mathbf
{d}$ there exists a degree $mathbf{v}$ which is\nn
ot $K$-trivial and $mathbf{a}>mathbf{v}\, mathbf{b
}>mathbf{v}$.\nThis work sheds light to the questi
on of the definability of the\n$K$-trivial degrees
in the c.e. degrees.\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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