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Cellular Automata as Models of Parallel Computation

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Computational Complexity

Article Outline

Glossary

Definition of the Subject

Introduction

Time and Space Complexity

Measuring and Controlling the Activities

Communication in CA

Future Directions

Bibliography

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Abbreviations

Cellular automaton:

The classical fine‐grained parallel model introduced by John von Neumann.

Hyperbolic cellular automaton:

A cellular automaton resulting from a tessellation of the hyperbolic plane.

Parallel Turing machine:

A generalization of Turing's classical model where several control units work cooperatively on the same tape (or set of tapes).

Time complexity:

Number of steps needed for computing a result. Usually a function \( { t\colon \mathbb{N}_+\to\mathbb{N}_+ } \), t(n) being the maximum (“worst case”) for any input of size n.

Space complexity:

Number of cells needed for computing a result. Usually a function \( { s\colon \mathbb{N}_+\to\mathbb{N}_+ } \), s(n) being the maximum for any input of size n.

State change complexity:

Number of proper state changes of cells during a computation. Usually a function \( sc\colon \mathbb{N}_+\to\mathbb{N}_+ \), sc(n) being the maximum for any input of size n.

Processor complexity:

Maximum number of control units of a parallel Turing machine which are simultaneously active during a computation. Usually a function \( { sc\colon \mathbb{N}_+\to\mathbb{N}_+ } \), sc(n) being the maximum for any input of size n.

\( { \mathbb{N}_+ } \) :

The set \( { \{1,2,3,\dots\} } \) of positive natural numbers.

ℤ:

The set \( { \{\dots, -3, -2, -1, 0, 1,2,3,\dots\} } \) of integers.

Q G :

The set of all (total) functions from a set G to a set Q.

Bibliography

  1. Achilles AC, Kutrib M, Worsch T (1996) On relations between arrays of processing elements of different dimensionality. In: Vollmar R, Erhard W, Jossifov V (eds) Proceedings Parcella ’96, no. 96 in Mathematical Research. Akademie, Berlin, pp 13–20

    Google Scholar 

  2. Blum M (1967) A machine‐independent theory of the complexity of recursive functions. J ACM 14:322–336

    Article  MATH  Google Scholar 

  3. Buchholz T, Kutrib M (1998) On time computability of functions in one-way cellular automata. Acta Inf 35(4):329–352

    Article  MathSciNet  MATH  Google Scholar 

  4. Chandra AK, Kozen DC, Stockmeyer LJ (1981) Alternation. J ACM 28(1):114–133

    Article  MathSciNet  MATH  Google Scholar 

  5. Delorme M, Mazoyer J, Tougne L (1999) Discrete parabolas and circles on 2D cellular automata. Theor Comput Sci 218(2):347–417

    Article  MathSciNet  MATH  Google Scholar 

  6. van Emde Boas P (1990) Machine models and simulations. In: van Leeuwen J (ed) Handbook of Theoretical Computer Science, vol A. Elsevier Science Publishers and MIT Press, Amsterdam, chap 1, pp 1–66

    Google Scholar 

  7. Fatès N, Thierry É, Morvan M, Schabanel N (2006) Fully asynchronous behavior of double‐quiescent elementary cellular automata. Theor Comput Sci 362:1–16

    Google Scholar 

  8. Garzon M (1991) Models of Massive Parallelism. Texts in Theoretical Computer Science. Springer, Berlin

    Google Scholar 

  9. Hemmerling A (1979) Concentration of multidimensional tape‐bounded systems of Turing automata and cellular spaces. In: Budach L (ed) International Conference on Fundamentals of Computation Theory (FCT ’79). Akademie, Berlin, pp 167–174

    Google Scholar 

  10. Hennie FC (1965) One-tape, off-line Turing machine computations. Inf Control 8(6):553–578

    Article  MathSciNet  Google Scholar 

  11. Iwamoto C, Margenstern M (2004) Time and space complexity classes of hyperbolic cellular automata. IEICE Trans Inf Syst E87-D(3):700–707

    Google Scholar 

  12. Iwamoto C, Hatsuyama T, Morita K, Imai K (2002) Constructible functions in cellular automata and their applications to hierarchy results. Theor Comput Sci 270(1–2):797–809

    Article  MathSciNet  MATH  Google Scholar 

  13. Iwamoto C, Tateishi K, Morita K, Imai K (2003) Simulations between multi‐dimensional deterministic and alternating cellular automata. Fundamenta Informaticae 58(3/4):261–271

    MathSciNet  MATH  Google Scholar 

  14. Kutrib M (2008) Efficient pushdown cellular automata: Universality, time and space hierarchies. J Cell Autom 3(2):93–114

    MathSciNet  MATH  Google Scholar 

  15. Kutrib M, Malcher A (2006) Fast cellular automata with restricted inter-cell communication: Computational capacity. In: Navarro YKG, Bertossi L (ed) Proceedings Theor Comput Sci (IFIP TCS 2006), pp 151–164

    Google Scholar 

  16. Kutrib M, Malcher A (2007) Real-time reversible iterative arrays. In: Csuhaj-Varjú E, Ésik Z (ed) Fundametals of Computation Theory 2007.LNCS vol 4639. Springer, Berlin, pp 376–387

    Chapter  Google Scholar 

  17. Mazoyer J, Terrier V (1999) Signals in one‐dimensional cellular automata.Theor Comput Sci 217(1):53–80

    Article  MathSciNet  MATH  Google Scholar 

  18. Merkle D, Worsch T (2002) Formal language recognition by stochastic cellular automata. Fundamenta Informaticae 52(1–3):181–199

    MathSciNet  Google Scholar 

  19. Mycielski J, Niwiński D (1991) Cellular automata on trees, a model for parallel computation. Fundamenta Informaticae XV:139–144

    Google Scholar 

  20. Nakamura K (1981) Synchronous to asynchronous transformation of polyautomata. J Comput Syst Sci 23:22–37

    Article  MATH  Google Scholar 

  21. von Neumann J (1966) Theory of Self‐Reproducing Automata. University of Illinois Press, Champaign. Edited and completed by Arthur W. Burks

    Google Scholar 

  22. Regnault D, Schabanel N, Thierry É (2007) Progress in the analysis of stochastic 2d cellular automata: a study of asychronous 2d minority. In: Csuhaj-Varjú E, Ésik Z (ed) Fundametals of Computation Theory 2007.LNCS vol 4639. Springer, Berlin, pp 376–387

    Google Scholar 

  23. Reischle F, Worsch T (1998) Simulations between alternating CA, alternating TM and circuit families. In: MFCS’98 satellite workshop on cellular automata, pp 105–114

    Google Scholar 

  24. Sanders P, Vollmar R, Worsch T (2002) Cellular automata: Energy consumption and physical feasibility. Fundamenta Informaticae 52(1–3):233–248

    MathSciNet  MATH  Google Scholar 

  25. Savitch WJ (1978) Parallel and nondeterministic time complexity classes. In: Proc. 5th ICALP, pp 411–424

    Google Scholar 

  26. Scheben C (2006) Simulation of \( { d^\prime } \)-dimensional cellular automata on d‑dimensional cellular automata. In: El Yacoubi S, Chopard B, Bandini S (eds) Proceedings ACRI 2006. LNCS, vol 4173. Springer, Berlin, pp 131–140

    Google Scholar 

  27. Smith AR (1971) Simple computation‐universal cellular spaces. J ACM 18(3):339–353

    Article  MATH  Google Scholar 

  28. Stoß HJ (1970) k‑Band‐Simulation von k‑Kopf‐Turing‐Maschinen. Computing 6:309–317

    Google Scholar 

  29. Stratmann M, Worsch T (2002) Leader election in d‑dimensional CA in time diam · log(diam). Future Gener Comput Syst 18(7):939–950

    Article  MATH  Google Scholar 

  30. Vollmar R (1982) Some remarks about the ‘efficiency’ of polyautomata.Int J Theor Phys 21:1007–1015

    Article  MathSciNet  MATH  Google Scholar 

  31. Wiedermann J (1983) Paralelný Turingov stroj – Model distribuovaného počítača. In: Gruska J (ed) Distribouvané a parallelné systémy. CRC, Bratislava, pp 205–214

    Google Scholar 

  32. Wiedermann J (1984) Parallel Turing machines. Tech. Rep. RUU-CS-84-11. University Utrecht, Utrecht

    Google Scholar 

  33. Worsch T (1997) On parallel Turing machines with multi-head control units. Parallel Comput 23(11):1683–1697

    Article  MathSciNet  Google Scholar 

  34. Worsch T (1999) Parallel Turing machines with one-head control units and cellular automata. Theor Comput Sci 217(1):3–30

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Lehrstuhl Informatik für Ingenieure und Naturwissenschaftler, Universität Karlsruhe , Karlsruhe, Germany

    Thomas Worsch

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  1. Thomas Worsch
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Editor information

Editors and Affiliations

  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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Worsch, T. (2012). Cellular Automata as Models of Parallel Computation. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_20

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