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On Radius 1 Nontrivial Reversible and Number-Conserving Cellular Automata

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Reversibility and Universality

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 30))

Abstract

Reversibiliity and number-conservation are widely studied physics-like constraints for cellular automata (CA). Although both seem to be ‘natural’ constraints for a CA, it was conjectured that one-dimensional reversible and number-conserving CA (RNCCA) only has a limited computing ability. Particularly in the case of radius 1/2 (2-neighbor), it was shown that the class of RNCCA is equal to a trivial class of CA, so called shift-identity product cellular automata (SIPCA). But recently it was also shown that a RNCCA of neighborhood size four is computation-universal. In this paper, we list radius 1 (3-neighbor) RNCCAs up to 4-state by exhaustive search. In contrast to the radius 1/2 case, there are three new types of nontrivial RNCCA rules in the case of 4-state. We also show that it is possible to compose new nontrivial RNCCAs by modifying a SIPCA even when the state number is larger than four.

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References

  1. Amoroso, S., Patt, Y.N.: Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures. J. Comput. Syst. Sci. 6(5), 448–464 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boccara, N., Fukś, H.: Number-conserving cellular automaton rules. Fundamenta Informaticae 52, 1–13 (2003)

    MathSciNet  MATH  Google Scholar 

  3. Durand, B., Formenti, E., Róka, Z.: Number conserving cellular automata I: decidability. Theor. Comput. Sci. 299(1–3), 523–535 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fukś, H., Sullivan, K.: Enumeration of number-conserving cellular automata rules with two inputs. J. Cell. Autom. 2(2), 141–148 (2007)

    MathSciNet  MATH  Google Scholar 

  5. García-Ramos, F.: Product decomposition for surjective 2-block NCCA. In: Proceedings of 17th International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2011), pp. 221–232 (2011)

    Google Scholar 

  6. Kari, J., Taati, S.: A particle displacement representation for conservation laws in two-dimensional cellular automata. In: Proceedings of Journ\(\acute{e}\)es Automates Cellulaires (JAC 2008) (Uzès), pp. 65–73 (2008)

    Google Scholar 

  7. Morita, K.: Universality of 1-D reversible number-conserving cellular automata. Resume for International meeting on cellular automata, pp. 1–8. Osaka (2012)

    Google Scholar 

  8. Schranko A., Oliveira P.: Derivation of one-dimensional, reversible, number-conserving cellular automata rules. In: Proceedings of 15th International Workshop On Cellular Automata and Discrete Complex Systems (Automata 2009). pp. 335–345 (2009)

    Google Scholar 

  9. Wolfram S.: A New Kind of Science, Wolfram Media (2002)

    Google Scholar 

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Acknowledgements

Katsunobu Imai thanks Artiom Alhazov from the Academy of Science of Moldova for the helpful discussions and gratefully acknowledges the support of the Japan Society for the Promotion of Science and the Grant-in-Aid for Scientific Research (C) 22500015. Bruno Martin was partially supported by the French ANR, project EMC (ANR-09-BLAN-0164).

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Authors and Affiliations

  1. Graduate School of Engineering, Hiroshima University, Higashihiroshima, Japan

    Katsunobu Imai & Ryohei Saito

  2. I3S-CNRS, Université Côte d’Azur, Nice, France

    Bruno Martin

Authors
  1. Katsunobu Imai
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  2. Bruno Martin
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  3. Ryohei Saito
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Corresponding author

Correspondence to Katsunobu Imai .

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Editors and Affiliations

  1. Unconventional Computing Centre, University of the West of England, Bristol, United Kingdom

    Andrew Adamatzky

Appendix: The List of RNCCAs

Appendix: The List of RNCCAs

Each rule is described in Wolfram numbering and its suffix is the radix of the number. The sets followed by \(\swarrow ,\downarrow \) and \(\searrow \) denote left propagating, stable and right propagating state sets, respectively.

1. Radius 1 / 2

\( \begin{array}{ll} n=4: \\ 2\text {-}4\text {-}1: 3210321032103210_4 \swarrow \{1,2,3\} \\ 2\text {-}4\text {-}2: 3232323210101010_4 \swarrow \{1\}\downarrow (2\} \\ 2\text {-}4\text {-}3: 3311220033112200_4 \swarrow \{2)\downarrow \{1\} \\ 2\text {-}4\text {-}4: 3333222211110000_4 \downarrow \{1,2,3\}\\ \end{array} \)

\( \begin{array}{ll} n=6:\\ 2\text {-}6\text {-}1: 543210543210543210543210543210543210_6 \swarrow \{1,2,3,4,5\}\\ 2\text {-}6\text {-}2: 543543543543543543210210210210210210_6 \swarrow \{1,2\}\downarrow \{3\}\\ 2\text {-}6\text {-}3: 545454545454323232323232101010101010_6 \swarrow \{1\}\downarrow \{2,4\}\\ 2\text {-}6\text {-}4: 553311442200553311442200553311442200_6 \swarrow \{2,4\}\downarrow \{1\}\\ 2\text {-}6\text {-}5: 555222444111333000555222444111333000_6 \swarrow \{3\}\downarrow \{1,2\}\\ 2\text {-}6\text {-}6: 555555444444333333222222111111000000_6 \downarrow \{1,2,3,4,5\}\\ \end{array}\)

\( \begin{array}{ll} n=8: \\ 2\text {-}8\text {-}1: &{} \mathtt{7654321076543210765432107654321076543210765432107654321076543210_8} \swarrow \{1,2,3,4,5,6,7\}\\ 2\text {-}8\text {-}2: &{} \mathtt{7654765476547654765476547654765432103210321032103210321032103210_8} \swarrow \{1,2,3\}\downarrow \{4\}\\ 2\text {-}8\text {-}3: &{} \mathtt{7676323276763232545410105454101076763232767632325454101054541010_8} \swarrow \{1,4,5\}\downarrow \{2\}\\ 2\text {-}8\text {-}4: &{} \mathtt{7676767676767676545454545454545432323232323232321010101010101010_8} \swarrow \{1\}\downarrow \{2,4,6\}\\ 2\text {-}8\text {-}5: &{} \mathtt{7755331166442200775533116644220077553311664422007755331166442200_8} \swarrow \{2,4,6\}\downarrow \{1\}\\ 2\text {-}8\text {-}6: &{} \mathtt{7755775566446644775577556644664433113311220022003311331122002200_8} \swarrow \{2\}\downarrow \{1,4,5\}\\ 2\text {-}8\text {-}7: &{} \mathtt{7777333366662222555511114444000077773333666622225555111144440000_8} \swarrow \{4\}\downarrow \{1,2,3\}\\ 2\text {-}8\text {-}8: &{} \mathtt{7777777766666666555555554444444433333333222222221111111100000000_8} \downarrow \{1,2,3,4,5,6,7\}\\ \end{array}\)

\(\begin{array}{ll} n=9: \\ 2\text {-}9\text {-}1: &{} \mathtt{876543210876543210876543210876543210876543210876543210876543210876543210876543210_9} \\ &{} \swarrow \{1,2,3,4,5,6,7,8\}\\ 2\text {-}9\text {-}2: &{} \mathtt{876876876876876876876876876543543543543543543543543543210210210210210210210210210_9} \\ &{} \swarrow \{1,2\}\downarrow \{3,6\}\\ 2\text {-}9\text {-}3: &{} \mathtt{888555222777444111666333000888555222777444111666333000888555222777444111666333000_9} \\ &{} \swarrow \{3,6\}\downarrow \{1,2\}\\ 2\text {-}9\text {-}4: &{} \mathtt{888888888777777777666666666555555555444444444333333333222222222111111111000000000_9} \\ &{} \downarrow \{1,2,3,4,5,6,7,8\}\\ \end{array}\)

\(\begin{array}{ll} n=10: \\ 2\text {-}10\text {-}1: &{} \mathtt{9876543210987654321098765432109876543210987654321098765432109876543210987654321098765}\\ &{} 432109876543210_{10}, \swarrow \{1,2,3,4,5,6,7,8,9\}\\ 2\text {-}10\text {-}2: &{} \mathtt{9876598765987659876598765987659876598765987659876543210432104321043210432104321043210} \\ &{} 432104321043210_{10}, \swarrow \{1,2,3,4\}\downarrow \{5\}\\ 2\text {-}10\text {-}3: &{} \mathtt{9898989898989898989876767676767676767676545454545454545454543232323232323232323210101} \\ &{} 010101010101010_{10}, \swarrow \{1\}\downarrow \{2,4,6,8\}\\ 2\text {-}10\text {-}4: &{} \mathtt{9977553311886644220099775533118866442200997755331188664422009977553311886644220099775} \\ &{} 533118866442200_{10}, \swarrow \{2,4,6,8\}\downarrow \{1\}\\ 2\text {-}10\text {-}5: &{} \mathtt{9999944444888883333377777222226666611111555550000099999444448888833333777772222266666} \\ &{} 111115555500000_{10}, \swarrow \{5\}\downarrow \{1,2,3,4\}\\ 2\text {-}10\text {-}6: &{} \mathtt{9999999999888888888877777777776666666666555555555544444444443333333333222222222211111} \\ &{} 111110000000000_{10}, \downarrow \{1,2,3,4,5,6,7,8,9\}\\ \end{array}\)

2. Radius 1

\( \begin{array}{llll} n=2: &{}\qquad n=3:\\ 3\text {-}2\text {-}1: \mathtt{10101010_2} \swarrow \{1\} &{}\qquad 3\text {-}3\text {-}1: &{} \mathtt{210210210210210210210210210_3} \swarrow \{1,2\}\\ 3\text {-}2\text {-}2: \mathtt{11001100_2} \downarrow \{1\} &{} \qquad 3\text {-}3\text {-}2: &{} \mathtt{222111000222111000222111000_3} \downarrow \{1,2\}\\ 3\text {-}2\text {-}3: \mathtt{11110000_2} \searrow \{1\} &{} \qquad 3\text {-}3\text {-}3: &{} \mathtt{222222222111111111000000000_3} \searrow \{1,2\}\\ \end{array}\)

\( \begin{array}{ll} n=4: \\ 3\text {-}4\text {-}1: &{} \mathtt{3210321032103210321032103210321032103210321032103210321032103210_4} \swarrow \{1,2,3\}\\ 3\text {-}4\text {-}2: &{} \mathtt{3230323212103232323032321210101032301010121010103230323212101010_4} \swarrow \{1\}\downarrow \{2\}\\ 3\text {-}4\text {-}3: &{} \mathtt{3232321210103010323232123232301032323212101030101010321210103010_4} \swarrow \{1,3\}\downarrow \{2\}\\ 3\text {-}4\text {-}4: &{} \mathtt{3232323210101010323232321010101032323232101010103232323210101010_4} \swarrow \{1\}\downarrow \{2\}\\ 3\text {-}4\text {-}5: &{} \mathtt{3232323232323232323232323232323210101010101010101010101010101010_4} \swarrow \{1\}\searrow \{2\}\\ \end{array}\)

\( \begin{array}{ll} 3\text {-}4\text {-}6: &{} \mathtt{3310221033113311331022102200220033102210331122003310221033112200_4} \swarrow \{2\}\downarrow \{1\}\\ 3\text {-}4\text {-}7: &{} \mathtt{3311220032113200331122003211320033113311321132002200220032113200_4} \swarrow \{2,3\}\downarrow \{1\}\\ 3\text {-}4\text {-}8: &{} \mathtt{3311220033112200331122003311220033112200331122003311220033112200_4} \swarrow \{2\}\downarrow \{1\}\\ 3\text {-}4\text {-}9: &{} \mathtt{3311331133113311220022002200220033113311331133112200220022002200_4} \swarrow \{2\}\searrow \{1\}\\ 3\text {-}4\text {-}10: &{} \mathtt{3331222213112222333122221311000033310000131100003331222213110000_4} \downarrow \{1,2\}\\ 3\text {-}4\text {-}11: &{} \mathtt{3331333313113333222022220200000033311111131111112220222202000000_4} \downarrow \{2\}\searrow \{1,3\}\\ 3\text {-}4\text {-}12: &{} \mathtt{3332223211111111333222320000000033322232111100003332223211110000_4} \downarrow \{1,2\}\\ 3\text {-}4\text {-}13: &{} \mathtt{3332223233333333333222322222222211100010111100001110001011110000_4} \downarrow \{1\}\searrow \{2,3\}\\ 3\text {-}4\text {-}14: &{} \mathtt{3333220211112000333322023333200033332202111120001111220211112000_4} \downarrow \{1,2\}\\ 3\text {-}4\text {-}15: &{} \mathtt{3333222210111000333322221011100033333333101110002222222210111000_4} \downarrow \{1,2\}\\ 3\text {-}4\text {-}16: &{} \mathtt{3333222211110000333322221111000033332222111100003333222211110000_4} \downarrow \{1,2,3\}\\ 3\text {-}4\text {-}17: &{} \mathtt{3333222232333222333322223233322211111111101110000000000010111000_4} \downarrow \{1\}\searrow \{2\}\\ 3\text {-}4\text {-}18: &{} \mathtt{3333222233332222333322223333222211110000111100001111000011110000_4} \downarrow \{1\}\searrow \{2\}\\ 3\text {-}4\text {-}19: &{} \mathtt{3333331311113111222222022222200033333313111131110000220200002000_4} \downarrow \{2\}\searrow \{1\}\\ 3\text {-}4\text {-}20: &{} \mathtt{3333333311111111222222220000000033333333111111112222222200000000_4} \downarrow \{2\}\searrow \{1\}\\ 3\text {-}4\text {-}21: &{} \mathtt{3333333333333333222222222222222211111111111111110000000000000000_4} \searrow \{1,2,3\}\\ \end{array}\)

\( \begin{array}{ll} n=5: \\ 3\text {-}5\text {-}1: &{} \mathtt{432104321043210432104321043210432104321043210432104321043210432104321043210432104321} \\ &{} \mathtt{04321043210432104321043210432104321043210_5} \\ 3\text {-}5\text {-}2: &{} \mathtt{444243333324202333330200044424333332420211111020004442411111242023333302000444243333} \\ &{} \mathtt{32420211111020004442411111242021111102000_5} \\ 3\text {-}5\text {-}3: &{} \mathtt{444443333322202111110200044444333332220211111020004444433333222023333302000444443333} \\ &{} \mathtt{32220211111020004444411111222021111102000_5} \\ 3\text {-}5\text {-}4: &{} \mathtt{444443333322222100111100044444333332222210011110004444433333222221001111000444444444} \\ &{} \mathtt{43333310011110003333322222222221001111000_5} \\ 3\text {-}5\text {-}5: &{} \mathtt{444443333322222110110100044444333332222211011010004444433333222221101101000444443333} \\ &{} \mathtt{33333311011010004444422222222221101101000_5} \\ 3\text {-}5\text {-}6: &{} \mathtt{444443330322222111113000044444333032222244444300004444433303222221111130000444443330} \\ &{} \mathtt{32222211111300001111133303222221111130000_5} \\ 3\text {-}5\text {-}7: &{} \mathtt{444443333321222201111000044444333332122220111100004444444444212222011110000333334444} \\ &{} \mathtt{42122220111100003333333333212222011110000_5} \\ 3\text {-}5\text {-}8: &{} \mathtt{444443333322222101111000044444333332222210111100004444433333222221011110000444444444} \\ &{} \mathtt{42222210111100003333333333222221011110000_5} \\ 3\text {-}5\text {-}9: &{} \mathtt{444443333122222113110000044444333312222211311222224444433331222221131100000444443333} \\ &{} \mathtt{10000011311000004444433331222221131100000_5} \\ 3\text {-}5\text {-}10: &{} \mathtt{444443313122222313110000044444331314444431311222224444433131222223131100000222223313} \\ &{} \mathtt{10000031311000004444433131222223131100000_5} \\ 3\text {-}5\text {-}11: &{} \mathtt{444413333322222141113333344441333332222214111000004444133333222221411100000444410000} \\ &{} \mathtt{02222214111000004444133333222221411100000_5} \\ 3\text {-}5\text {-}12: &{} \mathtt{444443313322222311110000044444331334444431111000004444433133222223111100000222223313} \\ &{} \mathtt{32222231111000004444433133222223111100000_5} \\ 3\text {-}5\text {-}13: &{} \mathtt{444443333321222211110000044444333332122221111000004444444444212222111100000333333333} \\ &{} \mathtt{32122221111000004444433333212222111100000_5} \\ 3\text {-}5\text {-}14: &{} \mathtt{444433334222232111111111144443333422223200000111114444333342222320000000000444433334} \\ &{} \mathtt{22223211111000004444333342222321111100000_5} \\ 3\text {-}5\text {-}15: &{} \mathtt{444443333222232111110000044444333322223211111111114444433332222320000000000444443333} \\ &{} \mathtt{22223211111000004444433332222321111100000_5} \\ 3\text {-}5\text {-}16: &{} \mathtt{444243333324222333330000044424333332422211111000004442411111242221111100000444243333} \\ &{} \mathtt{32422211111000004442433333242221111100000_5} \\ 3\text {-}5\text {-}17: &{} \mathtt{444433334322222111111111144443333432222200000000004444333343222221111100000444433334} \\ &{} \mathtt{32222211111000004444333343222221111100000_5} \\ 3\text {-}5\text {-}18: &{} \mathtt{444343343322222222220000044434334331111111111000004443433433222221111100000444343343} \\ &{} \mathtt{32222211111000004443433433222221111100000_5} \\ 3\text {-}5\text {-}19: &{} \mathtt{444443333322222111110000044444333332222211111000004444433333222221111100000444443333} \\ &{} \mathtt{32222211111000004444433333222221111100000_5} \\ 3\text {-}5\text {-}20: &{} \mathtt{444444444444444444444444433333333333333333333333332222222222222222222222222111111111} \\ &{} \mathtt{11111111111111110000000000000000000000000_5} \end{array}\)

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Imai, K., Martin, B., Saito, R. (2018). On Radius 1 Nontrivial Reversible and Number-Conserving Cellular Automata. In: Adamatzky, A. (eds) Reversibility and Universality. Emergence, Complexity and Computation, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-319-73216-9_12

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  • DOI: https://doi.org/10.1007/978-3-319-73216-9_12

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