A Novel Design of One-Sided Recursive Firing Squad

Synchronization Algorithms for One-Dimensional Cellular Arrays

Hayato YANASE†, Masaya HISAOKA† and Hiroshi UMEO†

†Univ. of Osaka Electro-Communication,Faculty of Information Science and Technology,Neyagawa-shi, Hatsu-cho, 18-8, Osaka, Japan

Abstract—The firing squad synchronization prob-lem in cellular automata has been studied extensivelyfor more that forty years, and a rich variety of syn-chronization algorithms have been proposed [1-14]. Inthe present paper, we introduce a new notion of one-and two-sided recursive properties to classify thosesynchronization algorithms proposed so far. We pro-pose a new scheme for designing synchronization algo-rithms with the one-sided recursive property operat-ing in optimum- and linear-time, respectively. We alsogive their implementations on a computer.

1. Introduction

In recent years cellular automata (CA) have beenestablishing increasing interests in the study of model-ing non-linear phenomena occurring in biology, chem-istry, ecology, economy, geology, mechanical engineer-ing, medicine, physics, sociology, public traffic, etc.Cellular automata are considered to be a nice model ofcomplex systems in which an infinite one-dimensionalarray of finite state machines (cells) updates itself insynchronous manner according to a uniform local rule.We study a synchronization problem which gives afinite-state protocol for synchronizing a large scale ofcellular automata. The synchronization in cellular au-tomata has been known as firing squad synchroniza-tion problem since its development, in which it wasoriginally proposed by J. Myhill to synchronize allparts of self-reproducing cellular automata [8]. Thefiring squad synchronization problem has been stud-ied extensively for more than forty years [1-14].

In this paper, we introduce a new notion of one- andtwo-sided recursive properties to classify those syn-chronization algorithms proposed so far. It is shownthat optimum-time synchronization algorithms devel-oped by Balzer [1], Gerken [3], and Waksman [13] aretwo-sided ones and an algorithm proposed by Mazoyer[6] is an only synchronization algorithm with the one-sided recursive property. We propose a new schemefor designing synchronization algorithms with the one-sided recursive property operating in optimum- andlinear-time, respectively. We also give their implemen-tations on a computer. Due to the space available, we

C1 C2 C3 C4 Cn

Figure 1: A one-dimensional cellular automaton.

omit the proofs of the theorems presented.

2. Firing Squad Synchronization Problem forOne-Dimensional Cellular Automata

The firing squad synchronization problem is formal-ized in terms of the model of cellular automata. Figure1 shows a finite one-dimensional cellular array consist-ing of n cells, denoted by Ci, where 1 ≤ i ≤ n. Allcells (except the end cells) are identical finite state au-tomata. The array operates in lock-step mode suchthat the next state of each cell (except the end cells)is determined by both its own present state and thepresent states of its right and left neighbors. All cells(soldiers), except the left end cell, are initially in thequiescent state at time t = 0 and have the propertywhereby the next state of a quiescent cell having quies-cent neighbors is the quiescent state. At time t = 0 theleft end cell (general) is in the fire-when-ready state,which is an initiation signal to the array. The firingsquad synchronization problem is stated as follows.Given an array of n identical cellular automata, in-cluding a general on the left end which is activated attime t = 0, we want to give the description (state setand next-state function) of the automata so that, atsome future time, all of the cells will simultaneouslyand, for the first time, enter a special firing state. Theset of states must be independent of n. Without lossof generality, we assume n ≥ 2. The tricky part of theproblem is that the same kind of soldier having a fixednumber of states must be synchronized, regardless ofthe length n of the array.

3. One-Sided vs. Two-Sided Recursive Syn-chronization Algorithms

Firing squad synchronization algorithms have beendesigned on the basis of parallel divide-and-conquer

2004 International Symposium on NonlinearTheory and its Applications (NOLTA2004)

Fukuoka, Japan, Nov. 29 - Dec. 3, 2004

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cell

timet = 0

t = n-1

t = 2n-2

nG0

G1

G2

G3

cell

timet = 0 n

t = n-1

t = 2n-2

G0

G1

G2

G3

Figure 2: One-sided recursive synchronization scheme (left) andtwo-sided recursive synchronization scheme (right).

time

t = 0cell

G0

w0

n

w1

w3

r

d

w2

G2

r

d

G3

G1

t = n-1

t = 2n-2

xy

w0 :

w1 :

w2 :

w3:

y

y

2x +4xy+y

y

2x +6x y+6xy +y

wi :y

(x+y) (1+ )l

2x+y

1

1r :

1

1

d : y-x

x

Speed of signals

2 23 3

3

2 2

2

l=1

i

∑(x+y)

l-1xyi

i

w1

w3

r

w2

r

Figure 3: Time-space diagram for optimum-time one-sided recur-sive synchronization algorithm.

strategy that calls itself recursively in parallel. Thoserecursive calls are implemented by generating manyGenerals that are responsible for synchronizing di-vided small areas in the cellular space. Initially a Gen-eral located at the left end is responsible for synchro-nizing the whole cellular space consisting of n cells. InFig. 2 (left), the General Gi, i = 2, 3, ..., is respon-sible for synchronizing the cellular space between Gi

and Gi−1, respectively. G1 synchronizes the subspacebetween G1 and the right end of the array. Thus, allof the Generals generated by G0 are located at theleft end of the divided cellular spaces to be synchro-nized by them independently. On the other hand, inFig. 2 (right), the General G0 generates General Gi,i = 1, 2, 3, ...,. Each Gi, i = 1, 2, 3, ..., synchronizesthe divided space between Gi and Gi+1, respectively.In addition, Gi, i = 2, 3, ..., does the same operationsas G0. Thus, in Fig. 2 (right) we find Generals lo-cated at either end of the subspace for which they areresponsible.

If all of the recursive calls are issued by Gener-

als located at one (two) end(s) of partitioned cellu-lar spaces for which the General is responsible, thealgorithm is said to have one-sided (two-sided) recur-sive property. We call synchronization algorithm withone-sided (two-sided) recursive property as one-sided(two-sided) recursive synchronization algorithm. Fig-ure 2 illustrates a time-space diagram for one-sided(Fig. 2 (left)) and two-sided (Fig. 2 (right)) recur-sive synchronization algorithms operating in optimum2n − 2 steps.

It is noted that optimum-time synchronization algo-rithms developed by Balzer [1], Gerken [3], and Waks-man [13] are two-sided ones and an algorithm proposedby Mazoyer [6] is an only synchronization algorithmwith the one-sided recursive property. In addition, itis also observed that all of the 3n-step synchroniza-tion algorithms developed so far are in the class oftwo-sided algorithms.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 G - - - - - - - - - - - - - - -

1 G A2 - - - - - - - - - - - - - -

2 G A2 A3 - - - - - - - - - - - - -

3 G A2 B A4 - - - - - - - - - - - -

4 G A2 A3 B A1 - - - - - - - - - - -

5 G A2 B A4 C A2 - - - - - - - - - -

6 G A2 A3 A4 C B A3 - - - - - - - - -

7 G A2 A3 A4 B C B A4 - - - - - - - -

8 G A2 A3 B A1 B C B A1 - - - - - - -

9 G A2 B A4 C A2 B C C A2 - - - - - -

10 G A2 A3 A4 C B A3 C C B A3 - - - - -

11 G A2 A3 A4 B C A3 C B C B A4 - - - -

12 G A2 A3 B A1 C A3 B C B C B A1 - - -

13 G A2 B A4 A1 C B A4 B C B C C A2 - -

14 G A2 A3 A4 A1 B C B A1 B C C C B A3 -

15 G A2 A3 A4 C A2 B C C A2 C C B C B D40

16 G A2 A3 A4 C B A3 C C A2 C B C B D40D41

17 G A2 A3 A4 B C A3 C C A2 B C B D40D40 -

18 G A2 A3 B A1 C A3 C C B A3 B D40D40D41 -

19 G A2 B A4 A1 C A3 C B C B G4 D40D40 - -

20 G A2 A3 A4 A1 C A3 B C B D40 G4 D40D41 - -

21 G A2 A3 A4 A1 C B A4 B D40D40 G4 D40 - - -

22 G A2 A3 A4 A1 B C B G1 D40D41 G4 D41 - - -

23 G A2 A3 A4 C A2 B D10 G1 D40 - G A2 - - -

24 G A2 A3 A4 C B G3 D10 G1 D41 - G A2 A3 - -

25 G A2 A3 A4 B D30 G3 D11 G A2 - G A2 B A4 -

26 G A2 A3 B G1 D31 G - G A2 A3 G A2 A3 B D10

27 G A2 B G4 G1 A2 G A2 G A2 B D40 A2 B G4 D11

28 G A2 G3 G G A2 D31 A2 D31 A2 G3 D41 A2 G3 G4 -

29 G G G G G G G G G G G G G G G G

30 F F F F F F F F F F F F F F F F

Figure 4: Configurations of a 22-state 498-rule imple-mentation of one-sided recursive optimum-time syn-chronization algorithm on 16 cells.

[Observation 1] Optimum-time synchronization al-gorithms developed by Balzer [1], Gerken [3], andWaksman [13] are two-sided ones. The algorithm pro-posed by Mazoyer [6] is one-sided one.[Observation 2] Synchronization algorithms operat-ing in 3n±O(log n)+O(1) steps developed by Minskyand MacCarthy [7], Fischer [4], and Yunes [14] aretwo-sided ones.

4. Design of Time-Optimum One-Sided Recur-sive Synchronization Algorithms

In this section we propose a design scheme for one-sided recursive optimum-time synchronization algo-rithms that can synchronize any n cells in 2n−2 steps.Figure 3 is a time-space diagram for an optimum-timeone-sided recursive synchronization algorithm. TheGeneral G0 generates an infinite number of signals

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w0, w1, w2, .., to generate Generals G1, G2, .., by di-viding the array recursively with the ratio x/y, wherex, y is any positive integer such that 2x ≤ y. Propa-gation speed of the i-th signal wi, i ≥ 1 is as follows:

yi/(x + y)i(1 +i∑

l=1

xyl−1/(x + y)l)).

When the first signal w0 hits the right end of thearray, an r-signal is generated that propagates at speed1/1 in the left direction. At the same time, a d-signalpropagating at speed x/(y − x) in the left direction isgenerated. The w1- and r-signals meets on cell Cm,m = �ny/(x + y)�, and a special mark is printed as apotential General. When the d-signal arrives at thecell Cm, a new General G1 is generated. Its generationis delayed for �n(y − 2x)/(x + y)� steps. The G1 doesthe same procedures as G0 to the subspace betweenCm and Cn.

When x = 1, y = 2, the scheme given above co-incides with the Mazoyer’s algorithm [6]. It is notedthat, in this case, we need no delay for the generationof Generals. We have implemented the scheme in thecase where x = 1, y = 3 on a computer and got a22-state 498-rule cellular automaton that realizes theone-sided recursive synchronization. Figure 4 is con-figurations of the 22-state implementation of one-sidedrecursive optimum-time synchronization algorithm on16 cells.[Theorem 3] There exists a one-sided 22-state 498-rule cellular automaton that can synchronize any ncells in 2n − 2 optimum steps.

w0 :

w1 :

w2 :

w3:

w0

w1

w2

w3

r

1

1

3k-2

1

7k-6

wl :1

(k-1)2 -k+2l

ncell

t = n-1

s

k

1

1r :

1

k-2

s : k

1time

t = 0

r

2

n-2+2 MOD( +1 ,2)

t = kn O(1)

t = 2(MOD(n+1,2))+-2

t =

2

(k-2)n

2

(k-2)n+

n+

( )( )i=1

2

i-1 i 2

ni-1∑

+

2

n-2+2 MOD( +1 ,2)t =

2

(k-2)n+ ( )( )

i=1=1

3

i-1 i 2

ni-1∑

G0

G1

G21 G22

w1

w2

w3

G1

G22

l = log2n

Speed of signals

Figure 5: Time-space diagram for kn+O(1)-step one-sided recursive synchronization algorithm.

5. Design of Linear-Time One-Sided RecursiveSynchronization Algorithms

In this section we study a class of synchronizationalgorithms operating in linear-time, that can synchro-nize any n cells in kn+O(1) steps, where k is any pos-itive integer such that k ≥ 3. The first synchroniza-tion algorithm developed by Minsky and MacCarthy[7] operates in 3n+O(1) steps. Yunes [14] gave its 14-and 15-state implementations. Fischer [4] also pre-sented a solution with 15 states that synchronizes ncells in 3n − 4 steps. In 1994 Yunes [14] proposed aseven-state solution operating in 3n ± 2θn log n+O(1)steps, where 0 ≤ θn ≤ 1. In addition, the 3n-step syn-chronization algorithm is often used as a subroutinein the design of other optimum-time synchronizationalgorithms. Thus, a class of linear-time synchroniza-tion algorithms is interesting and important in its ownright.

We propose an efficient way to cause a General cellto generate infinite signals w0, w1, w2, ..., w� propagat-ing at speeds of 1/1, 1/k, 1/(3k− 2), .., 1/((k− 1)2� −k + 2), where � = �log2 n�. These signals play an im-portant role in dividing the array into two, four, eight,. . . , equal parts synchronously. The end cell in eachpartition takes a special prefiring state so that whenthe last partition occurs, where all cells have a left andright neighbor in this state.

Figure 5 is a time-space diagram for kn+O(1)-stepone-sided recursive synchronization algorithm. Attime t = 0, the initial General G0 generates an in-finite right-going signals mentioned above. The firstw0-signal arrives at the right end of the array at timet = n − 1. Then, the right end cell sends out anreflected r-signal that proceeds towards the left endat speed 1/(k − 2). The r-signal meets with the sec-ond w1-signal at the center point of the array at timet = n + �(k − 2)n/2� − 2 + 2(n + 1 mod 2), and thesecond General G1 is generated there. The GeneralG1 does the same operations as G0 to the right halfof the array. Simultaneously, G1 generates a left-goings-signal that continues to propagates towards the leftend of the array at speed 1/k. The w2- and r-signalsmeet at the quarter point of the array and a specialmark, being kept until the s-signal arrives at the cell,is given as a potential General. The s-signal acts as adelay for the generation of General G21 so that bothG21 and G22 can be generated simultaneously to syn-chronize three quarters of the array. The G21 is de-layed for ∆t = n/2 steps. The readers can see howthese signals work and all of the Generals generatedare located at the left end of the divided subspaces.Let T (n) be time complexity for synchronizing n cells.Then, T (n) = kn/2 + T (n/2) = kn+O(1).

Thus, we have:[Theorem 4] Let k be any positive integer such

671

that k ≥ 3. The one-sided recursive synchroniza-tion scheme given above can synchronize any n cellsin kn + O(1) steps.

Figure 6 is a time-space diagram in the case of k =3. We have implemented the scheme on a 12-state236-rule cellular automaton. Configurations of a 12-state implementation of one-sided recursive (3n − 3)-step synchronization algorithm on 21 cells are shownin Fig. 7.[Theorem 5] There exists a one-sided recursive 12-state 236-rule cellular automaton that can synchronizeany n cells in 3n − 3 steps.

t = n-1

t = 2n-2

t = 3n-3

time

t = 0

1/1

1/3

1/7

1/151/1

1/3

1/1

2

n-2+2t =

2

n+ ( )( )

i=1

3

i-1 i 2

ni-1∑

n

t = 2(MOD(n+1,2))+-22

nn+

2

n-2+2t =

2

n+ ( )( )

i=1=1

2

i-1 i 2

ni-1∑ MOD( +1 ,2)

MOD( +1 ,2)

cellular space

1 2 ....

Figure 6: Time-spacediagram for (3n − 3)-step one-sided recur-sive synchronizationalgorithm.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

0 G - - - - - - - - - - - - - - - - - - - -

1 G A - - - - - - - - - - - - - - - - - - -

2 G C A - - - - - - - - - - - - - - - - - -

3 G A B A - - - - - - - - - - - - - - - - -

4 G A B C A - - - - - - - - - - - - - - - -

5 G A A C B A - - - - - - - - - - - - - - -

6 G A A B B C A - - - - - - - - - - - - - -

7 G A A B C C B A - - - - - - - - - - - - -

8 G A A A C B B C A - - - - - - - - - - - -

9 G A B A B B C C B A - - - - - - - - - - -

10 G A B A B C C B B C A - - - - - - - - - -

11 G A B A A C B B C C B A - - - - - - - - -

12 G A B C A B B C C B B C A - - - - - - - -

13 G A A C A B C C B B C C B A - - - - - - -

14 G A A C A A C B B C C B B C A - - - - - -

15 G A A C B A B B C C B B C C B A - - - - -

16 G A A B B A B C C B B C C B B C A - - - -

17 G A A B B A A C B B C C B B C C B A - - -

18 G A A B B C A B B C C B B C C B B C A - -

19 G A A B C C A B C C B B C C B B C C B A -

20 G A A A C C A A C B B C C B B C C B B C D

21 G A B A C C B A B B C C B B C C B B C DE -

22 G A B A C B B A B C C B B C C B B C DE - -

23 G A B A B B B A A C B B C C B B C DE - - -

24 G A B A B B B C A B B C C B B C DE - - - -

25 G A B A B B C C A B C C B B C DE - - - - -

26 G A B A B C C C A A C B B C DE - - - - - -

27 G A B A A C C C B A B B C DE - - - - - - -

28 G A B C A C C B B A B C DE - - - - - - - -

29 G A A C A C B B B A A DE - - - - - - - - -

30 G A A C A B B B B C G - - - - - - - - - -

31 G A A C A B B B C C H1 A - - - - - - - - -

32 G A A C A B B C C C DE C A - - - - - - - -

33 G A A C A B C C C C G- A B A - - - - - - -

34 G A A C A A C C C H1 G- A B C A - - - - - -

35 G A A C B A C C C DE G- A A C B A - - - - -

36 G A A B B A C C C - G- A A B B C A - - - -

37 G A A B B A C C H1 - G- A A B C C B A - - -

38 G A A B B A C C DE - G- A A A C B B C A - -

39 G A A B B A C C - - G- A B A B B C C B A -

40 G A A B B A C H1 - - G- A B A B C C B B C D

41 G A A B B A C DE - - G- A B A A C B B C DE -

42 G A A B B A C - - - G- A B C A B B C DE - -

43 G A A B B A H1 - - - G- A A C A B C DE - - -

44 G A A B B A DE - - - G- A A C A A DE - - - -

45 G A A B B G - - - - G- A A C B G - - - - -

46 G A A B B H1 A - - - G- A A B B H1 A - - - -

47 G A A B B DE C A - - G- A A B B DE C A - - -

48 G A A B B G- A B A - G- A A B B G- A B A - -

49 G A A B H2 G- A B C A G- A A B H2 G- A B C A -

50 G A A B E G- A A C B D A A B E G- A A C B D

51 G A A B - G- A A B D E A A B - G- A A B D E

52 G A A H2 - G- A A D E G- A A H2 - G- A A D E -

53 G A A E - G- A A E - G- A A E - G- A A E - -

54 G A G G - G- A G G - G- A G G - G- A G G - -

55 G A H1 G A G- A H1 G A G- A H1 G A G- A H1 G A -

56 G A DE G C D A DE G C D A DE G C D A DE G C D

57 G G- - G G- G- G- - G G- G- G- - G G- G- G- - G G- -

58 G H1 D G D G- H1 D G D G- H1 D G D G- H1 D G H1 D

59 G G G G G G G G G G G G G G G G G G G G G

60 F F F F F F F F F F F F F F F F F F F F F

Figure 7: Configu-rations of a 12-state236-rule implementa-tion of one-sided re-cursive (3n − 3)-stepsynchronization algo-rithm on 21 cells.

6. Conclusions

We have introduced a new notion of one- and two-sided recursive properties in the synchronization al-gorithms for one-dimensional cellular automata andproposed a new scheme for designing synchronizationalgorithms with the one-sided recursive property op-erating in optimum- and linear-time, respectively. Wehave also given their 22- and 12-state implementationsthat realize those algorithms on a computer. We omit-ted the complete lists of transitions rules designed dueto the space available. We have checked their valid-ity for cellular spaces of size n = 2 to 10000 through

our computer simulation. All of the synchronizationalgorithms proposed so far for two-dimensional arraysare designed based on two-sided recursive synchroniza-tion algorithms for one-dimensional arrays. It is aninteresting question whether we can design a real two-dimensional synchronization algorithm with one-sidedrecursive properties.

References

[1] R. Balzer: An 8-state minimal time solution to the firing squadsynchronization problem. Information and Control, vol. 10(1967), pp. 22-42.

[2] W. T. Beyer: Recognition of topological invariants by iterativearrays. Ph.D. Thesis, MIT, pp. 144 (1969).

[3] Hans-D., Gerken: Uber Synchronisations - Probleme bei Zel-lularautomaten. Diplomarbeit, Institut fur Theoretische Infor-matik, Technische Universitat Braunschweig, (1987), pp. 50.

[4] P. C. Fischer: Generation of primes by a one-dimensional real-time iterative array. J. of ACM, vol. 12, No. 3, pp. 388-394,(1965).

[5] A. Grasselli: Synchronization of cellular arrays: The firingsquad problem in two dimensions. Information and Control,28, pp. 113-124 (1975).

[6] J. Mazoyer: A six-state minimal time solution to the fir-ing squad synchronization problem. Theoretical Computer Sci-ence, vol. 50 (1987), pp. 183-238.

[7] M. Minsky: Computation: Finite and infinite machines. Pren-tice Hall, (1967), pp. 28-29.

[8] E. F. Moore: The firing squad synchronization problem. inSequential Machines, Selected Papers (E. F. Moore, ed.),Addison-Wesley, Reading MA.,(1964), pp. 213-214.

[9] I. Shinahr: Two- and three-dimensional firing squad synchro-nization problems. Information and Control, 24, pp. 163-180(1974).

[10] H. Umeo, M. Maeda and N. Fujiwara: An efficient mappingscheme for embedding any one-dimensional firing squad syn-chronization algorithm onto two-dimensional arrays. Proc. ofthe 5th International Conference on Cellular Automata forResearch and Industry, LNCS 2493, pp.69-81, (2002).

[11] H. Umeo, M. Hisaoka, and T. Sogabe: An investigation intotransition rule sets for optimum-time firing squad synchroniza-tion algorithms on one-dimensional cellular automata. Inter-disciplinary Information Sciences, Vol.8, No.2, pp.207-217,(2002).

[12] H. Umeo: A simple design of time-optimum firing squad syn-chronization algorithms with fault-tolerance. IEICE Trans. onInformation and Systems, Vol.E87-D, No.3, (2004), pp.733-739.

[13] A. Waksman: An optimum solution to the firing squad syn-chronization problem. Information and Control, vol. 9 (1966),pp. 66-78.

[14] J. B. Yunes: Seven-state solution to the firing squad synchro-nization problem. Theoretical Computer Science, 127, pp.313-332, (1994).

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