Space Complexity in Infinite Time Turing Machines
Joost Winter
Abstract:
In this thesis, we investigate space complexity issues for both
Infinite Time Turing Machines as originally defined by Hamkins and
Kidder, and for Ordinal Turing Machines as defined by Peter Koepke.
Analogues of the classical classes PSPACE and NPSPACE are defined for
these machines performing transfinite computations, following the
definition of P and NP classes given in a paper by Ralf Schindler.
After that, the question P =? PSPACE is investigated, as well as the
questions NP =? NPSPACE, and PSPACE =? NPSPACE.
In the case of the Hamkins-Kidder model, we make use of G-bĂ¶del's-A
constructible hierarchy in defining the space complexity classes,
whereas in the case of Koepke's machines we simply look at the number
of cells used. It turns out there is no uniform positive or negative
result for the main question: in the case of both models, for some
functions f we find P_f = PSPACE_f, and for others we find P_f \ne
PSPACE_f. Also for the questions NP =? NPSPACE, and PSPACE =?
NPSPACE, a number of (negative) results are proved.
Keywords: ITTM, Infinite Time Turing Machine, Ordinal Turing Machine,
space complexity, PSPACE, P=PSPACE