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Documents authored by Pchelina, Daria

Document
DOI: 10.4230/LIPIcs.SoCG.2023.32
When Ternary Triangulated Disc Packings Are Densest: Examples, Counter-Examples and Techniques

Authors: Thomas Fernique and Daria Pchelina

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each "hole" is bounded by three pairwise tangent discs are called triangulated. Connelly conjectured that when such packings exist, one of them maximizes the proportion of the covered surface: this holds for unary and binary disc packings. For ternary packings, there are 164 pairs (r, s), 1 > r > s, allowing triangulated packings by discs of radii 1, r and s. In this paper, we enhance existing methods of dealing with maximal-density packings in order to study ternary triangulated packings. We prove that the conjecture holds for 31 triplets of disc radii and disprove it for 40 other triplets. Finally, we classify the remaining cases where our methods are not applicable. Our approach is based on the ideas present in the Hales' proof of the Kepler conjecture. Notably, our proof features local density redistribution based on computer search and interval arithmetic.
Cite as

Thomas Fernique and Daria Pchelina. When Ternary Triangulated Disc Packings Are Densest: Examples, Counter-Examples and Techniques. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fernique_et_al:LIPIcs.SoCG.2023.32,
  author =	{Fernique, Thomas and Pchelina, Daria},
  title =	{{When Ternary Triangulated Disc Packings Are Densest: Examples, Counter-Examples and Techniques}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{32:1--32:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.32},
  URN =		{urn:nbn:de:0030-drops-178827},
  doi =		{10.4230/LIPIcs.SoCG.2023.32},
  annote =	{Keywords: Disc packing, density, interval arithmetic}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.51
Oritatami Systems Assemble Shapes No Less Complex Than Tile Assembly Model (ATAM)

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

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
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.
Cite as

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)

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@InProceedings{pchelina_et_al:LIPIcs.STACS.2022.51,
  author =	{Pchelina, Daria and Schabanel, Nicolas and Seki, Shinnosuke and Theyssier, Guillaume},
  title =	{{Oritatami Systems Assemble Shapes No Less Complex Than Tile Assembly Model (ATAM)}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{51:1--51:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.51},
  URN =		{urn:nbn:de:0030-drops-158618},
  doi =		{10.4230/LIPIcs.STACS.2022.51},
  annote =	{Keywords: Molecular Self-assembly, Co-transcriptional folding, Intrinsic simulation, Arithmetical hierarchy of real numbers, 2D Turing machines, Computability}
}
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