Reversible Causal Graph Dynamics

  • Pablo Arrighi
  • Simon MartielEmail author
  • Simon Perdrix
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9720)


Causal Graph Dynamics extend Cellular Automata to arbitrary, bounded-degree, time-varying graphs. The whole graph evolves in discrete time steps, and this global evolution is required to have a number of physics-like symmetries: shift-invariance (it acts everywhere the same) and causality (information has a bounded speed of propagation). We study a further physics-like symmetry, namely reversibility. We extend a fundamental result on reversible cellular automata by proving that the inverse of a causal graph dynamics is a causal graph dynamics. We also address the question of the evolution of the structure of the graphs under reversible causal graph dynamics, showing that any reversible causal graph dynamics preserves the size of all but a finite number of graphs.


Bijective Invertible Cayley graphs Hedlund Reversible cellular automata 



This work has been funded by the ANR-12-BS02-007-01 TARMAC grant, the ANR-10-JCJC-0208 CausaQ grant, and the John Templeton Foundation, grant ID 15619. The authors acknowledge enlightening discussions with Bruno Martin and Emmanuel Jeandel. This work has been partially done when PA was delegated at Inria Nancy Grand Est, in the project team Carte.


  1. 1.
    Arrighi, P., Dowek, G.: Causal graph dynamics. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 54–66. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Arrighi, P., Martiel, S., Nesme, V., Cayley, G.: Graphs, cellular automata over them submitted (long version) (2013). Pre-print arXiv:1212.0027
  3. 3.
    Arrighi, P., Nesme, V., Werner, R.: Unitarity plus causality implies localizability. J. Comput. Syst. Sci. 77, 372–378 (2010). QIP 2010 (long talk)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arrighi, P., Martiel, S., Perdrix, S.: Block representation of reversible causal graph dynamics. In: Kosowski, A., Walukiewicz, I. (eds.) FCT 2015. LNCS, vol. 9210, pp. 351–363. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  5. 5.
    Boehm, P., Fonio, H.R., Habel, A.: Amalgamation of graph transformations: a synchronization mechanism. J. Comput. Syst. Sci. 34(2–3), 377–408 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Danos, V., Laneve, C.: Formal molecular biology. Theoret. Comput. Sci. 325(1), 69–110 (2004). Computational Systems BiologyMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Durand-Lose, J.O.: Representing reversible cellular automata with reversible block cellular automata. Discret. Math. Theoret. Comput. Sci. 145, 154 (2001)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ehrig, H., Lowe, M.: Parallel and distributed derivations in the single-pushout approach. Theoret. Comput. Sci. 109(1–2), 123–143 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ferrari, G.-L., Hirsch, D., Lanese, I., Montanari, U., Tuosto, E.: Synchronised hyperedge replacement as a model for service oriented computing. In: Boer, F.S., Bonsangue, M.M., Graf, S., Roever, W.-P. (eds.) FMCO 2005. LNCS, vol. 4111, pp. 22–43. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Gromov, M.: Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1(2), 109–197 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hasslacher, B., Meyer, D.A.: Modelling dynamical geometry with lattice gas automata. In: Expanded Version of a Talk Presented at the Seventh International Conference on the Discrete Simulation of Fluids Held at the University of Oxford, June 1998Google Scholar
  12. 12.
    Hedlund, G.A.: Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theor. 3, 320–375 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kari, J.: Reversibility of 2D cellular automata is undecidable. In: Cellular Automata: Theory and Experiment, vol. 45, pp. 379–385. MIT Press (1991)Google Scholar
  14. 14.
    Kari, J.: Representation of reversible cellular automata with block permutations. Theor. Comput. Syst. 29(1), 47–61 (1996)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kari, J.: On the circuit depth of structurally reversible cellular automata. Fundamenta Informaticae 38(1–2), 93–107 (1999)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Klales, A., Cianci, D., Needell, Z., Meyer, D.A., Love, P.J.: Lattice gas simulations of dynamical geometry in two dimensions. Phys. Rev. E. 82(4), 046705 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Konopka, T., Markopoulou, F., Smolin, L.: Quantum graphity. Arxiv preprint arXiv:hep-th/0611197 (2006)
  18. 18.
    Morita, K.: Reversible simulation of one-dimensional irreversible cellular automata. Theoret. Comput. Sci. 148(1), 157–163 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sorkin, R.: Time-evolution problem in Regge calculus. Phys. Rev. D. 12(2), 385–396 (1975)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Taentzer, G.: Parallel and distributed graph transformation: formal description and application to communication-based systems. Ph.D. thesis, Technische Universitat Berlin (1996)Google Scholar
  21. 21.
    Taentzer, G.: Parallel high-level replacement systems. Theoret. Comput. Sci. 186(1–2), 43–81 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tomita, K., Kurokawa, H., Murata, S.: Graph automata: natural expression of self-reproduction. Physica D: Nonlinear Phenom. 171(4), 197–210 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Aix-Marseille University, LIFMarseille Cedex 9France
  2. 2.University Nice Sophia Antipolis, I3SSophia AntipolisFrance
  3. 3.CNRS, LORIA, Inria Project Team CARTEUniversity de LorraineNancyFrance

Personalised recommendations