Abstract
Infinite time register machines (ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands. At limit times register contents are defined by appropriate limit operations. In this paper, we examine the ITRMs introduced by the third and fourth author (Koepke and Miller in Logic and Theory of Algorithms LNCS, pp. 306–315, 2008), where a register content at a limit time is set to the lim inf of previous register contents if that limit is finite; otherwise the register is reset to 0. The theory of these machines has several similarities to the infinite time Turing machines (ITTMs) of Hamkins and Lewis. The machines can decide all \({\Pi^1_1}\) sets, yet are strictly weaker than ITTMs. As in the ITTM situation, we introduce a notion of ITRM-clockable ordinals corresponding to the running times of computations. These form a transitive initial segment of the ordinals. Furthermore we prove a Lost Melody theorem: there is a real r such that there is a program P that halts on the empty input for all oracle contents and outputs 1 iff the oracle number is r, but no program can decide for every natural number n whether or not \({n \in r}\) with the empty oracle. In an earlier paper, the third author considered another type of machines where registers were not reset at infinite lim inf’s and he called them infinite time register machines. Because the resetting machines correspond much better to ITTMs we hold that in future the resetting register machines should be called ITRMs.
Similar content being viewed by others
Rferences
Dimitriou, I., Hamkins, J.D., Koepke, P. (eds.): BIWOC—Bonn International Workshop on Ordinal Computability, Bonn Logic Reports (2007)
Hamkins J.D., Lewis A.: Infinite time turing machines. J. Symb. Log. 65(2), 567–604 (2000)
Hamkins, J.D., Linetsky, D., Miller, R.: The complexity of quickly ORM-decidable sets. In: Cooper, S.B., et al. (eds.) Computation and Logic in the Real World, LNCS, vol. 4497, pp. 488–496. Springer, Heidelberg (2007)
Hamkins, J.D., Miller, R.: Post’s problem for ordinal register machines. In: Cooper, S.B., et al. (eds.) Computation and Logic in the Real World, LNCS. vol. 4497, pp. 358–367. Springer, Heidelberg (2007)
Koepke P., et al.: Infinite time register machines. In: Beckmann, A., (eds) Logical Approaches to Computational Barriers, LNCS. vol. 3988, pp. 257–266. Springer, Heidelberg (2006)
Koepke P.: Turing computations on ordinals. B. Symb. Log. 11, 377–397 (2005)
Koepke, P., Miller, R.: An enhanced theory of infinite time register machines. In: Beckmann, A., et al. (eds.) Logic and Theory of Algorithms LNCS, vol. 5028, pp. 306–315. (2008)
Koepke P., Siders R.: Register computations on ordinals. Arch. Math. Log. 47, 529–548 (2008)
Koepke P., Seyfferth B.: Ordinal machines and admissible recursion theory. Ann. Pure Appl. Log. 160, 310–318 (2009)
Koepke, P., Siders, R.: Computing the recursive truth predicate on ordinal register machines. In: Beckmann, A., et al. (eds.) Logical Approaches to Computational Barriers, Computer Science Report Series, vol. 7, pp. 160–169. (2006)
Koepke P., Siders R.: Minimality considerations for ordinal computers modeling constructibility. Theoretical Comput. Sci. 394, 197–207 (2008)
Jech, T.: Set Theory, 3rd Millennium edition, revisited and expanded. Springer, Berlin (2002)
Jensen, R.B.: The fine structure of the constructible hierarchy. Ann. Math. Log. (1972)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Carl, M., Fischbach, T., Koepke, P. et al. The basic theory of infinite time register machines. Arch. Math. Logic 49, 249–273 (2010). https://doi.org/10.1007/s00153-009-0167-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-009-0167-x