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SUMMARY:Exact pairs for the ideal of the $K$-trivial sequences\n in the Tu
 ring degrees - Barmpalias\, G (Chinese Academy of Sciences)
DTSTART:20120705T100000Z
DTEND:20120705T110000Z
UID:TALK38873@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:The $K$-trivial sets form an ideal in the Turing degrees\, whi
 ch is\ngenerated by its computably enumerable (c.e.) members and has an ex
 act\npair below the degree of the halting problem. The question of whether
 \nit has an exact pair in the c.e. degrees was first raised in a published
 \nlist of questions in the Bulletin of Symbolic Logic in 2006 by Miller an
 d\nNies and later in Nies' book on computability and randomness. Moreover 
 it\nwas featured in several conferences in algorithmic randomness\, since 
 2005.\n\nWe give a negative answer to this question. In fact\, we show the
 \nfollowing stronger statement in the c.e. degrees. There exists a\n$K$-tr
 ivial degree $mathbf{d}$ such that for all degrees $mathbf{a}\,\nmathbf{b}
 $ which are not $K$-trivial and $mathbf{a}>mathbf{d}\,\nmathbf{b}>mathbf{d
 }$ there exists a degree $mathbf{v}$ which is\nnot $K$-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
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