The Group of Reversible Turing Machines

  • Sebastián Barbieri
  • Jarkko Kari
  • Ville Salo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9664)


We consider Turing machines as actions over configurations in \(\varSigma ^{\mathbb {Z}^d}\) which only change them locally around a marked position that can move and carry a particular state. In this setting we study the monoid of Turing machines and the group of reversible Turing machines. We also study two natural subgroups, namely the group of finite-state automata, which generalizes the topological full groups studied in the theory of orbit-equivalence, and the group of oblivious Turing machines whose movement is independent of tape contents, which generalizes lamplighter groups and has connections to the study of universal reversible logical gates. Our main results are that the group of Turing machines in one dimension is neither amenable nor residually finite, but is locally embeddable in finite groups, and that the torsion problem is decidable for finite-state automata in dimension one, but not in dimension two.


Cellular Automaton Turing Machine Local Rule Torsion Problem Sofic Shift 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The third author was supported by FONDECYT grant 3150552.


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© IFIP International Federation for Information Processing 2016

Authors and Affiliations

  1. 1.LIP, ENS de Lyon – CNRS – INRIA – UCBL – Université de LyonLyonFrance
  2. 2.University of TurkuTurkuFinland
  3. 3.Center for Mathematical ModelingUniversity of ChileSantiagoChile

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