NDLI: The basic theory of infinite time register machines

Content Provider	Springer Nature Link
Author	Miller, Russell Fischbach, Tim Koepke, Peter Weckbecker, Gregor Carl, Merlin Nasfi, Miriam
Copyright Year	2010
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.
Ending Page	273
Page Count	25
Starting Page	249
File Format	PDF
ISSN	09335846
e-ISSN	14320665
Journal	Archive for Mathematical Logic
Issue Number	2
Volume Number	49
Language	English
Publisher	Springer-Verlag
Publisher Date	2010-01-12
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Infinitary computation Register machine Hypercomputation Algebra Ordinal computability Mathematics Computability and recursion theory on ordinals, admissible sets, etc. Mathematical Logic and Foundations
Content Type	Text
Resource Type	Article
Subject	Philosophy Logic

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