Hostname: page-component-54dcc4c588-nx7b4 Total loading time: 0 Render date: 2025-10-01T02:08:56.461Z Has data issue: false hasContentIssue false
  • English
  • Français

Cellular automata over generalized Cayley graphs

Published online by Cambridge University Press:  29 May 2017

PABLO ARRIGHI ,
SIMON MARTIEL  and
VINCENT NESME
PABLO ARRIGHI
Affiliation:
Aix-Marseille Univ., CNRS, LIF, Marseille and IXXI, Lyon, France Email: pablo.arrighi@univ-amu.fr
SIMON MARTIEL
Affiliation:
INRIA Saclay, ENS Cachan, LSV, 61 avenue du président Wilson 94235 Cachan Email: martiel@lsv.ens-cachan.fr
VINCENT NESME
Affiliation:
Université de Grenoble, LIG, 220 rue de la chimie, 38400 Saint-Martin-d'Hères, France Email: vnesme@imag.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well-known that cellular automata can be characterized as the set of translation-invariant continuous functions over a compact metric space; this point of view makes it easy to extend their definition from grids to Cayley graphs. Cayley graphs have a number of useful features: the ability to graphically represent finitely generated group elements and their relations; to name all vertices relative to an origin; and the fact that they have a well-defined notion of translation. We propose a notion of graphs, which preserves or generalizes these features. Whereas Cayley graphs are very regular, generalized Cayley graphs are arbitrary, although of a bounded degree. We extend cellular automata theory to these arbitrary, bounded degree, time-varying graphs. The obtained notion of cellular automata is stable under composition and under inversion.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 28 , Special Issue 3: Mind, Mechanism and Mathematics: Computability Unchained , March 2018 , pp. 340 - 383
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Arrighi, P. and Dowek, G. (2012). Causal graph dynamics. In: Proceedings of ICALP 2012, Warwick, Lecture Notes in Computer Science, vol. 7392, 54–66.Google Scholar
Arrighi, P., Fargetton, R., Nesme, V. and Thierry, E. (2011). Applying causality principles to the axiomatization of probabilistic cellular automata. In: Proceedings of CiE 2011, Sofia, Lecture Notes in Computer Science, vol. 6735, 1–10.Google Scholar
Arrighi, P., Martiel, S. and Wang, Z. (2014). Causal dynamics over discrete surfaces. In: Ayala-Rincón, M., Bonelli, E. and Mackie, I. (eds.) Proceedings 9th International Workshop on Developments in Computational Models, Buenos Aires, Argentina, 26 August 2013, Electronic Proceedings in Theoretical Computer Science, vol. 144, Open Publishing Association, 30–40.Google Scholar
Arrighi, P., Nesme, V. and Werner, R.F. (2008). Quantum cellular automata over finite, unbounded configurations. In: Proceedings of LATA, Lecture Notes in Computer Science, vol. 5196, Springer, 64–75.Google Scholar
Boehm, P., Fonio, H. R. and Habel, A. (1987). Amalgamation of graph transformations: A synchronization mechanism. Journal of Computer and System Sciences 34 (2–3) 377–408.Google Scholar
Ceccherini-Silberstein, T. and Coornaert, M. (2010). Cellular Automata and Groups, Springer Verlag.CrossRefGoogle Scholar
Ceccherini-Silberstein, T., Fiorenzi, F. and Scarabotti, F. (2004). The Garden of Eden Theorem for cellular automata and for symbolic dynamical systems. In: Random Walks and Geometry. Proceedings of a Workshop at the Erwin Schrödinger Institute, Vienna, June 18–July 13, 2001. In collaboration with Klaus Schmidt and Wolfgang Woess. Collected papers, Berlin: de Gruyter, 73–108.Google Scholar
Danos, V., Feret, J., Fontana, W., Harmer, R., Hayman, J., Krivine, J., Thompson-Walsh, C. and Winskel, G. (2012). Graphs, rewriting and pathway reconstruction for rule-based models. In: FSTTCS 2012-IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, vol. 18, 276–288.Google Scholar
Derbel, B., Mosbah, M. and Gruner, S. (2008). Mobile agents implementing local computations in graphs. In: Ehrig, H., Heckel, R., Rozenberg, G. and Taentzer, G. (eds.) Graph Transformations: 4th International Conference, ICGT 2008, Leicester, United Kingdom, September 7–13, 2008. Proceedings, Springer Berlin Heidelberg, Berlin, Heidelberg, 99–114.Google Scholar
Dürr, C. and Santha, M. (1996). A decision procedure for unitary linear quantum cellular automata. In: Proceedings of the 37th IEEE Symposium on Foundations of Computer Science, IEEE, 38–45.Google Scholar
Ehrig, H. and Lowe, M. (1993). Parallel and distributed derivations in the single-pushout approach. Theoretical Computer Science 109 (1–2) 123–143.Google Scholar
Fedorchuk, V. V., Arkhangelskiui, A. V. and Pontriagin, L.S. (1990). General Topology I, vol. 1, Springer.Google Scholar
Giavitto, J.L. and Spicher, A. (2008). Topological rewriting and the geometrization of programming. Physica D: Nonlinear Phenomena 237 (9) 1302–1314.Google Scholar
Godsil, C. D., Royle, G. and Godsil, C. D. (2001). Algebraic Graph Theory, vol. 8, Springer-Verlag, New York.Google Scholar
Gromov, M. (April 1999). Endomorphisms of symbolic algebraic varieties. Journal of the European Mathematical Society 1 (2) 109–197.Google Scholar
Gruner, S. (2010). Mobile agent systems and cellular automata. Autonomous Agents and Multi-Agent Systems 20 198–233. 10.1007/s10458-009-9090-0.Google Scholar
Hasslacher, B. and Meyer, D. A. (June 1998). Modelling dynamical geometry with lattice gas automata. Expanded version of a talk presented at the Seventh International Conference on the Discrete Simulation of Fluids held at the University of Oxford.Google Scholar
Hedlund, G. A. (1969). Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory 3 320–375.Google Scholar
Kari, K. (2011). Cellular Automata, Lecture notes. Available at: http://users.utu.fi/jkari/ca/.Google Scholar
Klales, A., Cianci, D., Needell, Z., Meyer, D. A. and Love, P. J. (Oct. 2010). Lattice gas simulations of dynamical geometry in two dimensions. Physical Review E 82 (4) 046705.Google Scholar
Kreowski, H. J. and Kuske, S. (2007). Autonomous units and their semantics - The parallel case. In: Fiadeiro, J. L. and Schobbens, P.-Y. (eds.) Recent Trends in Algebraic Development Techniques: 18th International Workshop, WADT 2006, La Roche en Ardenne, Belgium, June 1–3, 2006, Revised Selected Papers, Springer Berlin Heidelberg, 56–73.Google Scholar
Kreowski, H. J. and Kuske, S. (2011). Graph multiset transformation: A new framework for massively parallel computation inspired by dna computing. Natural Computing 10 (2) 961–986.Google Scholar
Kurth, W., Kniemeyer, O. and Buck-Sorlin, G. (2005). Relational growth grammars – A graph rewriting approach to dynamical systems with a dynamical structure. In: Banâtre, J.-P., Fradet, P., Giavitto, J.-L. and Michel, O. (eds.) Unconventional Programming Paradigms: International Workshop UPP 2004, Le Mont Saint Michel, France, September 15–17, 2004, Revised Selected and Invited Papers, Springer Berlin Heidelberg, 56–72.Google Scholar
Löwe, M. (1993). Algebraic approach to single-pushout graph transformation. Theoretical Computer Science 109 (1–2) 181–224.Google Scholar
Martiel, S. and Martin, B. (2013). Intrinsic universality of causal graph dynamics. In: Neary, T. and Cook, M. (eds.) Proceedings Machines, Computations and Universality 2013, Zürich, Switzerland, 9/09/2013–11/09/2013, Electronic Proceedings in Theoretical Computer Science, vol. 128, Open Publishing Association, 137–149.Google Scholar
Martiel, S. and Martin, B. (2015). An intrinsically universal family of causal graph dynamics. In: Durand-Lose, J. and Nagy, B. (eds.) Machines, Computations, and Universality: 7th International Conference, MCU 2015, Famagusta, North Cyprus, September 9–11, 2015. Proceedings, Springer International Publishing, Cham, 129–148.Google Scholar
Métivier, Y. and Sopena, E. (1997). Graph relabelling systems: A general overview. Computers & Artificial Intelligence 16 167–185.Google Scholar
Papazian, C. and Remila, E. (2002). Hyperbolic recognition by graph automata. In: Proceedings of the Automata, Languages and Programming: 29th International Colloquium, ICALP 2002, Málaga, Spain, vol. 2380, Springer Verlag, 330.Google Scholar
Regge, T. (1961). General relativity without coordinates. Il Nuovo Cimento (1955–1965) 19 (3) 558–571.Google Scholar
Róka, Z. (1999). Simulations between cellular automata on Cayley graphs. Theoretical Computer Science 225 (1–2) 81–111.Google Scholar
Ryszka, I., Paszyska, A., Grabska, E., Sieniek, M. and Paszyski, M. (2015a). Graph transformation systems for modeling three dimensional finite element method: Part i. Fundamenta Informaticae 140 (2) 129–172.Google Scholar
Ryszka, I., Paszyska, A., Grabska, E., Sieniek, M. and Paszyski, M. (2015b). Graph transformation systems for modeling three dimensional finite element method: Part ii. Fundamenta Informaticae 140 (2) 173–203.Google Scholar
Schumacher, B. and Werner, R. (2004). Reversible quantum cellular automata. ArXiv pre-print quant-ph/0405174.Google Scholar
Sorkin, R. (1975). Time-evolution problem in Regge calculus. Physical Review D. 12 (2) 385–396.Google Scholar
Taentzer, G. (1996). Parallel and Distributed Graph Transformation: Formal Description and Application to Communication-Based Systems. PhD thesis, Technische Universitat Berlin.Google Scholar
Taentzer, G. (1997). Parallel high-level replacement systems. Theoretical Computer Science 186 (1–2) 43–81.Google Scholar
Tomita, K., Kurokawa, H. and Murata, S. (2002). Graph automata: Natural expression of self-reproduction. Physica D: Nonlinear Phenomena 171 (4) 197–210.Google Scholar
Tomita, K., Kurokawa, H. and Murata, S. (2009). Graph-rewriting automata as a natural extension of cellular automata. In: Gross, T. and Sayama, H. (eds.) Adaptive Networks, Understanding Complex Systems, vol. 51, Springer, Berlin/Heidelberg, 291–309.Google Scholar
Tomita, K., Murata, S., Kamimura, A. and Kurokawa, H. (2005). Self-description for construction and execution in graph rewriting automata. In: Capcarrère, M. S., Freitas, A. A., Bentley, P. J., Johnson, C. G. and Timmis, J. (eds.) Advances in Artificial Life: 8th European Conference, ECAL 2005, Canterbury, UK, September 5–9, 2005, Springer Berlin Heidelberg. Proceedings, 705–715.Google Scholar
Von Mammen, S., Phillips, D., Davison, T. and Jacob, C. (2010). A graph-based developmental swarm representation and algorithm. In: Dorigo, M., Birattari, M., Di Caro, G. A., Doursat, R., Engelbrecht, A. P., Floreano, D., Gambardella, L. M., Gross, R., Sahin, E., Stützle, Th. and Sayama, H. (eds.) Swarm Intelligence: 7th International Conference, ANTS 2010, Brussels, Belgium, September 8–10, 2010, Springer Berlin Heidelberg. Proceedings, 1–12.Google Scholar