Block Representation of 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 9210)


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. More precisely, we show that Reversible Causal Graph Dynamics can be represented as finite-depth circuits of local reversible gates.


Bijective Invertible Locality Cayley graphs 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., Grattage, J.: Partitioned quantum cellular automata are intrinsically universal. Nat. Comput. 11, 13–22 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Arrighi, P., Martiel, S., Nesme, V.: Generalized Cayley graphs and cellular automata over them. submitted (long version) (2013). Pre-print arXiv:1212.0027
  4. 4.
    Arrighi, P., Martiel, S., Perdrix, P.: Reversible Causal Graph Dynamics (2015). Pre-print arXiv:1502.04368
  5. 5.
    Arrighi, P., Nesme, V., Werner, R.: Unitarity plus causality implies localizability. J. Comput. Syst. Sci. 77, 372–378 (2010). QIP 2010 (long talk)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Boehm, P., Fonio, H.R., Habel, A.: Amalgamation of graph transformations: a synchronization mechanism. J. Comput. Syst. Sci. 34(2–3), 377–408 (1987)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Durand-Lose, J.O.: Representing reversible cellular automata with reversible block cellular automata. Discrete Math. Theor. Comput. Sci. 145, 154 (2001)Google Scholar
  8. 8.
    Ehrig, H., Lowe, M.: Parallel and distributed derivations in the single-pushout approach. Theor. Comput. Sci. 109(1–2), 123–143 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Hedlund, G.A.: Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory 3, 320–375 (1969)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Kari, J.: Representation of reversible cellular automata with block permutations. Theory Comput. Syst. 29(1), 47–61 (1996)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kari, J.: On the circuit depth of structurally reversible cellular automata. Fundamenta Informaticae 38(1–2), 93–107 (1999)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Konopka, T., Markopoulou, F., Smolin, L.: Quantum graphity. Arxiv preprint (2006). Pre-print arXiv:hep-th/0611197
  13. 13.
    Morita, K.: Computation-universality of one-dimensional one-way reversible cellular automata. Inf. Process. Lett. 42(6), 325–329 (1992)CrossRefzbMATHGoogle Scholar
  14. 14.
    Sorkin, R.: Time-evolution problem in Regge calculus. Phys. Rev. D. 12(2), 385–396 (1975)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Taentzer, G.: Parallel and distributed graph transformation: Formal description and application to communication-based systems. Ph.D. thesis, Technische Universitat Berlin (1996)Google Scholar
  16. 16.
    Taentzer, G.: Parallel high-level replacement systems. Theor. comput. sci. 186(1–2), 43–81 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Tomita, K., Kurokawa, H., Murata, S.: Graph automata: natural expression of self-reproduction. Phys. D: Nonlin. Phenom. 171(4), 197–210 (2002)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

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

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