Space bounds for infinitary computation
Benedikt Löwe
Abstract:
For an ordinary Turing machine that stops in a finite number $t of
steps, it is easy to define its space usage: during its computation,
it has used at most $t cells of the tape, possibly less. This finite
number of used cells can serve as a measure of space usage. A halting
computation will have used a finite amount of time and space; if,
however, time or space usage are infinite, then this corresponds to
usage of order type \omega and automatically implies that the
computation was non-halting.
In this paper, we shall consider both Hamkins-Kidder machines and
Koepke's Ordinal Machines. Koepke machines can not only extend their
computation into transfinite ordinal time, but they also have
ordinal-indexed cells on their tapes. Therefore, there is a natural
notion of space usage for computations on Koepke machines that
corresponds to the classical idea of space constraints on Turing
Machines: just count the number (order type) of cells being used.
This is very different for Hamkins-Kidder machines whose space is
constrained to a tape of order type \omega whereas time can have
arbitrary ordinals as order type. This asymmetry makes is hard to give
a definition of space usage that can be compared to time usage.
In this paper, we discuss the basics of possible definitions for space
constraints for the mentioned two types of infinitary Turing machine
computations. In section 1, we give all definitions needed in the
paper and then brie~y discuss space constraints for Koepke's machines
in section 2 and space constraints for Hamkins-Kidder machines in
section 3. Finally, in section 4, we discuss nondeterministic
computation.