Definition
Cellular Automata (CA) are discrete, spatially explicit extended dynamic systems. A CA system is composed of adjacent cells characterized by an internal state whose value belongs to a finite set. The updating of these states is made simultaneously according to a common local transition rule involving only a neighborhood of each cell. Thus, if \( { \sigma ^{(T)}_{i} } \) is taken to denote the value of cell i at time step T, the site values evolve by iteration of the mapping: \( { \sigma^{(T+1)}_{i}= \phi {\big (\{\sigma} ^{(T)}_{j}\}\in \mathcal{N}_{i}{\big)} } \), where ϕ is an arbitrary function which specifies the cellular automaton rule operating on \( { \mathcal{N}_{i} } \), i. e. the set of cells in the neighborhood of the cell i. Last but not least, standard CA are ahistoric (memoryless): i. e., the new state of a cell depends on the neighborhood configuration only at the preceding time step.
The standard framework of CA can be extended by the consideration of all...
Abbreviations
- Cellular automata:
-
Cellular Automata (CA) are discrete, spatially explicit extended dynamic systems composed of adjacent cells characterized by an internal state whose value belongs to a finite set. The updating of these states is made simultaneously according to a common local transition rule involving only a neighborhood of each cell.
- Memory:
-
Standard CA are ahistoric (memoryless): i. e., the new state of a cell depends on the neighborhood configuration only at the preceding time step. The standard framework of CA can be extended by the consideration of all past states (history) in the application of the CA rules by implementing memory capabilities in cells and links when topology is dynamic.
Bibliography
PrimaryLiterature
Adamatzky A (1994) Identification of Cellular Automata. Taylor and Francis
Adamatzky A (2001) Computing in Nonlinear Media and Automata Collectives. IoP Publishing, London
Adamatzky A, Holland O (1998) Phenomenology of excitation in 2-D cellular automata and swarm systems. Chaos Solit Fract 9:1233–1265
Aicardi F, Invernizzi S (1982) Memory effects in Discrete Dynamical Systems. Int J Bifurc Chaos 2(4):815–830
Alonso-Sanz R (1999) The Historic Prisoner’s Dilemma. Int J Bifurc Chaos 9(6):1197–1210
Alonso-Sanz R (2003) Reversible Cellular Automata with Memory. Phys D 175:1–30
Alonso-Sanz R (2004) One‐dimensional, \( { r=2 } \) cellular automata with memory. Int J Bifurc Chaos 14:3217–3248
Alonso-Sanz R (2004) One‐dimensional, \( { r=2 } \) cellular automata with memory. Int J BifurcChaos 14:3217–3248
Alonso-Sanz R (2005) Phase transitions in an elementary probabilistic cellular automaton with memory. Phys A 347:383–401 Alonso-Sanz R, Martin M (2004) Elementary Probabilistic Cellular Automata with Memory in Cells. Sloot PMA et al (eds) LNCS, vol 3305. Springer, Berlin, pp 11–20
Alonso-Sanz R (2005) The Paulov versus Anti‐Paulov contest with memory. Int J Bifurc Chaos 15(10):3395–3407
Alonso-Sanz R (2006) A Structurally Dynamic Cellular Automaton with Memory in the Triangular Tessellation. Complex Syst 17(1):1–15. Alonso-Sanz R, Martin, M (2006) A Structurally Dynamic Cellular Automaton with Memory in the Hexagonal Tessellation. In: El Yacoubi S, Chopard B, Bandini S (eds) LNCS, vol 4774. Springer, Berlin, pp 30-40
Alonso-Sanz R (2007) Reversible Structurally Dynamic Cellular Automata with Memory: a simple example. J Cell Automata 2:197–201
Alonso-Sanz R (2006) The Beehive Cellular Automaton with Memory. J Cell Autom 1:195–211
Alonso-Sanz R (2007) A Structurally Dynamic Cellular Automaton with Memory. Chaos Solit Fract 32:1285–1295
Alonso-Sanz R, Adamatzky A (2008) On memory and structurally dynamism in excitable cellular automata with defensive inhibition. Int J Bifurc Chaos 18(2):527–539
Alonso-Sanz R, Cardenas JP (2007) On the effect of memory in Boolean networks with disordered dynamics: the \( { K=4 } \) case. Int J Modrn Phys C 18:1313–1327
Alonso-Sanz R, Martin M (2002) One‐dimensional cellular automata with memory: patterns starting with a single site seed. Int J Bifurc Chaos 12:205–226
Alonso-Sanz R, Martin M (2002) Two‐dimensional cellular automata with memory: patterns starting with a single site seed. Int J Mod Phys C 13:49–65
Alonso-Sanz R, Martin M (2003) Cellular automata with accumulative memory: legal rules starting from a single site seed. Int J Mod Phys C 14:695–719
Alonso-Sanz R, Martin M (2004) Elementary cellular automata with memory. Complex Syst 14:99–126
Alonso-Sanz R, Martin M (2004) Three-state one‐dimensional cellular automata with memory. Chaos, Solitons Fractals 21:809–834
Alonso-Sanz R, Martin M (2005) One‐dimensional Cellular Automata with Memory in Cells of the Most Frequent Recent Value. Complex Syst 15:203–236
Alonso-Sanz R, Martin M (2006) Elementary Cellular Automata with Elementary Memory Rules in Cells: the case of linear rules. J Cell Autom 1:70–86
Alonso-Sanz R, Martin M (2006) Memory Boosts Cooperation. Int J Mod Phys C 17(6):841–852
Alonso-Sanz R, Martin MC, Martin M (2000) Discounting in the Historic Prisoner’s Dilemma. Int J Bifurc Chaos 10(1):87–102
Alonso-Sanz R, Martin MC, Martin M (2001) Historic Life Int J Bifurc Chaos 11(6):1665–1682
Alonso-Sanz R, Martin MC, Martin M (2001) The Effect of Memory in the Spatial Continuous‐valued Prisoner’s Dilemma. Int J Bifurc Chaos 11(8):2061–2083
Alonso-Sanz R, Martin MC, Martin M (2001) The Historic Strategist. Int J Bifurc Chaos 11(4):943–966
Alonso-Sanz R, Martin MC, Martin M (2001) The Historic‐Stochastic Strategist. Int J Bifurc Chaos 11(7):2037–2050
Alvarez G, Hernandez A, Hernandez L, Martin A (2005) A secure scheme to share secret color images. Comput Phys Commun 173:9–16
Fredkin E (1990) Digital mechanics. An informal process based on reversible universal cellular automata. Physica D 45:254–270
Grössing G, Zeilinger A (1988) Structures in Quantum Cellular Automata. Physica B 15:366
Hauert C, Schuster HG (1997) Effects of increasing the number of players and memory steps in the iterated Prisoner’s Dilemma, a numerical approach. Proc R Soc Lond B 264:513–519
Hooft G (1988) Equivalence Relations Between Deterministic and Quantum Mechanical Systems. J Statistical Phys 53(1/2):323–344
Ilachinski A (2000) Cellular Automata. World Scientific, Singapore
Ilachinsky A, Halpern P (1987) Structurally dynamic cellular automata. Complex Syst 1:503–527
Kaneko K (1986) Phenomenology and Characterization of coupled map lattices, in Dynamical Systems and Sigular Phenomena. World Scientific, Singapore
Kauffman SA (1993) The origins of order: Self‐Organization and Selection in Evolution. Oxford University Press, Oxford
Lindgren K, Nordahl MG (1994) Evolutionary dynamics of spatial games. Physica D 75:292–309
Love PJ, Boghosian BM, Meyer DA (2004) Lattice gas simulations of dynamical geometry in one dimension. Phil Trans R Soc Lond A 362:1667
Margolus N (1984) Physics‐like Models of Computation. Physica D 10:81–95
Martin del Rey A, Pereira Mateus J, Rodriguez Sanchez G (2005) A secret sharing scheme based on cellular automata. Appl Math Comput 170(2):1356–1364
Nowak MA, May RM (1992) Evolutionary games and spatial chaos. Nature 359:826
Nowak MA, Sigmund K (1993) A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner’s Dilemma game. Nature 364:56–58
Requardt M (1998) Cellular networks as models for Plank-scale physics. J Phys A 31:7797; (2006) The continuum limit to discrete geometries, arxiv.org/abs/math-ps/0507017
Requardt M (2006) Emergent Properties in Structurally Dynamic Disordered Cellular Networks. J Cell Aut 2:273
Ros H, Hempel H, Schimansky‐Geier L (1994) Stochastic dynamics of catalytic CO oxidation on Pt(100). Pysica A 206:421–440
Sanchez JR, Alonso-Sanz R (2004) Multifractal Properties of R90 Cellular Automaton with Memory. Int J Mod Phys C 15:1461
Stauffer D, Aharony A (1994) Introduction to percolation Theory. CRC Press, London
Svozil K (1986) Are quantum fields cellular automata? Phys Lett A 119(41):153–156
Toffoli T, Margolus M (1987) Cellular Automata Machines. MIT Press, Massachusetts
Toffoli T, Margolus N (1990) Invertible cellular automata: a review. Physica D 45:229–253
Vichniac G (1984) Simulating physics with Cellular Automata. Physica D 10:96–115
Watts DJ, Strogatz SH (1998) Collective dynamics of Small-World networks. Nature 393:440–442
Wolf-Gladrow DA (2000) Lattice‐Gas Cellular Automata and Lattice Boltzmann Models. Springer, Berlin
Wolfram S (1984) Universality and Complexity in Cellular Automata. Physica D 10:1–35
Wuensche A (2005) Glider dynamics in 3-value hexagonal cellular automata: the beehive rule. Int J Unconv Comput 1:375–398
Wuensche A, Lesser M (1992) The Global Dynamics of Cellular Automata. Addison‐Wesley, Massachusetts
Books and Reviews
Alonso-Sanz R (2008) Cellular Automata with Memory. Old City Publising, Philadelphia (in press)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag
About this entry
Cite this entry
Alonso-Sanz, R. (2009). Cellular Automata with Memory. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_55
Download citation
DOI: https://doi.org/10.1007/978-0-387-30440-3_55
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-75888-6
Online ISBN: 978-0-387-30440-3
eBook Packages: Physics and AstronomyReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics