An Overview of Recent Solutions to and Lower Bounds for the Firing Synchronization Problem
aa r X i v : . [ c s . F L ] J a n An Overview of Recent Solutions to and Lower Bounds forthe Firing Synchronization Problem
Thiago Correa, Breno Gustavo, Lucas Lemos, Amber SettleAugust 13, 2018
Complex systems in a wide variety of areas such as biological modeling, image processing, andlanguage recognition can be modeled using networks of very simple machines called finite automata.A finite automaton is an idealized machine that follows a fixed set of rules and changes its state torecognize a pattern, accepting or rejecting the input based on the state in which it finds itself at theconclusion of the computation. In general a subsystem can be modeled using a finite automaton ifit can be described by a finite set of states and rules. Connecting subsystems modeled using finiteautomata into a network allows for more computational power. One such network, called a cellularautomaton, consists of an n -dimensional array for n > quiescent state, except the initiator whichstarts in the general state and is the one initiating the synchronization. The next state of each cellis determined by the cell’s state and the state of its neighbors, with a required rule being that a ma-chine that is quiescent and has two neighbors that are quiescient must remain so in the next roundof the computation. The goal of the problem is to synchronize the array of automata, which meansthat each machine enters a unique designated firing state for the first time and simultaneously atsome point during the execution.As might be expected with a problem that has existed as long as the firing synchronizationproblem, there are a wide range of variants for the network and its initial configuration. In theoriginal problem the initiator is placed at one end of the array, either on the left or the right. Aproblem that is more complex than the original problem is the generalized problem, defined byMoore and Langdon in 1968 [12]. In this version the initiator can be placed anywhere in the array,although there is a restriction that there is a single initiator. The problem can be made more1omplex by requiring that the solution be symmetric, a requirement first introduced by Szwerinskiin 1982 [18]. In this type of solution an automaton cannot distinguish its right and left neighbors,eliminating any directional information provided to the automaton and making the problem moredifficult to solve.Another set of variants of the problem consider alterations to the underlying structure of thenetwork. All of the problems described above operate on a one-dimensional array, which is thesimplest configuration for a cellular automaton. In that configuration the end machines do nothave neighbors, so that the end machines make state transitions based only on their own state andone existing neighbor. An alternate approach in one dimension is to connect the two end machinesto each other, forming a ring rather than an array. The network can also be expanded to more thanone dimension, producing a grid (two dimensions), a cube (three dimensions), or more generally ahybercube (four or more dimensions).Once a variant and configuration for the network has been established, it is important to measurethe complexity of the solution to the problem. One measure of a solution is the time in which thesolution causes the network to synchronize. A minimal-time solution is one that synchronizes thenetwork as quickly as possible, and a non-minimal-time solution is one that synchronizes the networkat some point but requires more steps/rounds than the minimal-time solution. For example, theminimal time required for the original problem on a one-dimensional array is 2 n − n machines. Any solution sychronizing in 2 n − n − While the focus of this paper is on recent solutions to and lower bounds for the firing synchro-nization problem (FSP), it is useful to summarize earlier results to provide context for more recentones. In this section we summarize results providing solutions to the FSP using a (relatively) smallnumber of states as well as known lower bounds on the number of states required to solve the FSP.This section focuses on solutions and lower bounds published prior to 1999.2 .1 Minimal-time results
The first minimal-time solution to the original one-dimensional FSP was produced in 1962when Goto gave a solution with over a thousand states [6]. Once it became clear that a minimal-time solution was possible work quickly turned to minimizing the number of states required by thesolution. In 1966 Waksman [30] gave a 16-state minimal-time solution, and Balzer [1] independentlyproduced an 8-state solution using the same ideas. In 1987 Mazoyer produced a 6-state solution toa restricted version of the problem in which the initiator is always located at the left endpoint ofthe array [10].In 1968 Moore and Langdon introduced the generalized problem in which the initiator can belocated anywhere in the array and gave a 17-state minimal-time solution [12]. An improvement onthis solution was produced in 1970 with the publication of a 10-state solution [28]. Further workwas done by Szwerinski who produced a 10-state symmetric solution [18]. Szwerinski’s solution wassubsequently improved to a 9-state solution [16].There were many fewer early solutions for the firing synchronization problem on a ring. In 1989Culik modified Waksman’s solution to function on the one-dimensional ring [3]. Like Waksman’ssolution Culik’s ring solution uses 16 states.
An early non-minimal-time solution to the FSP was created by Fischer who gave a 15-statesolution that synchronized an array of n automata in 3 n -4 steps[4]. Later Mazoyer suggested thatall solutions with few states must necessarily be minimal-time, a conjecture based on the idea thatthe simplest solution will naturally be the fastest. Yun´es disproved the conjecture in 1994 by givingan implementation of a divide and conquer solution originally given by McCarthy and Minsky [11]that requires only 7 states and synchronizes in time t ( n ) = 3 n ± θ n logn + C where 0 ≤ θ n < n automatain 2 n − n + 1 rounds if the initiator islocated at the right endpoint.The same authors produced a 7-state non-minimal-time solution to the generalized problem[16]. The 7-state solution requires 2 n − k rounds to synchronize an array of length n when theinitiator is located in position k of the array.Also in that work is an 11-state non-minimal-time solution to the ring variant of the FSP [16].To synchronize a ring of length n the solution requires 2 n − n − Early work on finding lower bounds for the number of states for solutions to the firing synchro-nization problem is limited. The earliest state lower bound for the FSP was claimed in 1967 byBalzer [1] and confirmed by Sanders in 1994 [14]. Sanders showed that there is no 4-state minimal-time solution to the restricted original problem where the initiator must be located at the right3ndpoint of the array. Sander’s (and Balzer’s) technique involved a modified exhaustive search inwhich a program examined all possible 4-state solutions to demonstrate that none of them correctlysolved the problem. It is crucial for the programs that only minimal-time solutions are examinedsince it allows the program to discard any solution that has not achieved synchronization of n automata by time step 2 n − The purpose of this paper is to document and classify recent results about the firing synchro-nization problem. Our main focus is on results published between 1998 and 2015. In this sectionwe first describe the methodology we used to find the recent results. We then describe each of theresults, categorizing them by type of problem and type of solution.
In order to summarize current research on minimal-state solutions to the firing synchronizationproblem we conducted a literature review. We began by identifying pre-1998 authors who hadpublished on the problem and from that the conferences and journals in which their work had beenpublished. That resulted in the following list of conferences and journals: • Journals – Information and Computation – Information and Control – Information Processing Letters – International Journal of Unconventional Computing – Journal of Algorithms – Journal of Computer and System Sciences – Parallel Processing Letters – SIAM Journal on Computing – Theoretical Computer Science • Conferences 4
International Colloquium on Structural Information and Communication Complexity(SIROCCO) – International Conference on Cellular Automata for Research and Industry (ACRI) – International Conference on Membrane Computing – International Conference on Networking and Computing (ICNC) – international Workshop on Cellular Automata and Discrete Complex Systems – Symposium of Principles of Distributed ComputingWe then examined all of the journals and conferences looking for articles with relevant titles be-tween 1998 and 2015. Among the journals only Information Processing Letters, Parallel ProcessingLetters, and Theoretical Computer Science yielded articles in the specified time period. In the spec-ified time period only the International Colloquium on Structural Information and CommunicationComplexity (SIROCCO) and the International Conference on Cellular Automata for Research andIndustry (ACRI) yielded relevant articles.We also searched for more recent articles written by pre-1998 authors. When we were aware ofthem, we included papers that had referenced pre-1998 articles. Any article found via any meansalso added to the list of authors included in the search. We also did searches for papers includingrelevant keywords, such as cellular automata, firing synchronization, and firing squad.In the remainder of the paper we summarize the articles containing results that minimize thenumber of states used to solve the firing synchronization problem. While our literature review mayhave missed some articles, either because they were published in venues of which we were unawareor by authors who were not in any way connected to previous results, we believe that the majorityof articles containing state minimization results are represented.
In order to understand some of the results discussed later, some background information isnecessary. We describe in this section Wolfram’s rules as well as discuss a solution techniquecommon to many non-minimal-time results.
Stephen Wolfram is a scientist known for his work on cellular automaton and his contribution totheoretical physics. In 2002 he published A New Kind of Science[31] in which he presents systems hecalls simple programs. Generally these simple programs have a very simple abstract framework likesimple cellular automata, Turing machines, and combinators. The focus of A New Kind of Scienceis to use these simple programs to study simple abstract rules, which are, essentially, elementarycomputer programs.In our survey, we found solutions based on some of Wolfram’s rules. Here we describe the basicstructure of the rules and discuss two of them in particular. In Wolfram’s text elementary cellularautomata have two possible values for each cell which are colored white (for 0) or black (for 1).The rules for a transition are based on triples, with the next color of a cell depending on its color5nd the colors of its two immediate neighbors. The configuration of the transition rules are alwaysthe same and are given in the following order:1. three black squares (111)2. two black squares followed by a white square (110)3. a black square, a white square, and a black square (101)4. a black square, and two white squares (100)5. a white square and two black squares (011)6. a white square, a black square, and a white square (010)7. two white squares and a black square (001)8. three white squares (000)Thus each set of transition rules can be thought of as an eight-digit binary number since each ruleneeds to transition either to white (0) or black (1). The rules are named based on the representationof the state produced in the automaton in binary. So rule 60 is the one that assigns the transitionsthe binary value 00111100. This means that it defines the transitions as 111 →
0, 110 →
0, 101 →
1, 100 →
1, 011 →
1, 010 →
1, 001 →
0, and 000 → The 3n-step algorithm is an interesting approach to synchronizing a two-dimensional cellularautomaton due to its simplicity and straightforwardness. This type of approach is used when aminimal-time solution is not required.In the design of any 3n-step approach the most crucial step is to find the center cell of thearray to be synchronized. The basic mechanism for doing this is to use two signals moving atdifferent speeds. The first signal is called a-signal and it moves at the speed of one cell per unitstep. The second signal is called b-signal and it moves at a speed of one cell per three unit steps.When the a-signal finds the end of the array it returns to meet the b-signal at the half of thearray, but the reflected a-signal is called r-signal . If the length of the array is odd, the cell C ⌈ n/ ⌉ becomes the general, if the length is even two generals are created and each general is responsiblefor synchronizing it is left and right halves of the cellular space.Recursion is an important part of this algorithm. After finding the center cell, the process hasto be started over with the new general, or generals if it is an even array. And the simplest wayto start over is to apply recursion on the left and right side of the now divided array. After manysteps of recursion, which depends on the size of the array, the problem is reduced to a problem ofsize two which is the last step before firing. 6 .3 Overview of results In the remainder of the paper we describe and categorize results for the FSP, with a focus onresults published between 1998 and 2015. To provide an overview of the work we summarize thetable below gives information about the results discussed.Table 1: Summary of results for the FSPAuthor Year Algorithm n − n + 1 left or right -Settle [16] 1999 Mazoyer +3 n -step 7 127 2 n − k arbitrary -Noguchi [13] 2004 Balzer 8 119 2 n − n -step 6 115 max(k, n k+ 1) +2n+ O(logn) arbitrary symmetricUmeo andYanagihara[25] 2009 3 n -step 5 71 3 n − n = k Umeo,Yunes andKamikawa[27] 2008 Wolfram’srule 60 4 - 2 n − N = 2 n Umeo,Yun`es andKamikawa[27] 2008 Wolfram’srule 150 4 - 2 n − N = 2 n + 1Yun`es [33] 2008 Wolfram’srule 60 4 32* 2 n +1 − N = 2 n Umeo,Yun`es andKamikawa[21] 2009 Computer-basedinvestigation 4 - 2 n − n = k , symmetricUmeo,Kamikawaand Nishioka[20] 2010 Computer-basedinvestigation 8 222 2 n − n automata, where the initiator is located when the synchronization isbegun, and any important notes about the solution. Recall that a minimal-time solution is one that takes 2 n − n automata. Each section below summarizes a paper by a set of authors providing a minimal-timesolution to the FSP. Settle and Simon [15] provide a range of different solutions to the firing synchronization problem.The first solution is a minimal-time solution to the generalized problem, where the initiator canbe located in any cell in the array. The 9-state minimal-time to the generalized problem is animprovement from the 10-states solution created by Szwerinski[18]. One common strategy to dealwith problems that use the generalized problem is first make the generalized problem look like theoriginal problem, and then solve the original problem. Szwerinski’s solution needed two states totransform the generalized problem into the original problem, and Settle and Simon and improvedthat to use only one more state.The second contribution of the paper is a 6-states non-minimal-time solution to the originalproblem, where the initiator must be on either the left or right side of the array. The 6-stateautomaton is based on Mazoyer’s 6-state solution, but the version by Settle and Simon the initiatormay be on either the left and right side of the array. Mazoyer’s solution works by dividing the arrayin unequal parts of 2 / n and 1 / n to create subproblems that will also be divided recursively, andSettle and Simon’s solution works the same way.The third solution in the paper is a 7-state non-minimal-time solution to the generalized problem.The solution also based on the optimum time 6-state solution from Mazoyer. One state is added tothe Mazoyer solution to transform it into a solution to the generalized problem. To do this Settleand Simon used the 3n-Step algorithm to allow the initiator to be put in any place of the array.The new state is used to find the middle of the array, from which point the work continues withthe Mazoyer solution. The focus of Noguchi’s work is to provide a more straightforward solution with fewer rules tothe 8-state minimal time problem on rings[13]. His 8-state solution uses a strategy inspired byWaksman[30] and Balzer[1] where waves are used to gather, store, and pass information about thesystem, typically by halving the array and placing initiators at the midpoints. This is not a solutionto the generalized problem, so the general must be at the beginning of the line. The strategy usedin the solution has a main wave that travels at full speed, and a reverse wave that goes back alsoat maximum speed. These waves are used to detect the middle point, quarter point, and so on ofthe line. Doing that reduces the problem to sub-problems of the original problem, and every sub-8roblem can also be reduced recursively. For every sub-problem, a new general and consequently anew main wave and reverse wave are created.The algorithm works sending multiple waves in different speeds. The primary wave travels aone cell per time, and after the primary wave, multiple middle waves are sent. This multiple waveswill be responsible for the creation of the future middle points. The n-middle waves that are sentafter the primary wave moves one machine in the time that the primary wave moves 2 n machines.This will ensure that when the primary wave reflects it will find the middle waves in the respectivemiddle spots of the array, 1 /
2, 1 /
4, 1 /
16, and so on. The primary wave also is responsible forsending backward waves. These backward waves are responsible for passing-through the middlewaves. This contact serves as instruction for the middle waves to start the process again.
In the work by Umeo, Yun`es and Kamikawa[27], some elements of a new family of time-optimalsolutions to a less restrictive firing squad synchronization problem are presented. The solutions arebased on elementary algebraic cellular automata. The authors present some 4-state solutions whichsynchronize every line whose length is 2 n or 2 n + 1 based on Wolfram’s rules 60 and 150 describedin Section 3.2.1. Umeo, Yun`es and Kamikawa were able to construct different 4-state solutions tothe FSP which synchronizes every line of length 2 n or 2 n + 1. Three of these solutions use only 33transitions and one uses only 30 transitions.Rule 60 when run on a configuration of length a power of two where the left end cell is 1 (black)led to something that looked like a synchronization, although it should be noted that rule 60 isnot a solution to the problem. So to create something able to synchronize every power of two theauthors use a simple folding of the space-time diagram of rule 60 running on a line of length 2 n +1 ,resulting in a space-time diagram of a 4-state automata running on a line of length 2 n . Using thismodification they could obtain the synchronization at the time 2 n for a line of length n . Using thesame concept, they also present a solution that synchronizes every line of length N = 2 n + 1 attime 2 N − N − nlog ( n ) for a lineof length n , neither it is in the order of n , but something in between. In this paper Umeo, Kamikawa and Yun`es[21] provide a partial solution to the FSP by presentinga family of 4-state solutions to synchronize any one-dimensional ring of length n = 2 k where k represents any positive integer.The authors consider only symmetrical 4-state solutions for the ring. In their approach theyuse a computer to search the transition rule set in order to find a FSP solution. They did this bygenerating a symmetrical 4-state transition table and computing the configuration of each transitiontable for a ring of length n . They assume that Q is a quiescent state, A and G are auxiliary statesand F is the firing state. Their program starts from an initial configuration: G n − z }| { Q, ...Q where2 ≤ n ≤
32 and checks if each transition table yields a synchronized configuration: n z }| { F F, ...F during9he time t where n ≤ t ≤ n and the state F never appears before that. By doing this they foundthat there were 6412 successful synchronized configurations. Most of them included redundantentries, so they removed the redundancies and compared and classified the valid solutions intosmall groups. After that process they obtained seventeen solutions: four optimum-time solutions,four nearly-optimum time solutions, and nine non-optimum time solutions. All of these seventeensolutions can synchronize rings of length n = 2 k for any positive integer k .They also converted the solutions into solutions for an open ring, that is, a conventionalone-dimensional array. They found that the converted 4-state protocol can synchronize any one-dimensional array of length n = 2 k + 1 with the left-end general both in state G and A in optimum2 n − k ≥ In this paper the authors Umeo, Kamikawa, Nishioka and Akiguchi [20] present a computerstudy on different solutions to the generalized firing synchronization problem (GFSP). Recall thatthe GFSP may be described as the original problem with the general on the far left or right ofthe array at time t = − ( k − k is the number of cells between the general and thenearest end. This study reveals inaccuracies in previous solutions, such as redundant rules andunsuccessful synchronizations. The paper also introduces a new eight-state solution to the GFSP,which surpasses the previous best solution that had nine states. Their use of a computer-assistedapproach helped the transition table of their new solution to not have flaws in redundancy orunsuccessful synchronizations. A six-state solution is also examined in the study.The study presented in this paper takes into account the state transition tables for each solutionof the GFSP being analyzed. The first transition table studied is Moore and Langdon’s 17-stateoptimum-time solution [12]. This solution had problems synchronizing a relatively large numberof arrays with several positions of the initial general. The second transition table studied is Var-shavsky, Marakhovsky and Peschansky’s 10-state optimum time solution [28]. This solution alsohas unsuccessful synchronizations. The third table studied is Szwerinski’s optimum-time ten-statesolution [18]. This solution had no errors in the table, so it didn’t present any unsuccessful syn-chronization; however, it had 21 redundant rules. The fourth table studied is Settle and Simon’s9-state optimum time solution [15]. This table had no errors, but it presented 16 redundant rules.The 8-state optimum-time solution is then presented. This solution has 222 rules none of thembeing redundant. The last table studied is Umeo, Maeda and Hongyo’s 6-state non-optimum-timesolution [22]. This solution has 115 states none of them being redundant and is considered a 3n-stepsolution. Recall that a non-minimal-time solution is one that takes more than 2 n − n automata. Each section below summarizes a paper by a set of authors providing anon-minimal-time solution to the FSP. 10 .5.1 Umeo, Maeda and Hongyo: 2006 Umeo, Maeda and Hongyo [22] give a new 3n-step algorithm that improves the lower bound ofthe generalized firing synchronization to a 6-state symmetric solution. The paper has two parts.In the first part it provides a 6-state solution to the original problem with the initiator on the leftside. In the second part it gives a 6-state solution to the generalized problem, where the initiatorcan be placed anywhere on the array. The main difference between the two solutions is that on thesecond one, more rules are used to transform the solution to the original problem on the solutionto the generalized problem.This solution like other that uses the 3n - step algorithm starts with the propagation of thea and b signals, on this solution the propagation of the a- signal is represented by the state P.While P is going away on the right direction at speed of one cell per time, it takes a a state R andM alternatively at each step until either the b-signal or the r-signal arrives at the cell itself. Theb-signal is represented by the propagation of a 1/3 speed signal where the cells take a state R, Rand Z for each three steps. And finally, the R signal is represented as a 1/1 speed signal of the Zstate.One of the key ideas used on this paper to improve the 3n-step algorithm is based on the useof the quiescent cells of the zone T. The zone T is the triangle area circled by a-, b-, and r-signalsin the time-space diagram. In the implementations of Fischer and Yun`es, all cells in zone T keepquiescent state and are always inactive during the computations. These authors use a strategy thatdepends on making all cells inside the zone T active. Using the quiescent cells from this zone theauthors successfully reduced the number of states by reusing the Q states to help the r- signal (Zstate) find the center of the array.Finally, note that the result presented in this paper is an improvement of Yun`e’s 7-state solution.Further, the 6-state solution is the smallest one known at present in the class of non-trivial 3n-stepsynchronization algorithms. The authors also achieved a increase in the number of working cellsfrom O ( nlogn ) to O ( n Here Umeo and Yanagihara [24] provide a partial 5-state solution to the firing synchronizationproblem. Using a 3n-step algorithm, this solution can synchronize any one-dimensional cellulararray of length n = 2 k in 3n - 3 steps, for any positive integer k and with the initiator positionedon the right side. The first solution using this algorithm was from Fischer[4] who gave a 15-stateimplementation.This 5-state solution is a small but partial solution to the original problem, since it has some lim-itations regarding the length of the array n. Other authors like Settle and Simon[15] and Umeo[26]provided complete solutions to the generalized problem using the same kind of algorithm but witha higher number of states. Settle and Simon provided a solution using only 7 states[15], and Umeodid using 6 states[26]. Unlike this one in both solutions the initiator can be placed in any part ofthe array.The first two states of the solution are used to create the ripple drivers that enable the prop-agation of the b- signal at 1/3 speed (state S). The a- signal is also realized using the first twostates (state R). Every two steps the a- signal generates a 1/1 speed signal in state Q, this signal11s transmitted in the reverse direction. Using the reverse State Q, a third state S is added thatwill be responsible for the b- signal. Each ripple driver can be used to drive the propagation ofthe b-signal to its right neighbor. The three-step separation of two consecutive ripples enables theb-signal to propagate at 1/3 speed. Finally, state L is used for a reflected R-signal. The returnsignal propagates left at 1/1 speed. Any cell where the return signal passes remains in a quiescentstate. At time t=3n/2, the b-signal and the return signal meet at Cn/
Cn/
Cn/
Cn/
In this paper Yun`es[33] presents a solution to the FSP based on Wolfram’s rule 60 discussedin Section 3.2.1. He accomplishes his result using an algebraic approach instead of geometricalconstructions. This solution solves the problem on an infinite number of lines but not all possiblelines. Its state complexity is the lowest possible (4 states and 32 transitions).As mentioned before, running the rule 60 on a configuration where the left end cell is 1 (black)leads to something that looks like synchronization of lines of length which are powers of two, butit is not a solution to the problem. Yun`es points that whatever the solution, if it synchronizes aninfinite number of lines then for some N the synchronization can only appear at time 2 n − n for aline of length n > N . Then he designed the algorithm so that rule 60 ran sending a signal wave 1from the left most cell. When the leading 1 reaches the right end cell, another symmetric rule 60 islaunched. By doing that, a property of the global function is exploited so that the full interleavingof two basic configurations is reached and the synchronization appears naturally.This leads to the paper’s main theorem: There exists a 4-state solution to the firing squadsynchronization problem which synchronizes all lines of length a power of 2 in 2 n +1 − m = 2 n is 3 · m log (3) . They show that this theorem is optimal by proving that it is impossible to synchronizeany line of length n ≥ Umeo and Yanagihara present in this paper a partial solution to the firing synchronizationproblem with 5 states[25]. This solution only works with arrays of length n = 2 k , and it takes3 n − k is a positive integer. This solution uses a 3 n -step algorithm,which was explained in detail in Section 3.2.2. The advantage of the use of such algorithm is itssimplicity, which makes the approach easily understandable.The internal set of 5 states for this solution is represented by Q = { Q, L, R, S, F } . First theauthors used a 2-state implementation for the wave, which implements the a-signal and enables thefuture propagation of the b-signal. The 2-states are { Q, R } . The states { Q, R, S } implement thea- and b-signals which were explained in the subsection 3.2.2. The state L was used to implementthe search for the center cells. After all the center cells, for sections and subsections of the cellularspace, were found they could achieve the firing state F .12he solution described in this paper is proposed as not the smallest solution to the problem butinteresting in its own way. The authors achieved a 5-state partial solution, since this solution onlyworks for a certain set of arrays. Because of that we can consider it a small partial solution for theFSP. However, it is not the smallest, since a 4-state partial solution, presented in Section 3.5.3 alsoexists. In this section we summarize a survey article on the FSP and discuss an article that considers asolution for the FSP on a two-dimensional array. The two dimensional array is a cellular automatoncomposed of an array of m n cells. The state of any cell is not only influenced by the state of thecell on the both sides, but also the cells at north and south. Several synchronization algorithms ontwo-dimensional arrays have been proposed, including Grasselli [7], Kobayashi [8], Shinahr [17] andSzwerinski [18].
Umeo wrote a survey on solutions to the FSP that use a small number of states[19]. The solutionsdiscussed cover the FSP for one-dimensional arrays, two-dimensional arrays, multi-dimensionalarrays and the generalized FSP (GFSP).The first optimum-time solution to the FSP, developed by Goto in 1962 [6] had several thousandsof states. After that Waksman in 1966 [30] presented a 16-state optimum-time solution. After thatthere was a 8-state solution by Balzer in 1967 [1], a 7-state solution by Gerken in 1987 [5] andfinally a 6-state solution by Mazoyer in 1987 [10]. The GFSP has also been extensively studied,and the first optimum-time solution with a small number of states used 17. After that Varshavsky,Marakhovsky and Peschansky in 1970 [28], Szwerinski in 1982 [18], Settle and Simon in 2002 [15],Umeo, Hisaoka, Michisaka, Nishioka, and Maeda in 2002 [23] presented solution with 10 states and9 states. There is also a non-optimum-step GFSP solution by Umeo, Maeda, and Hongyo in 2006[22] that works with 6 states.The paper presents theorems that point that one-dimensional arrays need at least 2 n − The contribution of this paper [9] is a simple but efficient mapping scheme that enables theembedding any one-dimensional firing squad synchronization algorithm onto two dimensional arrayswithout introducing additional states. The paper gives a concise and small solution to this problem,where the rules from the 1D array can be easily converted to the 2D array.In the solution, the authors produce a series of conversion tables and rules to enable the em-bedding. First they split the m x n cells into groups g. This groups were formed by cells that wereon the same line when the m x n board is turned 45 degrees. Because now we have two array lines,we need a state w to be the right and left end state.13s there are more connections between the cells, to convert a cell from one dimension to twodimensions requires more rules. The authors provide a formula for the number of rules that atransformation (a, b, c) → d uses depending on the location on the grid. Although the firing synchronization problem has been studied for decades, there are still severalopen problems. We discuss open problems for the original problem on the one-dimensional array,for the generalized problem on the one-dimensional array where the initiator can be located in anycell, and for the ring.
The most significant open problem in the area is whether there exists a complete 5-state solutionto the FSP on a one-dimensional array.The smallest minimal-time solution so far for the original problem was created by Mazoyerwhich is a 6-state solution[10]. A 5-state solution would then optimal for the problem, since Balzershowed that there is no 4-state minimal-time full solution to the original problem[1], Which laterwas confirmed by Sanders [14]. Some partial 5-state solutions exist. Umeo gives a non-minimal-timepartial solution using 5 states[24].Yun`es has produced solutions based on Wolfram’s rules, finding that there exists a 4-statesolution to the FSP which synchronizes all lines of length a power of two[33]. He also proved thatthere is no 3-state solutions able to synchronize a line of length n ≥
5. Various 4-state partialsolutions as described by Yun`es can be found in Table 1.Since all the five-state solutions are partial solutions to the original problem, finding a completesolution would be a significant contribution to the area. It is still unknown whether it is easier totake an approach to the problem finding a minimal-time solution or a non-minimal-time solution.Evidence suggests that finding a non-minimal-time solution may lead to solutions with fewer states,as seen in Table 1.
Before 1998 the best solution for the minimal-time generalized problem was Mazoyer’s 10-statesolution[10]. Since then his work has been improved in 1999 by Settle to a 9-state solution[16].In 2010 Umeo, Kamikawa and Nishioka [20] presented a 8-state minimal-time solution which iscurrently the smallest minimal-time solution to the generalized problem.Work has also been done on finding non-minimal-time solutions to the generalized problem. In1999 Settle et al presented a 7-state non-minimal-time full solution to the generalized problem [16],which improves the previous minimal-time solution presented by the same authors by two states. In2006 Umeo, Maeda and Hongyo presented a 6-state non-minimal-time solution[22]. This evidencesuggests that non-minimal-time solutions may require fewer states than minimal-time solutions,although there is no proof of that claim. 14maller solutions to the generalized problem are still open problems. Considering that thesmallest solution has 6-states and that this number also applies to the original problem, bothversions of the problem still do not have a 5-state solution. The 6-state solution to the generalizedproblem, unlike the 6-state solution to the original problem, is not a minimal-time solution. Becauseof that, finding a 6-state minimal-time solution to the generalized problem is also an open problem.
There are some results on state lower bounds for the FSP on the ring. Berthiaume, Bittner,Perkovic, Settle and Simon showed that there is no 3-state full solution and no 4-state, symmetric,minimal-time full solution to the FSP for the ring[2]. Umeo, Kamikawa and Yun`es proved thatthere is no 3-state partial solution to the firing synchronization problem for the ring [21]. Thus it isopen whether or not there is an unrestricted 4-state minimal-time full solution to the FSP on thering.
Here we have summarized recent research on the firing synchronization problem, focusing pri-marily on results published between 1998 and 2015. We discussed results for the original problemon the one-dimensional array, the generalized problem where the initiator can be located in any cellof the array, the problem where the underlying network is a ring, and a paper which discusses howto modify one-dimensional solutions for multidimensional arrays. We also discuss the remainingopen problems for the original, generalized, and ring problems.