Highly Undecidable Problems about Recognizability by Tiling Systems

to appear in a Special Issue of the journal Fundamenta Informaticae on Machines, Computations and Universality.International audienceAltenbernd, Thomas and Wöhrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the Büchi and Muller ones [1]. It was proved in [9] that it is undecidable whether a Büchi-recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). We show here that these two decision problems are actually $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and ``highly undecidable". We give the exact degree of numerous other undecidable problems for Büchi-recognizable languages of infinite pictures. In particular, the non-emptiness and the infiniteness problems are $\Sigma^1_1$-complete, and the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, are all $\Pi^1_2$-complete. It is also $\Pi^1_2$-complete to determine whether a given Büchi recognizable language of infinite pictures can be accepted row by row using an automaton model over ordinal words of length $\omega^2$

The Complexity of Infinite Computations In Models of Set Theory

We prove the following surprising result: there exist a 1-counter B\"uchi automaton and a 2-tape B\"uchi automaton such that the \omega-language of the first and the infinitary rational relation of the second in one model of ZFC are \pi_2^0-sets, while in a different model of ZFC both are analytic but non Borel sets. This shows that the topological complexity of an \omega-language accepted by a 1-counter B\"uchi automaton or of an infinitary rational relation accepted by a 2-tape B\"uchi automaton is not determined by the axiomatic system ZFC. We show that a similar result holds for the class of languages of infinite pictures which are recognized by B\"uchi tiling systems. We infer from the proof of the above results an improvement of the lower bound of some decision problems recently studied by the author

Highly Undecidable Problems For Infinite Computations

We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all $\Pi_2^1$-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are also highly undecidable. These very surprising results provide the first examples of highly undecidable problems about the behaviour of very simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application

Highly Undecidable Problems about Recognizability by Tiling Systems

to appear in a Special Issue of the journal Fundamenta Informaticae on Machines, Computations and Universality.International audienceAltenbernd, Thomas and Wöhrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the Büchi and Muller ones [1]. It was proved in [9] that it is undecidable whether a Büchi-recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). We show here that these two decision problems are actually $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and ``highly undecidable". We give the exact degree of numerous other undecidable problems for Büchi-recognizable languages of infinite pictures. In particular, the non-emptiness and the infiniteness problems are $\Sigma^1_1$-complete, and the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, are all $\Pi^1_2$-complete. It is also $\Pi^1_2$-complete to determine whether a given Büchi recognizable language of infinite pictures can be accepted row by row using an automaton model over ordinal words of length $\omega^2$

Proceedings of JAC 2010. Journées Automates Cellulaires

The second Symposium on Cellular Automata “Journ´ees Automates 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 conference 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 distinguished invited speakers: Michel Coornaert (Universit´e de Strasbourg, France), Bruno Durand (Universit´e de Provence, Marseille, France), Dora Giammarresi (Universit` a di Roma Tor Vergata, Italy) and Martin Kutrib (Universit¨at Gie_en, Germany). 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.Siirretty Doriast