eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-03-09
51:1
51:23
10.4230/LIPIcs.STACS.2022.51
article
Oritatami Systems Assemble Shapes No Less Complex Than Tile Assembly Model (ATAM)
Pchelina, Daria
1
Schabanel, Nicolas
2
Seki, Shinnosuke
3
Theyssier, Guillaume
4
LIPN, Institut Galilée – Université Paris 13, France
École Normale Supérieure de Lyon (LIP UMR5668 and IXXI, MC2), France
University of Electro-Communications, Tokyo, Japan
Aix-Marseille Université, CNRS, I2M, Marseille, France
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol219-stacs2022/LIPIcs.STACS.2022.51/LIPIcs.STACS.2022.51.pdf
Molecular Self-assembly
Co-transcriptional folding
Intrinsic simulation
Arithmetical hierarchy of real numbers
2D Turing machines
Computability