1 research outputs found
Post's problem for supertasks has both positive and negative solutions
Recently we have introduced a new model of infinite computation by extending
the operation of ordinary Turing machines into transfinite ordinal time. In
this paper we will show that the infinite time Turing machine analogue of
Post's problem, the question whether there are supertask degrees between 0 and
the supertask jump 0^jump, has in a sense both positive and negative solutions.
Namely, in the context of the reals there are no degrees between 0 and 0^jump,
but in the context of SETS of reals, there are; indeed, there are incomparable
semi-decidable supertask degrees. Both arguments employ a kind of
transfinite-injury construction which generalizes canonically to oracles.Comment: 20 pages, submitted to the Archive for Mathematical Logic. See the
author's home pages at http://www.library.csi.cuny.edu/users/hamkins and
http://saturn.vcu.edu/~amlewi