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Universal Totalistic Asynchonous Cellular Automaton and Its Possible Implementation by DNA

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Unconventional Computation and Natural Computation (UCNC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9726))

Abstract

This paper presents a Cellular Automaton (CA) model designed for possible implementation by the reaction and diffusion of DNA strands. The proposed CA works asynchronously, whereby each cell undergoes its transitions independently from other cells and at random times. The state of a cell changes in a cyclic manner, rather than according to an any-to-any mapping. The transition rules are designed as totalistic, i.e., the next state of a cell is determined only by the number of states in the neighborhood of the cell, not by their relative positions. Universal circuit elements are designed for the CA as well as wires and crossings to connect them, which implies that the CA is Turing-complete.

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Acknowledgment

This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas “Molecular Robotics” (No. 15H00825) of The Ministry of Education, Culture, Sports, Science, and Technology, Japan.

Author information

Authors and Affiliations

  1. University of Hyogo, Himeji, Japan

    Teijiro Isokawa & Nobuyuki Matsui

  2. Center for Information and Neural Networks, National Institute of Information and Communications Technology, Osaka University, Osaka, Japan

    Ferdinand Peper

  3. Tohoku University, Sendai, Japan

    Ibuki Kawamata & Satoshi Murata

  4. The University of Tokyo, Tokyo, Japan

    Masami Hagiya

Authors
  1. Teijiro Isokawa
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  2. Ferdinand Peper
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  3. Ibuki Kawamata
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  4. Nobuyuki Matsui
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  5. Satoshi Murata
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  6. Masami Hagiya
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Corresponding author

Correspondence to Teijiro Isokawa .

Editor information

Editors and Affiliations

  1. Manchester Metropolitan University, Manchester, United Kingdom

    Martyn Amos

  2. University of British Columbia, Vancouver, BC, Canada

    ANNE CONDON

A Transition Rules Used in this Paper

A Transition Rules Used in this Paper

Three forms for each of rules used in this paper are shown, non-totalistic (NT), outer totalistic and inner-dependent (OT&ID), and inner totalistic (IT).

Rule No

NT form

OT&ID form

IT form

1

(1,0,1,0,2) \(\rightarrow \) Z

1

0(2) 1(1) 2(1) \(\rightarrow \) Z

0(2) 1(2) 2(1) \(\rightarrow \) Z

2

(2,0,Z,0,3) \(\rightarrow \) Y

2

0(2) 3(1) Z(1) \(\rightarrow \) Y

0(2) 2(1) 3(1) Z(1) \(\rightarrow \) Y

3

(3,0,Y,0,1) \(\rightarrow \) 1

3

0(2) 1(1) Y(1) \(\rightarrow \) 1

0(2) 1(1) 3(1) Y(1) \(\rightarrow \) 1

4

(Y,0,Z,0,1) \(\rightarrow \) 3

Y

0(2) 1(1) Z(1) \(\rightarrow \) 3

0(2) 1(1) Y(1) Z(1) \(\rightarrow \) 3

5

(Z,0,1,0,3) \(\rightarrow \) 2

Z

0(2) 1(1) 3(1) \(\rightarrow \) 2

0(2) 1(1) 3(1) Z(1) \(\rightarrow \) 2

6

(1,\(C_0\),1,\(C_0\),2) \(\rightarrow \) Z

1

1(1) 2(1) \(C_0\)(2) \(\rightarrow \) Z

1(2) 2(1) \(C_0\)(2) \(\rightarrow \) Z

7

(2,\(C_1\),Z,\(C_1\),3) \(\rightarrow \) Y

2

3(1) Z(1) \(C_1\)(2) \(\rightarrow \) Y

2(1) 3(1) Z(1) \(C_1\)(2) \(\rightarrow \) Y

8

(3,\(C_1\),Y,\(C_1\),1) \(\rightarrow \) 1

3

1(1) Y(1) \(C_1\)(2) \(\rightarrow \) 1

1(1) 3(1) Y(1) \(C_1\)(2) \(\rightarrow \) 1

9

(Y,\(C_1\),Z,\(C_1\),1) \(\rightarrow \) 3

Y

1(1) Z(1) \(C_1\)(2) \(\rightarrow \) 3

1(1) Y(1) Z(1) \(C_1\)(2) \(\rightarrow \) 3

10

(Z,\(C_1\),1,\(C_1\),3) \(\rightarrow \) 2

Z

1(1) 3(1) \(C_1\)(2) \(\rightarrow \) 2

1(1) 3(1) Z(1) \(C_1\)(2) \(\rightarrow \) 2

11

(1,1,1,1,2) \(\rightarrow \) Z

1

1(3) 2(1) \(\rightarrow \) Z

1(4) 2(1) \(\rightarrow \) Z

12

(2,1,Z,1,3) \(\rightarrow \) Y

2

1(2) 3(1) Z(1) \(\rightarrow \) Y

1(2) 2(1) 3(1) Z(1) \(\rightarrow \) Y

13

(3,1,Y,1,1) \(\rightarrow \) 1

3

1(3) Y(1) \(\rightarrow \) 1

1(3) 3(1) Y(1) \(\rightarrow \) 1

14

(Y,1,Z,1,1) \(\rightarrow \) 3

Y

1(3) Z(1) \(\rightarrow \) 3

1(3) Y(1) Z(1) \(\rightarrow \) 3

15

(Z,1,1,1,3) \(\rightarrow \) 2

Z

1(3) 3(1) \(\rightarrow \) 2

1(3) 3(1) Z(1) \(\rightarrow \) 2

16

(\(C_0\),0,1,Z,0) \(\rightarrow \) \(C_1\)

\(C_0\)

0(2) 1(1) Z(1) \(\rightarrow \) \(C_1\)

0(2) 1(1) Z(1) \(C_0\)(1) \(\rightarrow \) \(C_1\)

17

(\(C_1\),0,1,1,0) \(\rightarrow \) \(C_0\)

\(C_1\)

0(2) 1(2) \(\rightarrow \) \(C_0\)

0(2) 1(2) \(C_1\)(1) \(\rightarrow \) \(C_0\)

18

(1,2,0,1,K) \(\rightarrow \) Z

1

0(1) 1(1) 2(1) K(1) \(\rightarrow \) Z

0(1) 1(2) 2(1) K(1) \(\rightarrow \) Z

19

(Z,3,0,1,K) \(\rightarrow \) 2

Z

0(1) 1(1) 3(1) K(1) \(\rightarrow \) 2

0(1) 1(1) 3(1) Z(1) K(1) \(\rightarrow \) 2

20

(2,3,0,Z,K) \(\rightarrow \) Y

2

0(1) 3(1) Z(1) K(1) \(\rightarrow \) Y

0(1) 2(1) 3(1) Z(1) K(1) \(\rightarrow \) Y

21

(Y,1,0,Z,K) \(\rightarrow \) 3

Y

0(1) 1(1) Z(1) K(1) \(\rightarrow \) 3

0(1) 1(1) Y(1) Z(1) K(1) \(\rightarrow \) 3

22

(3,1,0,Y,K) \(\rightarrow \) 1

3

0(1) 1(1) Y(1) K(1) \(\rightarrow \) 1

0(1) 1(1) 3(1) Y(1) K(1) \(\rightarrow \) 1

23

(1,2,1,1,L) \(\rightarrow \) Z

1

L(1) 1(2) 2(1) \(\rightarrow \) Z

L(1) 1(3) 2(1) \(\rightarrow \) Z

24

(Z,3,1,1,L) \(\rightarrow \) X

Z

L(1) 1(2) 3(1) \(\rightarrow \) X

L(1) 1(2) 3(1) Z(1) \(\rightarrow \) X

25

(X,3,Z,1,L) \(\rightarrow \) Y

X

L(1) 1(1) 3(1) Z(1) \(\rightarrow \) Y

L(1) 1(1) 3(1) X(1) Z(1) \(\rightarrow \) Y

26

(Y,1,Z,1,L) \(\rightarrow \) 3

Y

L(1) 1(2) Z(1) \(\rightarrow \) 3

L(1) 1(2) Y(1) Z(1) \(\rightarrow \) 3

27

(3,1,Y,1,L) \(\rightarrow \) 1

3

L(1) 1(2) Y(1) \(\rightarrow \) 1

L(1) 1(2) 3(1) Y(1) \(\rightarrow \) 1

28

(1,2,1,0,U) \(\rightarrow \) Z

1

0(1) 1(1) 2(1) U(1) \(\rightarrow \) Z

0(1) 1(2) 2(1) U(1) \(\rightarrow \) Z

29

(Z,3,1,0,U) \(\rightarrow \) 2

Z

0(1) 1(1) 3(1) U(1) \(\rightarrow \) 2

0(1) 1(1) 3(1) Z(1) U(1) \(\rightarrow \) 2

30

(2,3,Z,0,\(D'\)) \(\rightarrow \) Y

2

0(1) 3(1) Z(1) \(D'\)(1) \(\rightarrow \) Y

0(1) 2(1) 3(1) Z(1) \(D'\)(1) \(\rightarrow \) Y

31

(Y,1,Z,0,D) \(\rightarrow \) 3

Y

0(1) 1(1) Z(1) D(1) \(\rightarrow \) 3

0(1) 1(1) Y(1) Z(1) D(1) \(\rightarrow \) 3

32

(3,1,Y,0,D) \(\rightarrow \) 1

3

0(1) 1(1) Y(1) D(1) \(\rightarrow \) 1

0(1) 1(1) 3(1) Y(1) D(1) \(\rightarrow \) 1

33

(U,0,2,E,1) \(\rightarrow \) \(D'\)

U

0(1) 1(1) 2(1) E(1) \(\rightarrow \) \(D'\)

0(1) 1(1) 2(1) E(1) U(1) \(\rightarrow \) \(D'\)

34

(\(D'\),0,Y,E,1) \(\rightarrow \) D

\(D'\)

0(1) 1(1) Y(1) E(1) \(\rightarrow \) D

0(1) 1(1) Y(1) E(1) \(D'\)(1) \(\rightarrow \) D

35

(1,2,1,0,D) \(\rightarrow \) Z

1

0(1) 1(1) 2(1) D(1) \(\rightarrow \) Z

0(1) 1(2) 2(1) D(1) \(\rightarrow \) Z

36

(Z,3,1,0,D) \(\rightarrow \) 2

Z

0(1) 1(1) 3(1) D(1) \(\rightarrow \) 2

0(1) 1(1) 3(1) Z(1) D(1) \(\rightarrow \) 2

37

(2,3,Z,0,\(U'\)) \(\rightarrow \) Y

2

0(1) 3(1) Z(1) \(U'\)(1) \(\rightarrow \) Y

0(1) 2(1) 3(1) Z(1) \(U'\)(1) \(\rightarrow \) Y

38

(Y,1,Z,0,U) \(\rightarrow \) 3

Y

0(1) 1(1) Z(1) U(1) \(\rightarrow \) 3

0(1) 1(1) Y(1) Z(1) U(1) \(\rightarrow \) 3

39

(3,1,Y,0,U) \(\rightarrow \) 1

3

0(1) 1(1) Y(1) U(1) \(\rightarrow \) 1

0(1) 1(1) 3(1) Y(1) U(1) \(\rightarrow \) 1

40

(D,0,2,E,1) \(\rightarrow \) \(U'\)

D

0(1) 1(1) 2(1) E(1) \(\rightarrow \) \(U'\)

0(1) 1(1) 2(1) E(1) D(1) \(\rightarrow \) \(U'\)

41

(\(U'\),0,Y,E,1) \(\rightarrow \) U

\(U'\)

0(1) 1(1) Y(1) E(1) \(\rightarrow \) U

0(1) 1(1) Y(1) E(1) \(U'\)(1) \(\rightarrow \) U

42

(1,2,U,1,1) \(\rightarrow \) Z

1

1(2) 2(1) U(1) \(\rightarrow \) Z

1(3) 2(1) U(1) \(\rightarrow \) Z

43

(Z,3,U,1,1) \(\rightarrow \) u

Z

1(2) 3(1) U(1) \(\rightarrow \) u

1(2) 3(1) Z(1) U(1) \(\rightarrow \) u

44

(u,3,U,1,Z) \(\rightarrow \) Y

u

1(1) 3(1) Z(1) U(1) \(\rightarrow \) Y

1(1) 3(1) Z(1) U(1) u(1) \(\rightarrow \) Y

45

(Y,1,U,1,Z) \(\rightarrow \) 3

Y

1(2) Z(1) U(1) \(\rightarrow \) 3

1(2) Y(1) Z(1) U(1) \(\rightarrow \) 3

46

(3,1,U,1,Y) \(\rightarrow \) 1

3

1(2) Y(1) U(1) \(\rightarrow \) 1

1(2) 3(1) Y(1) U(1) \(\rightarrow \) 1

47

(1,0,u,0,1) \(\rightarrow \) Z

1

0(2) 1(1) u(1) \(\rightarrow \) Z

0(2) 1(2) u(1) \(\rightarrow \) Z

48

(1,2,D,1,1) \(\rightarrow \) Z

1

1(2) 2(1) D(1) \(\rightarrow \) Z

1(3) 2(1) D(1) \(\rightarrow \) Z

49

(Z,3,D,1,1) \(\rightarrow \) d

Z

1(2) 3(1) D(1) \(\rightarrow \) d

1(2) 3(1) Z(1) D(1) \(\rightarrow \) d

50

(d,3,D,Z,1) \(\rightarrow \) Y

d

1(1) 3(1) Z(1) D(1) \(\rightarrow \) Y

1(1) 3(1) Z(1) D(1) d(1) \(\rightarrow \) Y

51

(Y,1,D,Z,1) \(\rightarrow \) 3

Y

1(2) Z(1) D(1) \(\rightarrow \) 3

1(2) Y(1) Z(1) D(1) \(\rightarrow \) 3

52

(3,1,D,Y,1) \(\rightarrow \) 1

3

1(2) Y(1) D(1) \(\rightarrow \) 1

1(2) 3(1) Y(1) D(1) \(\rightarrow \) 1

53

(1,d,E,1,0) \(\rightarrow \) Z

1

0(1) 1(1) E(1) d(1) \(\rightarrow \) Z

0(1) 1(2) E(1) d(1) \(\rightarrow \) Z

54

(Z,3,E,1,0) \(\rightarrow \) 2

Z

0(1) 1(1) 3(1) E(1) \(\rightarrow \) 2

0(1) 1(1) 3(1) Z(1) E(1) \(\rightarrow \) 2

55

(2,3,E,Z,0) \(\rightarrow \) Y

2

0(1) 3(1) Z(1) E(1) \(\rightarrow \) Y

0(1) 2(1) 3(1) Z(1) E(1) \(\rightarrow \) Y

56

(Y,1,E,Z,0) \(\rightarrow \) 3

Y

0(1) 1(1) Z(1) E(1) \(\rightarrow \) 3

0(1) 1(1) Y(1) Z(1) E(1) \(\rightarrow \) 3

57

(3,1,E,Y,0) \(\rightarrow \) 1

3

0(1) 1(1) Y(1) E(1) \(\rightarrow \) 1

0(1) 1(1) 3(1) Y(1) E(1) \(\rightarrow \) 1

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Isokawa, T., Peper, F., Kawamata, I., Matsui, N., Murata, S., Hagiya, M. (2016). Universal Totalistic Asynchonous Cellular Automaton and Its Possible Implementation by DNA. In: Amos, M., CONDON, A. (eds) Unconventional Computation and Natural Computation. UCNC 2016. Lecture Notes in Computer Science(), vol 9726. Springer, Cham. https://doi.org/10.1007/978-3-319-41312-9_15

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