Oritatami Systems Assemble Shapes No Less Complex Than Tile Assembly Model (ATAM)

Authors Daria Pchelina, Nicolas Schabanel, Shinnosuke Seki, Guillaume Theyssier

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Daria Pchelina
  • LIPN, Institut Galilée – Université Paris 13, France
Nicolas Schabanel
  • École Normale Supérieure de Lyon (LIP UMR5668 and IXXI, MC2), France
Shinnosuke Seki
  • University of Electro-Communications, Tokyo, Japan
Guillaume Theyssier
  • Aix-Marseille Université, CNRS, I2M, Marseille, France

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Daria Pchelina, Nicolas Schabanel, Shinnosuke Seki, and Guillaume Theyssier. Oritatami Systems Assemble Shapes No Less Complex Than Tile Assembly Model (ATAM). In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 51:1-51:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Different models have been proposed to understand natural phenomena at the molecular scale from a computational point of view. Oritatami systems are a model of molecular co-transcriptional folding: the transcript (the "molecule") folds as it is synthesized according to a local energy optimisation process, in a similar way to how actual biomolecules such as RNA fold into complex shapes and functions. We introduce a new model, called turedo, which is a self-avoiding Turing machine on the plane that evolves by marking visited positions and that can only move to unmarked positions. Any oritatami can be seen as a particular turedo. We show that any turedo with lookup radius 1 can conversely be simulated by an oritatami, using a universal bead type set. Our notion of simulation is strong enough to preserve the geometrical and dynamical features of these models up to a constant spatio-temporal rescaling (as in intrinsic simulation). As a consequence, turedo can be used as a readable oritatami "higher-level" programming language to build readily oritatami "smart robots", using our explicit simulation result as a compiler. As an application of our simulation result, we prove two new complexity results on the (infinite) limit configurations of oritatami systems (and radius-1 turedos), assembled from a finite seed configuration. First, we show that such limit configurations can embed any recursively enumerable set, and are thus exactly as complex as aTAM limit configurations. Second, we characterize the possible densities of occupied positions in such limit configurations: they are exactly the Π₂-computable numbers between 0 and 1. We also show that all such limit densities can be produced by one single oritatami system, just by changing the finite seed configuration. None of these results is implied by previous constructions of oritatami embedding tag systems or 1D cellular automata, which produce only computable limit configurations with constrained density.

Subject Classification

ACM Subject Classification
  • Computer systems organization → Molecular computing
  • Computing methodologies → Molecular simulation
  • Applied computing → Molecular structural biology
  • Theory of computation → Computability
  • Theory of computation → Complexity theory and logic
  • Molecular Self-assembly
  • Co-transcriptional folding
  • Intrinsic simulation
  • Arithmetical hierarchy of real numbers
  • 2D Turing machines
  • Computability


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