aState Key Laboratory of Computer Science Institute of Software ChineseAcademy of Sciences Beijing 100190 P.O. Box 8718 China; bInstitute for Logic Language and Computation Universiteit van Amsterdam P.O. Box 94242 1090 GE Amsterdam The Netherlands
Abstract:
The sequences which have trivial prefix-free initial segment complexity are known as K -trivial sets, and form a cumulative hierarchy of length ω . We show that the problem of finding the number of K -trivial sets in the various levels of the hierarchy is Δ 3 0 . This answers a question of Downey/Miller/Yu (see Downey (2010)?[7,?Section 10.1.4]) which also appears in Nies (2009)?[17,?Problem 5.2.16]. We also show the same for the hierarchy of the low for K sequences, which are the ones that (when used as oracles) do not give a shorter initial segment complexity compared to the computable oracles. In both cases the classification Δ 3 0 is sharp.
English Abstract:
The sequences which have trivial prefix-free initial segment complexity are known as K -trivial sets, and form a cumulative hierarchy of length ω . We show that the problem of finding the number of K -trivial sets in the various levels of the hierarchy is Δ 3 0 . This answers a question of Downey/Miller/Yu (see Downey (2010)?[7,?Section 10.1.4]) which also appears in Nies (2009)?[17,?Problem 5.2.16]. We also show the same for the hierarchy of the low for K sequences, which are the ones that (when used as oracles) do not give a shorter initial segment complexity compared to the computable oracles. In both cases the classification Δ 3 0 is sharp.
George Barmpalias,T.F. Sterkenburg. on the number of infinite sequences with trivial initial segment complexity[J]. Theoretical Computer Science,2011-01-01,412(52):7133-7146.
