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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

We introduce an extension of classical cellular automata (CA) to arbitrary labeled graphs, and show that FO logic on CA orbits is equivalent to MSO logic. We deduce various results from that equivalence, including a characterization of finitely generated groups on which FO model checking for CA orbits is undecidable, and undecidability of satisfiability of a fixed FO property for CA over finite graphs. We also show concrete examples of FO formulas for CA orbits whose model checking problem is equivalent to the domino problem, or its seeded or recurring variants respectively, on any finitely generated group. For the recurring domino problem, we use an extension of the FO signature by a relation found in the well-known Garden of Eden theorem, but we also show a concrete FO formula without the extension and with one quantifier alternation whose model checking problem does not belong to the arithmetical hierarchy on group ℤ².

Guillaume Theyssier. FO Logic on Cellular Automata Orbits Equals MSO Logic. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 154:1-154:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{theyssier:LIPIcs.ICALP.2024.154, author = {Theyssier, Guillaume}, title = {{FO Logic on Cellular Automata Orbits Equals MSO Logic}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {154:1--154:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.154}, URN = {urn:nbn:de:0030-drops-202972}, doi = {10.4230/LIPIcs.ICALP.2024.154}, annote = {Keywords: MSO logic, FO logic, cellular automata, domino problem, Cayley graphs} }

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**Published in:** LIPIcs, Volume 238, 28th International Conference on DNA Computing and Molecular Programming (DNA 28) (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.

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)

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@InProceedings{nalin_et_al:LIPIcs.DNA.28.6, author = {Nalin, Samuel and Theyssier, Guillaume}, title = {{On Turedo Hierarchies and Intrinsic Universality}}, booktitle = {28th International Conference on DNA Computing and Molecular Programming (DNA 28)}, pages = {6:1--6:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-253-2}, ISSN = {1868-8969}, year = {2022}, volume = {238}, editor = {Ouldridge, Thomas E. and Wickham, Shelley F. J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.28.6}, URN = {urn:nbn:de:0030-drops-167915}, doi = {10.4230/LIPIcs.DNA.28.6}, annote = {Keywords: Turedos, intrinsic universality, Higher-dimensional Turing machines, Oritatami systems} }

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

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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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

We prove general complexity lower bounds on automata networks, in the style of Rice’s theorem, but in the computable world. Our main result is that testing any fixed first-order property on the dynamics of an automata network is either trivial, or NP-hard, or coNP-hard. Moreover, there exist such properties that are arbitrarily high in the polynomial-time hierarchy. We also prove that testing a first-order property given as input on an automata network (also part of the input) is PSPACE-hard. Besides, we show that, under a natural effectiveness condition, any nontrivial property of the limit set of a nondeterministic network is PSPACE-hard. We also show that it is PSPACE-hard to separate deterministic networks with a very high and a very low number of limit configurations; however, the problem of deciding whether the number of limit configurations is maximal up to a polynomial quantity belongs to the polynomial-time hierarchy.

Guilhem Gamard, Pierre Guillon, Kevin Perrot, and Guillaume Theyssier. Rice-Like Theorems for Automata Networks. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gamard_et_al:LIPIcs.STACS.2021.32, author = {Gamard, Guilhem and Guillon, Pierre and Perrot, Kevin and Theyssier, Guillaume}, title = {{Rice-Like Theorems for Automata Networks}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {32:1--32:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus 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.2021.32}, URN = {urn:nbn:de:0030-drops-136770}, doi = {10.4230/LIPIcs.STACS.2021.32}, annote = {Keywords: Automata networks, Rice theorem, complexity classes, polynomial hierarchy, hardness} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

Cellular automata (CA) are dynamical systems defined by a finite local rule but they are studied for their global dynamics. They can exhibit a wide range of complex behaviours and a celebrated result is the existence of (intrinsically) universal CA, that is CA able to fully simulate any other CA. In this paper, we show that the asymptotic density of universal cellular automata is 1 in several families of CA defined by local symmetries. We extend results reviously established for captive cellular automata in two significant ways. First, our results apply to well-known families of CA (e.g. the family of outer-totalistic CA containing the Game of Life) and, second, we obtain such density results with both increasing number of states and increasing neighbourhood. Moreover, thanks to universality-preserving encodings, we show that the universality problem remains undecidable in some of those families.

Laurent Boyer and Guillaume Theyssier. On Local Symmetries and Universality in Cellular Automata. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 195-206, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{boyer_et_al:LIPIcs.STACS.2009.1836, author = {Boyer, Laurent and Theyssier, Guillaume}, title = {{On Local Symmetries and Universality in Cellular Automata}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {195--206}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1836}, URN = {urn:nbn:de:0030-drops-18369}, doi = {10.4230/LIPIcs.STACS.2009.1836}, annote = {Keywords: Cellular automata, Universality, Asymptotic density} }

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