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Jarkko Kari (Editor) Proceedings of JAC 2010 Journées Automates Cellulaires Turku Centrefor Computer Science TUCS Lecture Notes No 13, Dec 2010 Proceedings of JAC 2010 Journ´eesAutomates Cellulaires December 15-17, 2010, Turku, Finland Editor: Jarkko Kari TUCS Lecture Notes 13 December 2010 ISBN 978-952-12-2503-1 (printed) ISBN 978-952-12-2504-8 (electronic) ISSN 1797-8823 (printed) ISSN 1797-8831 (electronic) Foreword The second Symposium on Cellular Automata “Journ´eesAutomates Cellulaires” (JAC 2010) took place in Turku, Finland, on December 15-17, 2010. The ﬁrst two conference days were held in the Educarium building of the University of Turku, while the talks of the third day were given onboard passenger ferry boats in the beautiful Turku archipelago, along the route Turku–Mariehamn–Turku. The con- ference was organized by FUNDIM, the Fundamentals of Computing and Discrete Mathematics research center at the mathematics department of the University of Turku. The program of the conference included 17 submitted papers that were selected by the international program committee, based on three peer reviews of each paper. These papers form the core of these proceedings. I want to thank the members of the program committee and the external referees for the excellent work that have done in choosing the papers to be presented in the conference. In addition to the submitted papers, the program of JAC 2010 included four dis- tinguished invited speakers: Michel Coornaert (Universit´ede Strasbourg, France), Bruno Durand (Universit´ede Provence, Marseille, France), Dora Giammarresi (Uni- versit`adi Roma Tor Vergata, Italy) and Martin Kutrib (Universit¨atGießen, Ger- many). I sincerely thank the invited speakers for accepting our invitation to come and give a plenary talk in the conference. The invited talk by Bruno Durand was eventually given by his co-author Alexander Shen, and I thank him for accepting to make the presentation with a short notice. Abstracts or extended abstracts of the invited presentations appear in the ﬁrst part of this volume. The program also included several informal presentations describing very recent developments and ongoing research projects. I wish to thank all the speakers for their contribution to the success of the symposium. I also would like to thank the sponsors and our collaborators: the Finnish Academy of Science and Letters, the French National Research Agency project EMC (ANR-09-BLAN-0164), Turku Centre for Computer Science, the University of Turku, and Centro Hotel. Finally, I sincerely thank the members of the local organizing committee for making the conference possible. These proceedings are published both in an electronic format and in print. The electronic proceedings are available on the electronic repository HAL, managed by several French research agencies. The printed version is published in the general publications series of TUCS, Turku Centre for Computer Science. We thank both HAL and TUCS for accepting to publish the proceedings. Turku, December 2010 Jarkko Kari Program Committee Enrico Formenti (Universit´ede Nice-Sophia Antipolis, France) Jarkko Kari (chair, University of Turku, Finland) Jean Mairesse (Universit´eParis 7, France) Ivan Rapaport (Universidad de Chile, Santiago, Chile) Klaus Sutner (Carnegie Mellon University, Pittsburgh, USA) Siamak Taati (University of Groningen, the Netherlands) External referees Florent Becker, Raimundo Brice˜no, Silvio Capobianco, Romain Demangeon, Al- berto Dennunzio, Amir F. Dana, Anah´ıGajardo, Pierre Guillon, Edmund Harriss, Petr K˚urka, Bruno Martin, Pierre-Etienne´ Meunier, Christophe Papazian, Julien Provillard, Eric´ R´emila,Ville Salo, Matthias Schulz, V´eroniqueTerrier, Guillaume Theyssier, Michael Weiss, Jean-Baptiste Yun`es,Charalampos Zinoviadis. Organizing Committee Alexis Ballier Pierre Guillon Timo Jolivet Jarkko Kari (chair) Arto Lepist¨o Charalampos Zinoviadis Table of Contents Invited Talks: Abstracts and Extended Abstracts Some Extensions of the Moore-Myhill Garden of Eden Theorem . 1 M. Coornaert 1D Eﬀectively Closed Subshifts and 2D Tilings . 2 B. Durand, A. Romashchenko, and A. Shen Tiling-recognizable Two-dimensional Languages: from Non-determinism to Determinism through Unambiguity . 8 D. Giammarresi Measuring Communication in Cellular Automata . 13 M. Kutrib and A. Malcher Regular Papers A Quantum Game of Life. 31 P. Arrighi and J. Grattage The Block Neighborhood . 43 P. Arrighi and V. Nesme Computing (or not) Quasi-periodicity Functions of Tilings . 54 A. Ballier and E. Jeandel A Simulation of Oblivious Multi-head One-way Finite Automata by Real-time Cellular Automata . 65 A. Borello Construction of µ-limit Sets . 76 L. Boyer, M. Delacourt, and M. Sablik A Categorical Outlook on Cellular Automata . 88 S. Capobianco and T. Uustalu Combinatorial Substitutions and Soﬁc Tilings. 100 Th. Fernique and N. Ollinger Inﬁnite Time Cellular Automata: a Real Computation Model . 111 F. Givors, G. Laﬁtte, and N. Ollinger Computational Complexity of Avalanches in the Kadanoﬀ Two-dimensional Sandpile Model. 121 E. Goles and B. Martin Clandestine Simulations in Cellular Automata . 133 P. Guillon, P.-E. Meunier, and G. Theyssier Slopes of Tilings. 145 E. Jeandel and P. Vanier Transductions Computed by Iterative Arrays . 156 M. Kutrib and A. Malcher An Upper Bound on the Number of States for a Strongly Universal Hyperbolic Cellular Automaton on the Pentagrid . 168 M. Margenstern Time-symmetric Cellular Automata . 180 A. Moreira and A. Gajardo Yet Another Aperiodic Tile Set . 191 V. Poupet Decomposition Complexity . 203 A. Shen Real-time Sorting of Binary Numbers on One-dimensional CA . 214 Th. Worsch and H. Nishio Journees´ Automates Cellulaires 2010 (Turku), pp. 1-1 SOME EXTENSIONS OF THE MOORE-MYHILL GARDEN OF EDEN THEOREM MICHEL COORNAERT Institut de Recherche Math´ematique Avanc´ee Universit´e de Strasbourg, 7 rue Ren´e Descartes, 67084 Strasbourg Cedex, France E-mail address: [email protected] Abstract. The Moore-Myhill Garden of Eden theorem asserts that a cellular automaton with ﬁnite alphabet over a free abelian group of rank 2 is surjective if and only if it is pre-injective. Here, pre-injectivity means that two conﬁgurations which coincide outside of a ﬁnite subset of the group must coincide everywhere if they have the same image under the cellular automaton. The Garden of Eden theorem has been extended to amenable groups by Ceccherini-Silberstein, Machi and Scarabotti and to cellular automata deﬁned over strongly irreducible subshifts of ﬁnite type by Fiorenzi. I will discuss these extensions as well as linear versions jointly obtained with Ceccherini-Silberstein. c 1 Journees´ Automates Cellulaires 2010 (Turku), pp. 2-7 1D EFFECTIVELY CLOSED SUBSHIFTS AND 2D TILINGS BRUNO DURAND 1, ANDREI ROMASHCHENKO 2, AND ALEXANDER SHEN 2 1 LIF, CNRS & Aix–Marseille Universit´e E-mail address: [email protected] URL: http://www.lif.univ-mrs.fr/~bdurand 2 LIF, CNRS & Aix–Marseille Universit´e,on leave from IITP RAS E-mail address: {Andrei.Romashchenko,Alexander.Shen}@lif.univ-mrs.fr Abstract. Michael Hochman showed that every 1D eﬀectively closed subshift can be simulated by a 3D subshift of ﬁnite type and asked whether the same can be done in 2D. It turned out that the answer is positive and necessary tools were already developed in tilings theory. We discuss two alternative approaches: ﬁrst, developed by N. Aubrun and M. Sablik, goes back to Leonid Levin; the second one, developed by the authors, goes back to Peter Gacs. 1. Simulation Let A be a ﬁnite alphabet and let F be an enumerable set of A-strings. Con- sider all biinﬁnite A-sequences (i.e., mappings of type Z A) that do not contain substrings from F . The set of these sequences is eﬀectively→ closed (its complement is a union of an enumerable set of intervals in Cantor topology) and invariant under (left and right) shifts. Sets constructed in this way are called eﬀectively closed 1D subshifts. Eﬀectively closed 2D subshifts are deﬁned in a similar way; instead of biinﬁnite sequences we have conﬁgurations, i.e., mappings of type Z2 A, and instead of forbidden strings we have forbidden patterns (rectangles ﬁlled with→ A-letters). Given the set F of forbidden patterns, we consider the set of all conﬁgurations where no elements of F appear. This set of conﬁgurations is closed under vertical and horizontal shifts. If F is enumerable, we get eﬀectively closed 2D subshifts; if F is ﬁnite, we get 2D subshifts of ﬁnite type. 2D subshifts of ﬁnite type are closely related to tilings. A tile is a square with colored sides (colors are taken from some ﬁnite set C). A tile set is a set of tiles, i.e., a subset of C4, since each tile is determined by four colors (upper, lower, left, and right). For a tile set τ, we consider all τ-tilings, i.e., the tilings of the entire plane by translated copies of τ-tiles with matching colors. Key words and phrases: eﬀectively closed subshifts, subshifts of ﬁnite type, tilings. The paper was supported part by NAFIT ANR-08-EMER-008-01 grant, part by EMC ANR- 09-BLAN-0164-01. c 2 1D EFFECTIVELY CLOSED SUBSHIFTS AND 2D TILINGS 3 Tilings can be considered as a special case of 2D subshifts of ﬁnite type. In- deed, subshift is governed by local rules (forbidden pattern say what is not allowed according to these rules). In tilings the rules are extremely local: they say that the neighbor tiles should have matching colors, i.e., 1 2 rectangles where the colors do not match, are forbidden. × So every tile set determines a subshift of ﬁnite type (the alphabet is a tile set). The reverse statement is also true if we allow the extension of an alphabet. (It is natural since we have to simulate any local rule by a more restricted class of matching rules.) Formally, for every alphabet A and subshift S of ﬁnite type we can ﬁnd: a set of colors C; • a tile set τ C4; • a mapping d⊂: τ A such• that every τ-tiling→ after applying d to each tile becomes an element of S and every element of S can be obtained in this way from some τ-tiling.