On Turedo Hierarchies and Intrinsic Universality

Authors Samuel Nalin, Guillaume Theyssier

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Samuel Nalin
  • Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, FR-45067 Orléans, France
Guillaume Theyssier
  • I2M, CNRS, Université Aix-Marseille, France

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Samuel Nalin and Guillaume Theyssier. On Turedo Hierarchies and Intrinsic Universality. In 28th International Conference on DNA Computing and Molecular Programming (DNA 28). Leibniz International Proceedings in Informatics (LIPIcs), Volume 238, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


This paper is about turedos, which are Turing machines whose head can move in the plane (or in a higher-dimensional space) but only in a self-avoiding way, by putting marks (letters) on visited positions and moving only to unmarked, therefore unvisited, positions. The turedo model has been introduced recently as a useful abstraction of oritatami systems, which where established a few years ago as a theoretical model of RNA co-transcriptional folding. The key parameter of turedos is their lookup radius: the distance up to which the head can look around in order to make its decision of where to move to and what mark to write. In this paper we study the hierarchy of turedos according to their lookup radius and the dimension of space using notions of simulation up to spatio-temporal rescaling (a standard approach in cellular automata or self-assembly systems). We establish that there is a rich interplay between the turedo parameters and the notion of simulation considered. We show in particular, for the most liberal simulations, the existence of 3D turedos of radius 1 that are intrinsically universal for all radii, but that this is impossible in dimension 2, where some radius 2 turedo are impossible to simulate at radius 1. Using stricter notions of simulation, intrinsic universality becomes impossible, even in dimension 3, and there is a strict radius hierarchy. Finally, when restricting to radius 1, universality is again possible in dimension 3, but not in dimension 2, where we show however that a radius 3 turedo can simulate all radius 1 turedos.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Formal languages and automata theory
  • Turedos
  • intrinsic universality
  • Higher-dimensional Turing machines
  • Oritatami systems


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