Weaker variants of infinite time Turing machines
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Infinite time Turing machines represent a model of computability that extends the operations of Turing machines to transfinite ordinal time by defining the content of each cell at limit steps to be the \(\limsup \) of the sequences of previous contents of that cell. In this paper, we study a computational model obtained by replacing the \(\limsup \) rule with an ‘eventually constant’ rule: at each limit step, the value of each cell is defined if and only if the content of that cell has stabilized before that limit step and is then equal to this constant value. We call these machines weak infinite time Turing machines (wITTMs). We study different variants of wITTMs adding multiple tapes, heads, or bidimensional tapes. We show that some of these models are equivalent to each other concerning their computational strength. We show that wITTMs decide exactly the arithmetic relations on natural numbers.
KeywordsOrdinal computability Infinite time Turing machine Transfinite computation Supertask Arithmetic hierarchy Real arithmetic
Mathematics Subject Classification03D10 03D60 03D78 68Q01
This work is largely a part of my master thesis  under the supervision of Julia F. Knight. I am very grateful to Julia F. Knight for her skillful guidance and insightful comments at every stage of the project. I would like to warmly thank Quinn Culver for numerous, extremely helpful discussions. I am also grateful to Merlin Carl, Dan Turetsky, and Greg Igusa for useful discussions and to an anonymous reviewer for valuable suggestions. I am also grateful to everyone that attended to presentations of earlier versions of this work either at the Notre Dame Computability seminar or at the CiE 2015 meeting.
- 2.Bianchetti, M.: Infinite time computation: strong and weak infinite time turing machines. Master’s thesis, University of Notre Dame (2017)Google Scholar
- 3.Carl, M.: Ordinal computability. An introduction to infinitary machines (forthcoming)Google Scholar
- 4.Descartes, R.: Discours de la methode pour bien conduire sa raison, & chercher la verité dans les sciences: plus la doptrique, les meteores, et la geometrie, qui sont des essais de cete methode. Maire, Leiden (1637)Google Scholar