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Solution to the predecessors and Gardens-of-Eden problems for synchronous systems over directed graphs

Author

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  • Aledo, Juan A.
  • Diaz, Luis G.
  • Martinez, Silvia
  • Valverde, Jose C.

Abstract

In this work, we give a solution to the predecessors problems for synchronous systems over directed graphs, so extending the results given for systems over undirected graphs. In this same sense, we provide a characterization to have at least one predecessor for any given state and, consequently, a characterization of the Garden-of-Eden states. Furthermore, we solve the unique predecessor problem, the coexistence of predecessors problem and the number of predecessors problem, providing the best bounds for such a number and for the number of Garden-of-Eden states.

Suggested Citation

  • Aledo, Juan A. & Diaz, Luis G. & Martinez, Silvia & Valverde, Jose C., 2019. "Solution to the predecessors and Gardens-of-Eden problems for synchronous systems over directed graphs," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 22-28.
    • Handle: RePEc:eee:apmaco:v:347:y:2019:i:c:p:22-28
    DOI: 10.1016/j.amc.2018.10.077
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    References listed on IDEAS

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    1. Barrett, Chris L & Chen, William Y.C & Zheng, Michelle J, 2004. "Discrete dynamical systems on graphs and Boolean functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 66(6), pages 487-497.
    2. Chiaselotti, G. & Gentile, T. & Oliverio, P.A., 2014. "Parallel and sequential dynamics of two discrete models of signed integer partitions," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 1249-1261.
    3. Juan A. Aledo & Silvia Martinez & Jose C. Valverde, 2015. "Parallel Dynamical Systems over Graphs and Related Topics: A Survey," Journal of Applied Mathematics, Hindawi, vol. 2015, pages 1-14, March.
    4. Toroczkai, Zoltán & Guclu, Hasan, 2007. "Proximity networks and epidemics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 378(1), pages 68-75.
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    Cited by:

    1. Juan A. Aledo & Luis G. Diaz & Silvia Martinez & Jose C. Valverde, 2019. "Predecessors Existence Problems and Gardens of Eden in Sequential Dynamical Systems," Complexity, Hindawi, vol. 2019, pages 1-10, March.
    2. Juan A. Aledo & Luis G. Diaz & Silvia Martinez & Jose C. Valverde, 2020. "Coexistence of Periods in Parallel and Sequential Boolean Graph Dynamical Systems over Directed Graphs," Mathematics, MDPI, vol. 8(10), pages 1-14, October.

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