# PP-2006-27: Space bounds for infinitary computation

PP-2006-27: Löwe, Benedikt (2006) Space bounds for infinitary computation. [Report]

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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.