Skip to main content
SpringerLink
Log in

Nonclassical Computation — A Dynamical Systems Perspective

  • Reference work entry
Handbook of Natural Computing

Abstract

In this chapter, computation is investigated from a dynamical systems perspective. A dynamical system is described in terms of its abstract state space, the system’s current state within its state space, and a rule that determines its motion through its state space. In a classical computational system, that rule is given explicitly by the computer program; in a physical system, that rule is the underlying physical law governing the behavior of the system. Therefore, a dynamical systems approach to computation allows one to take a unified view of computation in classical discrete systems and in systems performing nonclassical computation. In particular, it gives a route to a computational interpretation of physical embodied systems exploiting the natural dynamics of their material substrates.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 999.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 1,199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Abraham RH (1987) Dynamics and self-organization. In: Yates FE (ed) Self-organizing systems: the emergence of order. Plenum, New York, pp 599–613

    Google Scholar 

  • Abraham RH, Shaw CD (1987) Dynamics: a visual introduction. In: Yates FE (ed) Self-organizing systems: the emergence of order. Plenum, New York, pp 543–597

    Google Scholar 

  • Adamatzky A (ed) (2002) Collision-based computing. Springer, London

    MATH  Google Scholar 

  • Adamatzky A, Arena P, Basile A, Carmona-Galan R, Costello B, Fortuna L, Frasca M, Rodriguez-Vazquez A (2004) Reaction-diffusion navigation robot control: from chemical to VLSI analogic processors. IEEE Trans Circuits Syst 51(5):926–938

    Article  Google Scholar 

  • Adamatzky A, Costello BDL, Asai T (2005) Reaction-diffusion computers. Elsevier, Boston, MA

    Google Scholar 

  • Agladze K, Magome N, Aliev R, Yamaguchi T, Yoshikawa K (1997) Finding the optimal path with the aid of chemical wave. Physica D 106:247–254

    Article  Google Scholar 

  • Andersson C, Nordahl MG (1998) Evolving coupled map lattices for computation. In: EuroGP'98, Paris, April 1998. LNCS, vol 1391. Springer, New York, pp 151–162

    Google Scholar 

  • Baguelin M, LeFèvre J, Richard JP (2003) A formalism for models with a metadynamically varying structure. In: Proceedings of the European Control Conference, Cambridge, UK

    Google Scholar 

  • Beer RD (1995) A dynamical systems perspective on agent-environment interaction. Artif Intell 72:173–215

    Article  Google Scholar 

  • Berlekamp ER, Conway JH, Guy RK (1982) Winning ways for your mathematical plays, vol 2. Academic, New York

    MATH  Google Scholar 

  • Bossomaier T, Sibley-Punnett L, Cranny T (2000) Basins of attraction and the density classification problem for cellular automata. In: Virtual Worlds 2000, Paris, July 2000. LNAI, vol 1834. Springer, Heidelberg, pp 245–255

    Google Scholar 

  • Brooks RA (1991) Intelligence without representation. Artif Intell 47:139–159

    Article  Google Scholar 

  • Cariani P (1993) To evolve an ear: epistemological implications of Gordon Pask's electrochemical devices. Syst Res 10(3):19–33

    Article  Google Scholar 

  • Cook M (2004) Universality in elementary cellular automata. Complex Syst 15(1):1–40

    MATH  Google Scholar 

  • Cooper D, Stepney S, Woodcock J (2002) Derivation of Z refinement proof rules: forwards and backwards rules incorporating input/output refinement. Tech. Rep. YCS-2002-347, Department of Computer Science, University of York

    Google Scholar 

  • Crutchfield JP (1994) The calculi of emergence. Physica D 75:1154

    Google Scholar 

  • Crutchfield JP, Kaneko K (1987) Phenomenology of spatio-temporal chaos. In: Bin-Lin H (ed) Direction in Chaos, World Scientific, Singapore, pp 272–353

    Google Scholar 

  • Culik K II, Yu S (1988) Undecidability of CA classification schemes. Complex Syst 2(2):177–190

    MathSciNet  MATH  Google Scholar 

  • Danks G, Stepney S, Caves L (2007) Folding protein-like structures with open L-systems. In: ECAL 2007, Lisbon, Portugal, September 2007. LNAI, vol 4648. Springer, Heidelberg, pp 1100–1109

    Google Scholar 

  • Danks G, Stepney S, Caves L (2008) Protein folding with stochastic L-systems. In: ALife XI, Winchester, UK, MIT Press, Boston, MA, pp 150–157

    Google Scholar 

  • Dasmahapatra S, Werner J, Zauner KP (2006) Noise as a computational resource. Int J Unconventional Comput 2(4):305–319

    Google Scholar 

  • Ditto WL, Murali K, Sinha S (2008) Chaos computing: ideas and implementations. Phil Trans R Soc A 366:653–664

    Article  MathSciNet  MATH  Google Scholar 

  • Drossel B (2008) Random Boolean Networks. In: Schuster HG (ed) Reviews of nonlinear dynamics and complexity, vol 1, Wiley, Weinheim, arXiv:0706.3351v2 [cond-mat.stat-mech]

    Google Scholar 

  • Durand B, Formenti E, Varouchas G (2003) On undecidability of equicontinuity classification for cellular automata. In: Morvan M, Rémila E (eds) Discrete models for complex systems, DMCS'03, Lyon, France, June 2003. DMTCS, vol AB, pp 117–128

    Google Scholar 

  • Gács P, Kurdyumov GL, Levin LA (1978) One dimensional uniform arrays that wash out finite islands. Probl Peredachi Inf 12:92–98

    Google Scholar 

  • Giavitto JL, Michel O (2002) Data structure as topological spaces. In: Unconventional models of computation, Kobe, Japan, October 2002. LNCS, vol 2509. Springer, Heidelberg, pp 137–150

    Chapter  Google Scholar 

  • Giavitto JL, Michel O, Cohen J, Spicher A (2005) Computation in space and space in computations. In: UPP 2004, France, September 2004. LNCS, vol 3566. Springer, Berlin, pp 137–152

    Google Scholar 

  • Hammel SM, Yorke JA, Grebogi C (1988) Numerical orbits of chaotic processes represent true orbits. Bull Am Math Soc 19(2):465–469

    Article  MathSciNet  MATH  Google Scholar 

  • Hao BL, Zheng WM (1998) Applied symbolic dynamics and chaos. World Scientific, Singapore

    Book  MATH  Google Scholar 

  • Harding SL, Miller JF (2004) A tone discriminator in liquid crystal. In: CEC 2004, Portland, Oregon. IEEE Press, pp 1800–1807

    Google Scholar 

  • Harding SL, Miller JF (2005) Evolution in materio: A real-time robot controller in liquid crystal. In: Proc. NASA/DoD Conference on Evolvable Hardware, Washington, DC. IEEE Press, pp 229–238

    Google Scholar 

  • Harding SL, Miller JF, Rietman EA (2006) Evolution in materio: exploiting the physics of materials for computation. arXiv:cond-mat/0611462

    Google Scholar 

  • Hayes W, Jackson KR (2005) A survey of shadowing methods for numerical solutions of ordinary differential equations. Appl Numer Math 53:299–321

    Article  MathSciNet  MATH  Google Scholar 

  • Holden AV, Tucker JV, Zhang H, Poole MJ (1992) Coupled map lattices as computational systems. Chaos 2(3):367–376

    Article  MathSciNet  MATH  Google Scholar 

  • Hornby GS, Pollack JB (2001) Body-brain coevolution using L-systems as a generative encoding. In: GECCO 2001, Morgan Kaufmann, San Francisco, CA, pp 868–875

    Google Scholar 

  • Ito K, Aoki T, Higuchi T (2004) Design of an excitable digital reaction-diffusion system for shortest path search. In: ITC-CSCC 2004, Japan

    Google Scholar 

  • Kaneko K (1983) Transition from torus to chaos accompanied by frequency lockings with symmetry breaking. Prog Theor Phys 69(5):1427–1442

    Article  MATH  Google Scholar 

  • Kaneko K (1984) Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled logistic lattice. Prog Theor Phys 72(3):480–486

    Article  MATH  Google Scholar 

  • Kaneko K (1985) Spatiotemporal intermittency in coupled map lattices. Prog Theor Phys 74(5):1033–1044

    Article  MATH  Google Scholar 

  • Kaneko K (1986) Lyapunov analysis and information flow in coupled map lattices. Physica D 23:436–447

    Article  MathSciNet  Google Scholar 

  • Kanso A, Smaoui N (2009) Logistic chaotic maps for binary numbers generations. Chaos, Solitons & Fractals 40(5):2557–2568

    Google Scholar 

  • Kauffman SA (1990) Requirements for evolvability in complex systems. Physica D 42:135–152

    Article  Google Scholar 

  • Kauffman SA (1993) The origins of order. Oxford University Press, New York

    Google Scholar 

  • Kelso JAS (1995) Dynamic Patterns. MIT Press, Cambridge, MA

    Google Scholar 

  • Kitano H (2004) Biological robustness. Nat Rev Genet 5:826–837

    Article  Google Scholar 

  • Kocarev L, Jakimoski G (2001) Logistic map as a block encryption algorithm. Phys Lett A 289(4–5):199–206

    Article  MathSciNet  MATH  Google Scholar 

  • Kuhnert L, Agladze K, Krinsky V (1989) Image processing using light-sensitive chemical waves. Nature 337:244–247

    Article  Google Scholar 

  • Kunishima W, Nishiyama A, Tanaka H, Tokihiro T (2004) Differential equations for creating complex cellular automaton patterns. J Phys Soc Jpn 73:2033–2036

    Article  MathSciNet  MATH  Google Scholar 

  • Land M, Belew RK (1995) No perfect two-state cellular automata for density classification exists. Phys Rev Lett 74(25):5148–5150

    Article  Google Scholar 

  • Li TY, Yorke JA (1975) Period three implies chaos. Am Math Monthly 82(10):985–992

    Article  MathSciNet  MATH  Google Scholar 

  • Lorenz EN (1993) The essence of chaos. UCL Press, London

    Book  MATH  Google Scholar 

  • Maturana HR, Varela FJ (1980) Autopoiesis and Cognition. D. Reidel, Boston, MA

    Book  Google Scholar 

  • May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261:459–467

    Article  Google Scholar 

  • Melby P, Kaidel J, Weber N, Hübler A (2000) Adaptation to the edge of chaos in a self-adjusting logistic map. Phys Rev Lett 84(26):5991–5993

    Article  Google Scholar 

  • Melby P, Weber N, Hübler A (2002) Robustness of adaptation in controlled self-adjusting chaotic systems. Fluctuation Noise Lett 2(4):L285–L292

    Article  Google Scholar 

  • Melby P, Weber N, Hübler A (2005) Dynamics of self-adjusting systems with noise. Chaos 15:033902

    Article  Google Scholar 

  • Metropolis N, Stein M, Stein P (1973) On finite limit sets for transformations on the unit interval. J Comb Theory 15(1):25–43

    Article  MathSciNet  MATH  Google Scholar 

  • Mills JW (2008a) The architecture of an extended analog computer core. In: UCAS-4, Austin, TX, USA

    Google Scholar 

  • Mills JW (2008b) The nature of the extended analog computer. Physica D 237(9):1235–1256

    Article  MathSciNet  MATH  Google Scholar 

  • Mills JW, Parker M, Himebaugh B, Shue C, Kopecky B, Weilemann C (2006) “Empty Space” computes: The evolution of an unconventional supercomputer. In: Proceedings of the 3rd ACM computing frontiers conference, New York, pp. 115–126

    Google Scholar 

  • Mitchell M, Crutchfield JP, Das R (1996) Evolving cellular automata with genetic algorithms: a review of recent work. In: Goodman ED, Uskov VL, Punch WF (eds) Evolutionary computation and its applications: EvCA'96, Moscow

    Google Scholar 

  • Motoike IN, Adamatzky A (2004) Three-valued logic gates in reaction-diffusion excitable media. Chaos Solitons Fractals 24:107–114

    MathSciNet  Google Scholar 

  • Moulay E, Baguelin M (2005) Meta-dynamical adaptive systems and their application to a fractal algorithm and a biological model. Physica D 207:79–90

    Article  MathSciNet  MATH  Google Scholar 

  • Murray JD (1993) Mathematical biology, 2nd edn. Springer, New York

    Book  MATH  Google Scholar 

  • Měch R, Prusinkiewicz P (1996) Visual models of plants interacting with their environment. In: SIGGRAPH '96, ACM, New Orleans, LA pp 397–410

    Chapter  Google Scholar 

  • Omohundro S (1984) Modelling cellular automata with partial differential equations. Physica D 10:128–134

    Article  MathSciNet  Google Scholar 

  • Ott E, Grebogi C, Yorke JA (1990) Controlling chaos. Phys Rev Lett 64(11):1196–1199

    MathSciNet  MATH  Google Scholar 

  • Ott E, Sauer T, Yorke JA (eds) (1994) Coping with chaos. Wiley, New York

    MATH  Google Scholar 

  • Pareeka NK, Patidara V, Sud KK (2006) Image encryption using chaotic logistic map. Image Vis Comput 24(9):926–934

    Article  Google Scholar 

  • Pask G (1960) The natural history of networks. In: Yovits MC, Cameron S (eds) Self-organizing systems. Pergamon, New York

    Google Scholar 

  • Păun G (2000) Computing with membranes. J Comput Syst Sci 61(1):108–143

    Article  MATH  Google Scholar 

  • Phatak SC, Rao SS (1995) Logistic map: a possible random-number generator. Phys Rev E 51(4):3670–3678

    Article  Google Scholar 

  • Powley EJ, Stepney S (2009a) Automorphisms of transition graphs for elementary cellular automata. J Cell Autom 4(2):125–136

    MathSciNet  MATH  Google Scholar 

  • Powley EJ, Stepney S (2009b) Automorphisms of transition graphs for linear cellular automata. J Cell Autom 4(4):293–310

    Google Scholar 

  • Prusinkiewicz P, Lindenmayer A (1990) The algorithmic beauty of plants. Springer, New York

    Book  MATH  Google Scholar 

  • Prusinkiewicz P, James M, Měch R (1994) Synthetic topiary. In: SIGGRAPH '94, ACM, Orlando, FL, pp 351–358

    Chapter  Google Scholar 

  • Rendell P (2002) Turing universality of the game of life. In: Adamatzky A (ed) Collision-based computing. Springer, London, chap 18

    Google Scholar 

  • Robinson JC (2001) Infinite-dimensional dynamical systems: an introduction to dissipative parabolic PDEs and the theory of global attractors. Cambridge University Press, Cambridge, MA

    MATH  Google Scholar 

  • Roselló-Merino M, Bechmann M, Sebald A, Stepney S (2010) Classical computing in nuclear magnetic resonance. Int J Unconventional Comput 6(3–4):163–195

    Google Scholar 

  • Rothemund PWK, Papadakis N, Winfree E (2004) Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol 2(12):e424

    Article  Google Scholar 

  • Rubel LA (1993) The extended analog computer. Adv Appl Math 14:39–50

    Article  MathSciNet  MATH  Google Scholar 

  • Sielewiesiuk J, Gorecki J (2001) Logical functions of a cross-junction of excitable chemical media. J Phys Chem A 105(35):8189–8195

    Article  Google Scholar 

  • Sinha S, Ditto WL (1998) Dynamics based computation. Phys Rev Lett 81(10):2156–2159

    Article  Google Scholar 

  • Sinha S, Ditto WL (1999) Computing with distributed chaos. Phys Rev E 60(1):363–377

    Article  Google Scholar 

  • Sinha S, Ditto WL (2006) Exploiting the controlled responses of chaotic elements to design configurable hardware. Phil Trans R Soc A 364:2483–2494

    Article  Google Scholar 

  • Sloane NJA (2008) The on-line encyclopedia of integer sequences. http://www.research.att.com/~njas/sequences/ (accessed 10 November 2008)

  • Spicher A, Michel O, Giavitto JL (2004) A topological framework for the specification and the simulation of discrete dynamical systems. In: ACRI 2004, Amsterdam, October 2004. LNCS, vol 3305. Springer, Heidelberg, pp 238–247

    Google Scholar 

  • Steinbock O, Tóth A, Showalter K (1995) Navigating complex labyrinths: optimal paths from chemical waves. Science 267:868–871

    Article  Google Scholar 

  • Stepney S (2007) Embodiment. In: Flower D, Timmis J (eds) In silico immunology, Springer, New York, chap 12, pp 265–288

    Chapter  Google Scholar 

  • Stepney S (2008) The neglected pillar of material computation. Physica D 237(9):1157–1164

    Article  MathSciNet  MATH  Google Scholar 

  • Stepney S (2009) Visualising random Boolean network dynamics. In: GECCO 2009, ACM, New York

    Google Scholar 

  • Stepney S, Polack F, Turner H (2006) Engineering emergence. In: ICECCS 2006, IEEE, Stanford, CA pp 89–97

    Google Scholar 

  • Strogatz SH (1994) Nonlinear dynamics and chaos. Westview, Boulder, CO

    Google Scholar 

  • Sutner K (2005) Universality and cellular automata. In: Machines, computations, and universality 2004, Saint Petersburg, September 2004. LNCS, vol 3354. Springer, Heidelberg, pp 50–59

    Google Scholar 

  • Teman R (1997) Infinite-dimensional dynamical systems in mechanics and physics, 2nd edn. Springer, New York

    Google Scholar 

  • Tokihiro T, Takahashi D, Matsukidaira J, Satsuma J (1996) From soliton equations to integrable cellular automata through a limiting procedure. Phys Rev Lett 76(18):3247–3250

    Article  Google Scholar 

  • Tolmachiev D, Adamatzky A (1996) Chemical processor for computation of Voronoi diagram. Adv Mater Opt Electron 6(4):191–196

    Article  Google Scholar 

  • Toth A, Showalter K (1995) Logic gates in excitable media. J Chem Phys 103:2058–2066

    Article  Google Scholar 

  • Ulam SM, von Neumann J (1947) On combination of stochastic and deterministic processes. Bull Am Math Soc 53(11):1120, (abstract 403)

    Google Scholar 

  • Waddington CH (1957) The strategy of the genes. Allen and Unwin, London

    Google Scholar 

  • Weimar JR, Boon JP (1994) Class of cellular automata for reaction-diffusion systems. Phys Rev E 49(2):1749–1752

    Article  Google Scholar 

  • Weyl H (1949) Philosophy of mathematics and natural science. Princeton University Press, Princeton, NJ

    MATH  Google Scholar 

  • Winfree E (2003) DNA computing by self-assembly. The Bridge 33(4):31–38

    Google Scholar 

  • Wolfram S (1984a) Computation theory of cellular automata. Commun Math Phys 96:15–57

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Wolfram S (1985) Twenty problems in the theory of cellular automata. Phys Scr T9:170–183

    Article  MathSciNet  MATH  Google Scholar 

  • Wolfram S (1986a) Cellular automata fluids: basic theory. J Stat Phys 45:471–526

    Article  MathSciNet  MATH  Google Scholar 

  • Wolfram S (1986b) Random sequence generation by cellular automata. Adv Appl Math 7:123–169

    Article  MathSciNet  MATH  Google Scholar 

  • Wolz D, de Oliveira PPB (2008) Very effective evolutionary techniques for searching cellular automata rule spaces. J Cell Autom 3(4):289–312

    MathSciNet  MATH  Google Scholar 

  • Wuensche A (2002) Finding gliders in cellular automata. In: Adamatzky A (ed) Collision-based computing. Springer, London, chap 13

    Google Scholar 

  • Wuensche A, Lesser M (1992) The global dynamics of cellular automata. Addison-Wesley, Reading

    Google Scholar 

  • Yates FE (ed) (1987) Self-organizing systems: the emergence of order. Plenum, New York

    Google Scholar 

Download references

Acknowledgments

My thanks to Ed Powley for providing Figs. 1, 3, 4, and 18, and for drawing attention to the work on exact correspondences between CAs and PDEs in Sect. 2.3.4. My thanks also to Andy Adamatzky, Leo Caves, Ed Clark, Simon Hickinbotham, Mic Lones, Adam Nellis, Ed Powley, and Jon Timmis for comments on a previous draft.

Author information

Authors and Affiliations

  1. Department of Computer Science, University of York, York, UK

    Susan Stepney

Authors
  1. Susan Stepney
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. LIACS, Leiden University, Leiden, The Netherlands

    Grzegorz Rozenberg

  2. Computer Science Department, University of Colorado, Boulder, USA

    Grzegorz Rozenberg

  3. LIACS, Leiden University, Leiden, The Netherlands

    Thomas Bäck

  4. LIACS, Leiden University, Leiden, The Netherlands

    Joost N. Kok

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this entry

Cite this entry

Stepney, S. (2012). Nonclassical Computation — A Dynamical Systems Perspective. In: Rozenberg, G., Bäck, T., Kok, J.N. (eds) Handbook of Natural Computing. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92910-9_59

Download citation

Publish with us

Policies and ethics

Search

Navigation