Abstract
This is an extended abstract about the research regarding theory of cellular automata: an overall overview of the past and current investigations along with an outlook on some promising research directions.
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References
Acerbi, L., Dennunzio, A., Formenti, E.: Surjective multidimensional cellular automata are non-wandering: a combinatorial proof. Inf. Process. Lett. 113(5–6), 156–159 (2013)
Acerbi, L., Dennunzio, A., Formenti, E.: Conservation of some dynamical properties for operations on cellular automata. Theoret. Comput. Sci. 410(38–40), 3685–3693 (2009)
Amoroso, S., Patt, Y.: Decision procedures for surjectivity and injectivity of parallel maps for tesselation structures. J. Comput. Syst. Sci. 6, 448–464 (1972)
Bandini, S., Mauri, G.: Multilayered cellular automata. Theor. Comput. Sci. 217(1), 99–113 (1999)
Bandini, S., Mauri, G., Pavesi, G., Simone, C.: Parallel simulation of reaction-diffusion phenomena in percolation processes: a model based on cellular automata. Future Gener. Comput. Syst. 17(6), 679–688 (2001)
Béaur, P., Kari, J.: Effective projections on group shifts to decide properties of group cellular automata. Int. J. Found. Comput. Sci. 35(1 &2), 77–100 (2024)
Bernardi, V., Durand, B., Formenti, E., Kari, J.: A new dimension sensitive property for cellular automata. Theor. Comput. Sci. 345, 235–247 (2005)
Blanchard, F., Kůrka, P., Maass, A.: Topological and measure-theoretic properties of one-dimensional cellular automata. Physica D 103, 86–99 (1997)
Blanchard, F., Tisseur, P.: Some properties of cellular automata with equicontinuity points. Ann. Inst. Henri Poincaré, Probabilité et Statistiques 36, 569–582 (2000)
Bouré, O., Fatès, N., Chevrier, V.: Probing robustness of cellular automata through variations of asynchronous updating. Nat. Comput. 11(4), 553–564 (2012)
Boyle, M., Kitchens, B.: Periodic points for cellular automata. Indag. Math. 10, 483–493 (1999)
Cattaneo, G., Dennunzio, A., Margara, L.: Chaotic subshifts and related languages applications to one-dimensional cellular automata. Fund. Inform. 52, 39–80 (2002)
Cattaneo, G., Finelli, M., Margara, L.: Investigating topological chaos by elementary cellular automata dynamics. Theor. Comput. Sci. 244, 219–241 (2000)
Cattaneo, G., Dennunzio, A., Margara, L.: Solution of some conjectures about topological properties of linear cellular automata. Theor. Comput. Sci. 325(2), 249–271 (2004)
Cattaneo, G., Formenti, E., Manzini, G., Margara, L.: Ergodicity, transitivity, and regularity for linear cellular automata over \(\mathbb{Z} _m\). Theor. Comput. Sci. 233(1–2), 147–164 (2000)
Chopard, B.: Cellular automata and lattice boltzmann modeling of physical systems. In: Rozenberg, G., Bäck, T., Kok, J.N. (eds.) Handbook of Natural Computing, pp. 287–331. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-540-92910-9_9
Chopard, B., Luthi, P.O.: Lattice boltzmann computations and applications to physics. Theor. Comput. Sci. 217(1), 115–130 (1999)
Chopard, B., Masselot, A.: Cellular automata and lattice boltzmann methods: a new approach to computational fluid dynamics and particle transport. Future Gener. Comput. Syst. 16(2–3), 249–257 (1999)
Codenotti, B., Margara, L.: Transitive cellular automata are sensitive. Am. Math. Mon. 103(1), 58–62 (1996)
Culik, K., Yu, S.: Undecidability of cellular automata classification schemes. Complex Syst. 2, 177–190 (1988)
Dennunzio, A., Formenti, E., Manzoni, L., Mauri, G.: m-Asynchronous cellular automata: from fairness to quasi-fairness. Natural Comput. 12, 561–572 (2013)
Dennunzio, A., Guillon, P., Masson, B.: Sand automata as cellular automata. Theor. Comput. Sci. 410, 3962–3974 (2009)
Dennunzio, A.: From one-dimensional to two-dimensional cellular automata. Fund. Inform. 115(1), 87–105 (2012)
Dennunzio, A., Di Lena, P., Formenti, E., Margara, L.: Periodic orbits and dynamical complexity in cellular automata. Fund. Inform. 126(2–3), 183–199 (2013)
Dennunzio, A., Formenti, E., Grinberg, D., Margara, L.: Chaos and ergodicity are decidable for linear cellular automata over \((\mathbb{Z} /m\mathbb{Z} )^n\). Inf. Sci. 539, 136–144 (2020)
Dennunzio, A., Formenti, E., Grinberg, D., Margara, L.: Dynamical behavior of additive cellular automata over finite abelian groups. Theor. Comput. Sci. 843, 45–56 (2020)
Dennunzio, A., Formenti, E., Grinberg, D., Margara, L.: Decidable characterizations of dynamical properties for additive cellular automata over a finite abelian group with applications to data encryption. Inf. Sci. 563, 183–195 (2021)
Dennunzio, A., Formenti, E., Grinberg, D., Margara, L.: An efficiently computable characterization of stability and instability for linear cellular automata. J. Comput. Syst. Sci. 122, 63–71 (2021)
Dennunzio, A., Formenti, E., Manzoni, L.: Computing issues of asynchronous CA. Fund. Inform. 120(2), 165–180 (2012)
Dennunzio, A., Formenti, E., Manzoni, L., Margara, L., Porreca, A.E.: On the dynamical behaviour of linear higher-order cellular automata and its decidability. Inf. Sci. 486, 73–87 (2019)
Dennunzio, A., Formenti, E., Manzoni, L., Mauri, G., Porreca, A.E.: Computational complexity of finite asynchronous cellular automata. Theor. Comput. Sci. 664, 131–143 (2017)
Dennunzio, A., Formenti, E., Margara, L.: An easy to check characterization of positive expansivity for additive cellular automata over a finite abelian group. IEEE Access 11, 121246–121255 (2023)
Dennunzio, A., Formenti, E., Margara, L.: An efficient algorithm deciding chaos for linear cellular automata over \((\mathbb{Z} /m\mathbb{Z} )^n\) with applications to data encryption. Inf. Sci. 657, 119942 (2024)
Dennunzio, A., Formenti, E., Provillard, J.: Non-uniform cellular automata: classes, dynamics, and decidability. Inf. Comput. 215, 32–46 (2012)
Dennunzio, A., Formenti, E., Provillard, J.: Local rule distributions, language complexity and non-uniform cellular automata. Theor. Comput. Sci. 504, 38–51 (2013)
Dennunzio, A., Formenti, E., Provillard, J.: Three research directions in non-uniform cellular automata. Theor. Comput. Sci. 559, 73–90 (2014)
Dennunzio, A., Formenti, E., Weiss, M.: Multidimensional cellular automata: closing property, quasi-expansivity, and (un)decidability issues. Theor. Comput. Sci. 516, 40–59 (2014)
Dennunzio, A., di Lena, P., Formenti, E., Margara, L.: On the directional dynamics of additive cellular automata. Theor. Comput. Sci. 410(47–49), 4823–4833 (2009)
Durand, B.: The surjectivity problem for 2D cellular automata. J. Comput. Syst. Sci. 49(3), 718–725 (1994)
Durand, B.: Global properties of cellular automata. In: Goles, E., Martinez, S. (eds.) Cellular Automata and Complex Systems. Kluwer (1998)
Durand, B., Formenti, E., Varouchas, G.: On undecidability of equicontinuity classification for cellular automata. Disc. Math. Theor. Comput. Sci. AB, 117–128 (2003)
Fatès, N.: Stochastic cellular automata solutions to the density classification problem - when randomness helps computing. Theory Comput. Syst. 53(2), 223–242 (2013)
Fatès, N.: A guided tour of asynchronous cellular automata. J. Cell. Autom. 9(5–6), 387–416 (2014)
Fatès, N., Thierry, E., Morvan, M., Schabanel, N.: Fully asynchronous behavior of double-quiescent elementary cellular automata. Theor. Comput. Sci. 362(1–3), 1–16 (2006)
Fuks, H.: Solving two-dimensional density classification problem with two probabilistic cellular automata. J. Cell. Autom. 10(1–2), 149–160 (2015)
Goles, E., Lobos, F., Montealegre, P., Ruivo, E.L.P., de Oliveira, P.P.B.: Computational complexity of the stability problem for elementary cellular automata. J. Cell. Autom. 15(4), 261–304 (2020)
Goles, E., Maldonado, D., Montealegre, P., Ollinger, N.: On the complexity of the stability problem of binary freezing totalistic cellular automata. Inf. Comput. 274, 104535 (2020)
Goles, E., Montalva-Medel, M., Montealegre, P., Ríos-Wilson, M.: On the complexity of generalized Q2R automaton. Adv. Appl. Math. 138, 102355 (2022)
Goles, E., Montealegre, P.: The complexity of the asynchronous prediction of the majority automata. Inf. Comput. 274, 104537 (2020)
Hedlund, G.A.: Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory 3, 320–375 (1969)
Hurd, L.P., Kari, J., Culik, K.: The topological entropy of cellular automata is uncomputable. Ergodic Theory Dyn. Syst. 12, 255–265 (1992)
Ito, M., Osato, N., Nasu, M.: Linear cellular automata over \(\mathbb{Z} _m\). J. Comput. Syst. Sci. 27, 125–140 (1983)
Kamilya, S., Kari, J.: Nilpotency and periodic points in non-uniform cellular automata. Acta Informatica 58(4), 319–333 (2021)
Kari, J.: The nilpotency problem of one dimensional cellular automata. SIAM J. Comput. 21, 571–586 (1992)
Kari, J.: Reversibility and surjectivity problems of cellular automata. J. Comput. Syst. Sci. 48, 149–182 (1994)
Kari, J.: Rice’s theorem for the limit set of cellular automata. Theor. Comput. Sci. 127(2), 229–254 (1994)
Kari, J.: Theory of cellular automata: a survey. Theor. Comput. Sci. 334(1–3), 3–33 (2005)
Kari, J.: The nilpotency problem of one-dimensional cellular automata. SIAM J. Comput. 21(3), 571–586 (1992)
Kari, J.: Linear cellular automata with multiple state variables. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 110–121. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-46541-3_9
Kůrka, P.: Languages, equicontinuity and attractors in cellular automata. Ergodic Theory Dyn. Syst. 17, 417–433 (1997)
Kůrka, P.: Topological and Symbolic Dynamics, vol. 11 of Cours Spécialisés, Société Mathématique de France (2004)
Lukkarila, V.: Sensitivity and topological mixing are undecidable for reversible one-dimensional cellular automata. J. Cell. Autom. 5(3), 241–272 (2010)
Mairesse, J., Marcovici, I.: Around probabilistic cellular automata. Theor. Comput. Sci. 559, 42–72 (2014)
Manenti, L., Manzoni, S., Bandini, S.: A stochastic cellular automata for modeling pedestrian groups distribution. J. Cell. Autom. 8(5–6), 321–332 (2013)
Manzini, G., Margara, L.: Attractors of linear cellular automata. J. Comput. Syst. Sci. 58(3), 597–610 (1999)
Manzini, G., Margara, L.: A complete and efficiently computable topological classification of d-dimensional linear cellular automata over \(\mathbb{Z} _m\). Theor. Comput. Sci. 221(1–2), 157–177 (1999)
Manzoni, L.: Asynchronous cellular automata and dynamical properties. Nat. Comput. 11(2), 269–276 (2012)
Manzoni, L., Umeo, H.: The firing squad synchronization problem on CA with multiple updating cycles. Theor. Comput. Sci. 559, 108–117 (2014)
Mariot, L.: Enumeration of maximal cycles generated by orthogonal cellular automata. Nat. Comput. 22(3), 477–491 (2023)
Mariot, L., Manzoni, L.: A classification of s-boxes generated by orthogonal cellular automata. Nat. Comput. 23(1), 5–16 (2024)
Maruoka, A., Kimura, M.: Conditions for injectivity of global maps for tessellation automata. Inf. Control 32, 158–162 (1976)
Meyerovitch, T.: Finite entropy for multidimensional cellular automata. Ergodic Theory Dyn. Syst. 28, 1243–1260 (2008)
Moore, E.F.: Machine models of self-reproduction. Proc. Symp. Appl. Math. 14, 13–33 (1962)
Myhill, J.: The converse to Moore’s garden-of-eden theorem. Proc. Am. Math. Soc. 14, 685–686 (1963)
de Oliveira, P.P.B., Formenti, E., Perrot, K., Riva, S., Ruivo, E.L.P.: Non-maximal sensitivity to synchronism in elementary cellular automata: exact asymptotic measures. Theor. Comput. Sci. 926, 21–50 (2022)
Plénet, T., Bagnoli, F., Yacoubi, S.E., Raïevsky, C., Lefèvre, L.: Synchronization of elementary cellular automata. Nat. Comput. 23(1), 31–40 (2024)
Plénet, T., Yacoubi, S.E., Raïevsky, C., Lefèvre, L.: Observability and reconstructibility of bounded cellular automata. Int. J. Syst. Sci. 53(14), 2901–2917 (2022)
Sutner, K.: De Bruijn graphs and linear cellular automata. Complex Syst. 5, 19–30 (1991)
Sutner, K.: On the computational complexity of finite cellular automata. J. Comput. Syst. Sci. 50(1), 87 (1995)
Theyssier, G., Sablik, M.: Topological dynamics of cellular automata: dimension matters. Theory Comput. Syst. 48, 693–714 (2011)
Thorimbert, Y., Lätt, J., Chopard, B.: Coupling of lattice Boltzmann shallow water model with lattice Boltzmann free-surface model. J. Comput. Sci. 33, 1–10 (2019)
Toupance, P., Chopard, B., Lefèvre, L.: System reduction: an approach based on probabilistic cellular automata. Nat. Comput. 23(1), 17–29 (2024)
Umeo, H.: How to synchronize cellular automata - recent developments -. Fund. Informaticae 171(1–4), 393–419 (2020)
Wacker, S., Worsch, T.: On completeness and decidability of phase space invertible asynchronous cellular automata. Fund. Informaticae 126(2–3), 157–181 (2013)
Wolnik, B., Dziemianczuk, M., Baets, B.D.: Non-uniform number-conserving elementary cellular automata. Inf. Sci. 626, 851–866 (2023)
Worsch, T.: Towards intrinsically universal asynchronous CA. Nat. Comput. 12(4), 539–550 (2013)
Yacoubi, S.E., Plénet, T., Dridi, S., Bagnoli, F., Lefèvre, L., Raïevsky, C.: Some control and observation issues in cellular automata. Complex Syst. 30(3), 391–413 (2021)
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Dennunzio, A. (2024). Theory of Cellular Automata: from the Past and Present to Some Path Towards the Future. In: Bagnoli, F., Baetens, J., Bandini, S., Matteuzzi, T. (eds) Cellular Automata. ACRI 2024. Lecture Notes in Computer Science, vol 14978. Springer, Cham. https://doi.org/10.1007/978-3-031-71552-5_1
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