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Structurally Dynamic Cellular Automata

  • Reference work entry
Computational Complexity

  • 245 Accesses

Article Outline

Glossary

Definition of the Subject

Introduction

The Basic Model

Emerging Patterns and Behaviors

SDCA as Models of Computation

Generalized SDCA Models

Related Graph Dynamical Systems

SDCA as Models of Fundamental Physics

Future Directions and Speculations

Bibliography

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Abbreviations

Adjacency matrix:

The adjacency matrix of a graph with N sites is an \( { N\times N } \) matrix \( { [a_{i\!j}] } \) with entries \( { a_{i\!j}=1 } \) if i and j are linked, and \( { a_{i\!j}=0 } \) otherwise. The adjacency matrix is symmetric (\( { a_{i\!j}=a_{ji} } \)) if the links in the graph are undirected.

Coupler link rules:

Coupler rules are local rules that act on pairs of next‐nearest sites of a graph at time t to decide whether they should be linked at \( { t+1 } \). The decision rules fall into one of three basic classes – totalistic (T), outer‐totalistic (OT) or restricted‐totalistic (RT) – but can be as varied as those for conventional cellular automata.

Decoupler link rules:

Decoupler rules are local rules that act on pairs of linked sites of a graph at time t to decide whether they should be unlinked at \( { t+1 } \). As for coupler rules, the decision rules fall into one of three basic classes – totalistic (T), outer‐totalistic (OT) or restricted‐totalistic (RT) – but can be as varied as those for conventional cellular automata.

Degree:

The degree of a node (or site, i) of a graph is equal to the number of distinct nodes to which i is linked, and where the links are assumed to possess no directional information. In general graphs, the in-degree (= number of incoming links towards i) is distinguished from the out-degree (= number of outgoing links originating at i).

Effective dimension:

A quantity used to approximate the dimensionality of a graph. It is defined as the ratio between the average number of next‐nearest neighbors to the average degree, both averaged over all nodes of the graph. The effective dimension equals the Euclidean dimension d, in cases where the graph is the familiar d‑dimensional hypercubic lattice.

Graph:

A graph is a finite, nonempty set of nodes (referred to as “sites” throughout this article), together with (a possibly empty) set of edges (or links). The links may be either directed (in which case the edge from a site i, say, is directed away from i toward another site j, and is considered distinct from another directed edge originating at j and pointed toward i) or undirected (in which case if a link exists between sites i and j it carries no directional information).

Graph grammar:

Graph grammars (sometimes also referred to as graph rewriting systems) apply formal language theory to networks. Each language specifies the space of “valid structures”, and the production (or “rewrite”) rules by which given graphs may be transformed into other valid graphs.

Graph metric function:

The graph metric function defines the distance between any two nodes, i and j. It is equal to the length of the shortest path between i and j. If no path exists (such as when i and j are on two disconnected components of the same graph), the distance is assumed to be equal to ∞.

Graph‐rewriting automata:

Graph‐rewriting automata are generalized CA-like systems in which both (the number of) nodes and links are allowed to change.

Next‐nearest neighbor:

Two sites i and j are next‐nearest neighbors in a graph if (1) they are not directly linked (so that \( { a_{i\!j}=0 } \); see adjacency matrix), and (2) there exists at least one other site k such that \( { k\notin \left\{{i,j}\right\} } \), and i and j are both lined to k.

Random dynamics approximation:

The long-term behavior of structurally dynamic cellular automata may be approximated in certain cases (in which the structure and value configurations are both sufficiently random and uncorrelated) by a random dynamics approximation: values of sites are replaced by the probability p σ of a site having value σ (and is assumed to be equal for all sites), and links between sites are replaced by the probability \( { p_{\ell} } \) of being linked (and also assumed to be the same for all pairs of sites).The approximation often yields qualitatively correct predictions about how the real system evolves under a specific set of rules; for example, to predict whether one expects unbounded growth or that the lattice will eventually settle onto a low periodic state or simply decay.

Restricted totalistic rules:

Restricted totalistic rules are a generalized class of link rules (operating on pairs of sites, i and j), analogous to “outer totalistic” rules (that operate on site values) used in conventional CA. The local neighborhood around i and j is first partitioned into three sets: (1) the two sites, i and j; (2) sites connected to either i or j, but not both; and (3) sites connected to both i and j. The restricted totalistic rule is then completely defined by associating a specific action with each possible 3-tuple of site-value sums (where the individual components represent a unique sum in each of the three neighborhoods).

Structurally dynamic cellular automata:

Structurally dynamic cellular automata are generalizations of conventional cellular automata models in which the underlying lattice structure is dynamically coupled to the local site-value configurations.

SDCA model hierarchy:

The SDCA model hierarchy is a set of eight related structurally dynamic cellular automata models, defined explicitly for studying their formal computational capabilities. The hierarchy is ordered (from lowest to highest level) according to their relative computational strength. For example, the SDCA model at the top of the hierarchy is capable of simulating a conventional CA with a speedup factor of two.

Bibliography

Primary Literature

  1. Adamatzky A (1995) Identification of Cellular Automata.Taylor and Francis, London

    Google Scholar 

  2. Applied Graph & Network Analysis software. http://benta.addr.com/agna/. Accessed 14 Oct 2008

  3. Albert J, Culik II K (1987) A simple universal cellular automaton and its one-way and totalistic version.Complex Syst 1:1–16

    MathSciNet  MATH  Google Scholar 

  4. Ali SM, Zimmer RM (1995) Games of Proto-Life in Masked Cellular Automata.Complexity International, vol 2. http://www.complexity.org.au

  5. Ali SM, Zimmer RM (2000) A Formal Framework for Emergent Panpsychism. In: Tucson 2000: Consciousness Research Abstracts. http://www.consciousness.arizona.edu/tucson2000/. Accessed 14 Oct 2008

  6. 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) Lecture Notes in Computer Science, vol 4173. Springer, New York, pp 30–40

    Google Scholar 

  7. Alonso-Sanz R (2006) The Beehive Cellular Automaton with Memory.J Cell Autom 1(3):195–211

    MathSciNet  MATH  Google Scholar 

  8. Alonso-Sanz R (2007) A structurally dynamic cellular automaton with memory.Chaos, Solitons, Fractals 32(4):1285–1304

    Article  MathSciNet  MATH  Google Scholar 

  9. Barabasi AL, Albert R (2002) Statistical mechanics of complex networks.Rev Mod Phys 74:47–97

    Article  MathSciNet  Google Scholar 

  10. Bollobas B (2002) Modern Graph Theory.Springer, New York

    Google Scholar 

  11. Borgatti SP (2002) NetDraw 1.0: Network visualization software.Analytic Technologies, Harvard

    Google Scholar 

  12. Chen C (2004) Graph Drawing Algorithms.In: Information Visualization. Springer, New York

    Google Scholar 

  13. Dadic I, Pisk K (1979) Dynamics of discrete‐space structure.Int J Theor Phys 18:345–358

    Article  Google Scholar 

  14. Doi H (1984) Graph theoretical analysis of cleavage pattern: graph developmental system and its application to cleavage pattern in ascidian egg.Dev Growth Differ 26(1):49–60

    Article  MathSciNet  Google Scholar 

  15. Durrett R (2006) Random Graph Dynamics.Cambridge University Press, New York

    Book  Google Scholar 

  16. Erdos P, Renyi A (1960) On the evolution of random graphs. Publ Math Inst Hung Acad Scie 5:11–61

    MathSciNet  Google Scholar 

  17. Ferber J (1999) Multi-Agent Systems: An Introduction to Distributed Artificial Intelligence. Addison-Wesley, New York

    Google Scholar 

  18. Ferreira A (2002) On models and algorithms for dynamic communication networks: the case for evolving graphs.In: 4th Recontres Francophones sur les Aspects Algorithmiques des Télécommunications (ALGOTEL 2002), Meze, France

    Google Scholar 

  19. Gerstner W, Kistler WM (2002) Spiking Neuron Models. Single Neurons, Populations, Plasticity.Cambridge University Press, New York

    Book  MATH  Google Scholar 

  20. Grzegorz R (1997) Handbook of Graph Grammars and Computing by Graph Transformation.World Scientific, Singapore

    Google Scholar 

  21. Halpern P (1989) Sticks and stones: a guide to structurally dynamic cellular automata.Amer J Phys 57(5):405–408

    Article  Google Scholar 

  22. Halpern P, Caltagirone G (1990) Behavior of topological cellular automata.Complex Syst 4:623–651

    MathSciNet  MATH  Google Scholar 

  23. Halpern P (1996) Genetic algorithms on structurally dynamic lattices.In: Toffo T, Biafore M, Leao J (eds) PhysComp96. New England Complex Systems Institute, Cambridge, pp 135–136

    Google Scholar 

  24. Halpern P (2003) Evolutionary Algorithms on a Self-Organized, Dynamic Lattice. In: Bar-Yam Y, Minai A (eds) Unifying Themes in Complex Systems, vol 2.Proceedings of the Second International Conference on Complex Systems. Westview Press, Cambridge

    Google Scholar 

  25. Harary F, Gupta G (1997) Dynamic graph models.Math Comp Model 25(7):79–87

    Article  MathSciNet  MATH  Google Scholar 

  26. Hasslacher B, Meyer D (1998) Modeling dynamical geometry with lattice gas automata.Int J Mod Phys C 9:1597

    Article  Google Scholar 

  27. Hillman D (1995) Combinatorial Spacetimes. Ph D Dissertation, University of Pittsburg

    Google Scholar 

  28. Ilachinski A (1986) Topological life-games I.Preprint. State University of New York at Stony Brook

    Google Scholar 

  29. Ilachinski A, Halpern P (1987) Structurally dynamic cellular automata.Preprint. State University of New York at Stony Brook

    Google Scholar 

  30. Ilachinski A, Halpern P (1987) Structurally dynamic cellular automata.Complex Syst 1(3):503–527

    MathSciNet  MATH  Google Scholar 

  31. Ilachinski A (1988) Computer Explorations of Self Organization in Discrete Complex Systems.Diss Abstr Int B 49(12):5349

    Google Scholar 

  32. Ilachinski A (2001) Cellular Automata: A Discrete Universe.World Scientific, Singapore

    MATH  Google Scholar 

  33. Jourjine AN (1985) Dimensional phase transitions: coupling of matter to the cell complex.Phys Rev D 31:1443

    Article  MathSciNet  Google Scholar 

  34. Kaplunovsky V, Weinstein M (1985) Space-time: arena or illusion?Phys Rev D 31:1879–1898

    Article  MathSciNet  Google Scholar 

  35. Kniemeyer O, Buck-Sorlin GH, Kurth W (2004) A Graph Grammar Approach to Artificial Life.Artif Life 10(4):413–431

    Article  Google Scholar 

  36. Krivovichev SV (2004) Crystal structures and cellular automata.Acta Crystallogr A 60(3):257–262

    Article  MathSciNet  Google Scholar 

  37. Lehmann KA, Kaufmann M (2005) Evolutionary algorithms for the self-organized evolution of networks. In: Proceedings of the 2005 Conference on Genetic and Evolutionary Computation, Washington DC. ACM Press, New York

    Google Scholar 

  38. Love P, Bruce M, Meyer D (2004) Lattice gas simulations of dynamical geometry in one dimension.Phil Trans Royal Soc A: Math Phys Eng Sci 362(1821):1667–1675

    MathSciNet  MATH  Google Scholar 

  39. Majercik S (1994) Structurally dynamic cellular automata.Master's Thesis, Department of Computer Science, University of Southern Maine

    Google Scholar 

  40. Makowiec D (2004) Cellular Automata with Majority Rule on Evolving Network. In: Lecture Notes in Computer Science, vol 3305. Springer, Berlin, pp 141–150

    Google Scholar 

  41. Mendes RV (2004) Tools for network dynamics.Int J Bifurc Chaos 15(4):1185–1213

    Google Scholar 

  42. Meschini D, Lehto M, Piilonen J (2005) Geometry, pregeometry and beyond.Stud Hist Phil Mod Phys 36:435–464

    Article  MathSciNet  MATH  Google Scholar 

  43. Miramontes O, Solé R, Goodwin B (1993) Collective behavior of random-activated mobile cellular automata.Physica D 63:145–160

    Google Scholar 

  44. Misner CW, Thorne KS, Wheeler JA (1973) Gravitation.W.H. Freeman, New York

    Google Scholar 

  45. Mitchell M (1998) An Introduction to Genetic Algorithms.MIT Press, Boston

    MATH  Google Scholar 

  46. Moore EF (1962) Sequential Machines: Selected Papers.Addison-Wesley, New York

    Google Scholar 

  47. Muhlenbein H (1991) Parallel genetic algorithm, population dynamics and combinatorial optimization.In: Schaffer H (ed) Third International Conference on Genetic Algorithms.Morgan Kauffman, San Francisco

    Google Scholar 

  48. Murata S, Tomita K, Kurokawa H (2002) System generation by graph automata.In: Ueda K (ed) Proceedings of the 4th International Workshop on Emergent Synthesis (IWES '02), Kobe University, Japan, pp 47–52

    Google Scholar 

  49. Mustafa S (1999) The Concept of Poiesis and Its Application in a Heideggerian Critique of Computationally Emergent Artificiality.Ph D Thesis, Brunel University, London

    Google Scholar 

  50. Newman M, Barabasi A, Watts DJ (2006) The Structure and Dynamics of Networks.Princeton University Press, New Jersey

    MATH  Google Scholar 

  51. Nochella J (2006) Cellular automata on networks. Talk given at the Wolfram Science Conference (NKS2006), Washington DC, USA, 16–18 June

    Google Scholar 

  52. Nooy W, Mrvar A, Batagelj V (2005) Exploratory Social Network Analysis with Pajek.Cambridge University Press, New York

    Book  Google Scholar 

  53. Nowotny T, Requardt M (1998) Dimension Theory of Graphs and Networks.J Phys A 31:2447–2463

    Article  MathSciNet  MATH  Google Scholar 

  54. Nowotny T, Requardt M (1999) Pregeometric Concepts on Graphs and Cellular Networks as Possible Models of Space-Time at the Planck-Scale. Chaos, Solitons, Fractals 10:469–486

    Article  MathSciNet  MATH  Google Scholar 

  55. Nowotny T, Requardt M (2006) Emergent Properties in Structurally Dynamic Disordered Cellular Networks. arXiv:cond-mat/0611427. Accessed 14 Oct 2008

  56. O'Sullivan D (2001) Graph-cellular automata: a generalized discrete urban and regional model.Environ Plan B: Plan Des 28(5):687–705

    Article  Google Scholar 

  57. Prusinkiewicz P, Lindenmayer (1990) The Algorithmic Beauty of Plants. Springer, New York

    Book  MATH  Google Scholar 

  58. Requardt M (1998) Cellular Networks as Models for Planck-Scale Physics.J Phys A 31:7997–8021

    Article  MathSciNet  MATH  Google Scholar 

  59. Requardt M (2003) A geometric renormalisation group in discrete quantum space-time.J Math Phys 44:5588–5615

    Article  MathSciNet  MATH  Google Scholar 

  60. Requardt M (2003) Scale free small world networks and the structure of quantum space-time. arXiv.org:gr-qc/0308089

  61. Rose H (1993) Topologische Zellulaere Automaten.Master's Thesis, Humboldt University of Berlin

    Google Scholar 

  62. Rose H, Hempel H, Schimansky-Geier L (1994) Stochastic dynamics of catalytic CO oxidation on Pt(100).Physica A 206:421–440

    Google Scholar 

  63. Saidani S (2003) Topodynamique de Graphe. Les Journées Graphes, Réseaux et Modélisation. ESPCI, Paris

    Google Scholar 

  64. Saidani S (2004) Self-reconfigurable robots topodynamic.In: IEEE International Conference on Robotics and Automation, vol 3. IEEE Press, New York, pp 2883–2887

    Google Scholar 

  65. Saidani S, Piel M (2004) DynaGraph: a Smalltalk Environment for Self-Reconfigurable Robots Simulation.European Smalltalk User Group Conference. http://www.esug.org/

  66. Schliecker G (1998) Binary random cellular structures.Phys Rev E 57:R1219–R1222

    Article  Google Scholar 

  67. Tomita K, Kurokawa H, Murata S (2002) Graph Automata: Natural Expression of Self-Reproduction.Physica D 171(4):197–210

    Article  MathSciNet  MATH  Google Scholar 

  68. Tomita K, Kurokawa H, Murata S (2005) Self-description for construction and execution in graph rewriting automata.In: Lecture Notes in Computer Science, vol 3630. Springer, Berlin, pp 705–715

    Google Scholar 

  69. Tomita K, Kurokawa H, Murata S (2006) Two-state graph-rewriting automata. NKS 2006 Conference, Washington, DC

    Google Scholar 

  70. Tomita K, Kurokawa H, Murata S (2006) Automatic generation of self-replicating patterns in graph automata.Int J Bifurc Chaos 16(4):1011–1018

    Article  MathSciNet  MATH  Google Scholar 

  71. Tomita K, Kurokawa H, Murata S (2006) Self-Description for Construction and Computation on Graph-Rewriting Automata.Artif Life 13(4):383–396

    Article  MathSciNet  Google Scholar 

  72. Weinert K, Mehnen J, Rudolph G (2002) Dynamic Neighborhood Structures in Parallel Evolution Strategies.Complex Syst 13(3):227–244

    MathSciNet  Google Scholar 

  73. Wheeler JA (1982) The computer and the universe.Int J Theor Phys 21:557

    Article  Google Scholar 

  74. Wolfram S (1984) Universality and complexity in cellular automata.Physica D 10:1–35

    Article  MathSciNet  Google Scholar 

  75. Wolfram S (2002) A New Kind of Science.Wolfram Media, Champaign, pp 508–545

    MATH  Google Scholar 

  76. Zuse K (1982) The Computing Universe. Int J Theor Phys 21:589–600

    Article  Google Scholar 

Books and Reviews

  1. Battista G, Eades P, Tamassia R, Tollis IG (1999) Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, New Jersey

    MATH  Google Scholar 

  2. Bornholdt S, Schuster HG (eds) (2003) Handbook of Graphs and Networks. Wiley-VCH

    Google Scholar 

  3. Breiger R, Carley K, Pattison P (2003) Dynamical Social Network Modeling and Analysis.The National Academy Press, Washington DC

    Google Scholar 

  4. Dogogovtsev SN, Mendes JF (2003) Evolution of Networks.Oxford University Press, New York

    Book  Google Scholar 

  5. Durrett R (2006) Random Graph Dynamics.Cambridge University Press, New York

    Book  Google Scholar 

  6. Gross JL, Yellen J (eds) (2004) Handbook of Graph Theory.CRC Press, Boca Raton

    MATH  Google Scholar 

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Authors and Affiliations

  1. Center for Naval Analyses, Alexandria, USA

    Andrew Ilachinski

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  1. Andrew Ilachinski
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Ilachinski, A. (2012). Structurally Dynamic Cellular Automata. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_194

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