Exploring Millions of 6-State FSSP Solutions: the Formal Notion of Local CA Simulation
EExploring Millions of 6-State FSSP Solutions:the Formal Notion of Local CA Simulation
Tien Thao Nguyen and Luidnel Maignan
LACL, Université Paris-Est Créteil, France
Abstract.
In this paper, we come back on the notion of local simulationallowing to transform a cellular automaton into a closely related one withdifferent local encoding of information. This notion is used to exploresolutions of the Firing Squad Synchronization Problem that are minimalboth in time ( n − for n cells) and, up to current knowledge, also instates (6 states). While only one such solution was proposed by Mazoyersince 1987, 718 new solutions have been generated by Clergue, Verel andFormenti in 2018 with a cluster of machines. We show here that, startingfrom existing solutions, it is possible to generate millions of such solutionsusing local simulations using a single common personal computer. Keywords: cellular automata · automata minimization · firing squad synchro-nization problem. The Firing Squad Synchronization Problem (FSSP) was proposed by John My-hill in 1957. The goal is to find a single cellular automaton that synchronizes anyone-dimensional horizontal array of an arbitrary number of cells. More precisely,one consider that at initial time, all cells are inactive (i.e. in the quiescent state )except for the leftmost cell which is in the general (i.e. in the general state ). Onewants the evolution of the cellular automaton to lead all cells to transition toa special state (i.e. the synchronization or firing state ) for the first time at thesame time. This time t s is called the synchronization time and it is known thatits minimal possible value is n − where n is the number of cells.For this problem, many minimum-time solutions were proposed using dif-ferent approaches. As indicated in [12], the first one was proposed by Goto in1962 [12] with many thousands of states, followed by Waksman in 1966 [14],Balzer in 1967 [1], Gerken in 1987 [4], and finally Mazoyer in 1987 [8] who pre-sented respectively a 16-state, 8-state, 7-state and 6-state minimum-time solu-tion, with no further improvements since 1987. Indeed, Balzer [1] already showsthat there are no 4-state minimal-time solutions, latter confirmed by Sanders [11]through an exhaustive search and some corrections to Balzer’s work. Whetherthere exist any 5-state minimumal-time solution or not is still an open question. a r X i v : . [ n li n . C G ] M a y T.T. Nguyen and L. Maignan
Note that all these solutions use a “divide and conquer” strategy. Goto’ssolution were pretty complex with two types of divisions. The following onesused a “mid-way” division but Mazoyer’s 6-state solution uses for the first time a“two-third” type of division . In 2018, Clergue, Verel and Formenti [2] generated718 new 6-state solutions using an Iterated Local Search algorithm to explore thespace of 6-state solutions on a cluster of heterogeneous machines: 717 of thesesolutions use a “mid-way” division, and only one use a “two-third” division. In 2012, Maignan and Yunes proposed the methodology of cellular fields to de-scribed formally the high-level implementation of a CA, and also formally thegeneration of the “low-level” transition table. One of the expected benefits wasto have an infinite CA cleanly modularized into many cellular fields with clearsemantical proof of correctness together with a correctness-preserving reductionprocedure into a finite state CA [5] using a particular kind of cellular field, “re-ductions”. This is very similar to what happen in usual computer programmingwhere one writes in a high-level language then transform the code into assemblyusing a semantic-preserving transformation, i.e. a compiler. In 2014, they madeprecise a particular reduction of the infinite CA into 21 states [6,7].From theses works and concepts, two intertwined research directions emerge.One direction is to ask whether a reduction to fewer states is firstly possible,and secondly automatically generable, in the same spirit as compiler optimiza-tion, with the possible application of reducing further the 21 states. The seconddirection is to build a map of as many FSSP solutions as possible and study howthey relate through the notion of “reduction” introduced, with application thediscovery of techniques used in hand-made transition table and also the factori-sation of correctness proofs. In 2018, Maignan and Nguyen [9] exhibited a few ofthese relations and in particular the fact the infinite Maignan-Yunes CA couldbe reduced to the 8-state solutions of Noguchi [10].
The initial motivations whose to complete the “map of reductions” by includ-ing the 718 solutions into the picture. In particular, a quick look at the 718solutions gave to the authors the feeling that they could be grouped into equiv-alence classes using the notion of “reduction”. Also, inspired by the idea of localsearch and exploration through small modifications used in [2] to generate the718 solutions, the first author tried such search algorithms to generate “reduc-tions” of existing solutions rather than transition tables directly. Although theidea of local search is to navigate randomly in a landscape with few actual so-lutions, the discovered landscape of reductions has so many solutions that a“best-effort-exhaustive” exploration have been tried, leading to many millions of 6-states solutions. Also, this space is much more easily explored because of itsnice computational properties. See Figure 1c for a mid-way division, and Figure 1a for a two-third divisionillions of 6-State FSSP Solutions through Local CA Simulations 3
In Section 2, we define formally cellular automata, local simulations, FSSP solu-tions and related objects. In Section 3, we present nice properties relating theseobjects and allowing the search algorithm to save a huge amount of time. InSection 4, we describe the exploration algorithm and continue in Section 5 withsome experimental results and a small analysis of the 718 solutions. We concludein Section 6 with some formal and experimental futur work.
In this section, we define formally cellular automata, local mappings and FSSPsolutions in a way suitable to the current study. Some objects have “incomplete”counterpart manipulated during the exploration algorithm. The material here isa considerable re-organization of the material found in [9]. A cellular automaton α consists of a finite set of states Σ α , a setof initial configurations I α ⊆ Σ α Z and a partial function δ α : Σ α (cid:55)→ Σ α calledthe local transition function or local transition table . The elements of Σ α Z arecalled (global) configurations and those of Σ α are called local configurations .For any c ∈ I α , its space-time diagram D α ( c ) : N × Z → Σ α is defined as: D α ( c )( t , p ) = (cid:40) c ( p ) if t = 0 ,δ α ( l − , l , l ) if t > with l i = D α ( c )( t − , p + i ) . The partial function δ α is required to be such that all space-time diagrams have tobe totally defined. When D α ( c )( t , p ) = s, we say that, for the cellular automaton α and initial configuration c, the cell at position p has state s at time t. Definition 2. A family of space-time diagrams D consists of a set of states Σ D and an arbitrary set D ⊆ Σ D N × Z of space-time diagram. The local transitionrelation δ D ⊆ Σ D × Σ D of D is defined as: (( l − , l , l ) , l ) ∈ δ D : ⇔ ∃ ( d , t , p ) ∈ D × N × Z s.t. l ji = d ( t + j , p + i ) . We call D a deterministic family if its local transition relation is functional.
Definition 3.
Given a deterministic family D, its associated cellular automaton Γ D is defined as having the set of states Σ Γ D = Σ D , the set of initial configura-tions I Γ D = { d (0 , − ) | d ∈ D } , and the local transition function δ Γ D = δ D . Definition 4.
Given a cellular automaton α , its associated family of space-timediagram (abusively denoted) D α is defined as having the set of states Σ D α =Σ α , and the set of space-time diagram D α := { D α ( c ) | c ∈ I α } and is clearlydeterministic. Here, d (0 , − ) is the function from Z to S defined as d (0 , − )( p ) = d (0 , p ) . T.T. Nguyen and L. Maignan These inverse constructions shows that deterministic families and cellularautomata are two presentations of the same object. For practical purposes, it isalso useful to note that, since δ D has a finite domain, there are finite subsets of D that are enough to specify it completely. Theses two concepts are more easily pictured with space-time diagrams. Givena space-time diagram d ∈ S N × Z , we build a new one d (cid:48) by determining eachstate d (cid:48) ( t , p ) from a little cone (cid:104) d ( t − dt , p + dp ) | dt ∈ { , } , dp ∈ [[ − dt , + dt ]] (cid:105) in d . This cone is simply a state for t = 0 , and when d is generated by a cellularautomaton, this cone is entirely determined by (cid:104) d ( t − , p + dp ) | dp ∈ [[ − , (cid:105) for t ≥ . Since the set of all these triplets is exactly dom ( δ α ) , the followingdefinitions suffice for the current study. We call this a local mapping, becausethe new diagram is determined locally by the original one. When transforming adeterministic family, the result might not be deterministic, but if it is, we speakof a local simulation between two CA. Definition 5. A local mapping h from a CA α to a finite set S consists of twofunctions h z : { d (0 , x ) | ( d , x ) ∈ D α × Z } → S and h s : dom ( δ α ) → S. Definition 6.
Given a local mapping h from a CA α to a finite set S, we defineits associated family of diagrams Φ h = { h ( d ) | d ∈ D α } where:h ( d )( t , p ) = (cid:40) h z ( d (0 , p )) if t = 0 , h s ( l − , l , l ) if t > with l i = d ( t − p + i ) . Definition 7.
A local mapping h from a CA α to a finite set S whose associatedfamily of diagrams Φ h is deterministic is called a local simulation from α to Γ Φ h . Proposition 1.
Equivalently, a local simulation h from a CA α to a CA β isa local mapping from α to the set Σ β such that { h z ( c ) | c ∈ I α } = I β and forall ( c , t , p ) ∈ I α × N × Z , we have h s ( l − , l , l ) = l (cid:48) with l i = D α ( c )( t , p + i ) and l (cid:48) = D β ( h z ( c ))( t + 1 , p ) . The details of these formula are more easily seengraphically. ... ... t + 2 t + 2 t + 1 t + 1 t tt − t − ... ... . . . . . . p − p − p − p − p pp + 1 p + 1 p + 2 p + 2 . . . . . . l − l l l (cid:48) D α ( c ) D β ( h z ( c )) illions of 6-State FSSP Solutions through Local CA Simulations 5 A cellular automaton is
FSSP-candidate if there are four specialstates (cid:63) α , G α , Q α , F α ∈ Σ α , if I α = { n α | n ≥ } with n α being the FSSP initialconfiguration of size n, i.e. n ( p ) = (cid:63) α , G α , Q α , (cid:63) α if p is respectively p ≤ ,p = 1 , p ∈ [[2 , n ]] , and p ≥ n + 1 . Moreover, (cid:63) α must be the outside state , i.e. forany ( l − , l , l ) ∈ dom ( δ α ) , we must have δ ( l − , l , l ) = (cid:63) α if and only if l = (cid:63) α .Also, Q α must be a quiescent state so δ α ( Q α , Q α , Q α ) = δ α ( Q α , Q α , (cid:63) α ) = Q α . The (cid:63) α state is not really counted as a state since it represents cells thatshould be considered as non-existing. Therefore, a FSSP-candidate cellular au-tomaton α will be said to have s states when | Σ α \ { (cid:63) α } | = s , and m transitionswhen | dom ( δ α ) \ Σ α × { (cid:63) α } × Σ α | = m . Definition 9.
A FSSP-candidate cellular automaton α is a minimal-time FSSPsolution if for any size n, D α ( n )( t , p ) = F α if and only if t ≥ n − andp ∈ [[1 , n ]] . We are only concerned with minimal-time solutions but sometimessimply write FSSP solution , or solution for short.
Our global strategy to find new FSSP solutions is to build them from localsimulations of already existing FSSP solution α . Taking the previous definitionslitteraly could lead to the following procedure for a given local mapping h . First,generates as many space-time diagrams of D α . Secondly, use h to transform eachdiagram d ∈ D α into a new one h ( d ) , thus producing a sub-family of Φ h . At thesame time, build δ Φ h by collecting all local transitions appearing in each h ( d ) and check for determinism and correct synchronization. If every thing goes fine,we have a new FSSP solution β = Γ Φ h .Such a procedure is time-consuming. We show here useful properties thatreduces drastically this procedure to a few steps. In fact, the space-time diagramsof Φ h never needs to be computed, neither to build the local transition relation δ Φ h (Section 3.1), nor to check that Γ Φ h is an FSSP solution as showed in thissection (Section 3.2). When trying to construct a CA β from a CA α and a local mapping h fromthe families of space-time diagrams as suggested by the formal definitions, thereis huge amount of redundancy. All entries of the local transition relation δ Φ h appear many times in Φ h , each of them being produced from the same recurringpatterns in the space-time diagrams of α . In fact, it is more efficient to simplycollect these recurring patterns that we may call super local transitions , andwork from them without constructing Φ h at all. It is specially useful because weconsider a huge number of local mappings from a single CA α . T.T. Nguyen and L. Maignan
Definition 10.
For a given CA α , the super local transition table ∆ α consistsof two sets ( ∆ α ) z ⊆ Σ α and ( ∆ α ) s ⊆ Σ α × Σ α defined as: ( s − , s , s ) ∈ ( ∆ α ) z : ⇔ ∃ ( d , p ) ∈ D α × Z s.t. s i = d (0 , p + i ) , (( s − , s − , s , s , s ) , ( s − , s , s )) ∈ ( ∆ α ) s : ⇔ ∃ ( d , t , p ) ∈ D α × N × Z s.t. s ji = d ( t + j , p + i ) Once all these patterns collected, it is possible to construct the local transi-tion relation δ Φ h as specified in the following proposition. Proposition 2.
Let h be a local mapping from a CA α to a set S. The localtransition relation δ Φ h of the family of space-time diagram Φ h generated by hand the super local transition function ∆ α of α obey: (( l − , l , l ) , l ) ∈ δ Φ h ⇔ ∃ ( s − , s , s ) ∈ ( ∆ α ) z s.t. l i = h z ( s i ) and l = h s ( s i − , s i , s i +1 ) ∨ ∃ (( s − , s − , s , s , s ) , ( s − , s , s )) ∈ ( ∆ α ) s s.t. l ji = h s ( s ji − , s ji , s ji +1 ) We know have an efficient way to build the local transition relation δ Φ h . Whenit is functional, it determines a cellular automaton β = Γ Φ h . For our purpose,we need to test or ensure in some way that β is an FSSP solution. We first note that the constraints put by the FSSP on space-time diagramsinduces constraints on local simulations between FSSP solutions. So we canrestrict our attention to local mappings respecting these constraints as formalizedby the following definition and proposition.
Definition 11.
A local mapping h from a FSSP solution α to the states Σ β ofa FSSP-candidate CA β is said to be FSSP-compliant if it is such that (0) h z maps (cid:63) α , G α , and Q α respectively to (cid:63) β , G β , and Q β , (1) h s ( l − , l , l ) = (cid:63) β ifand only if δ α ( l − , l , l ) = (cid:63) α (meaning simply l = (cid:63) α ), (2) h s ( l − , l , l ) = F β if and only if δ α ( l − , l , l ) = F α , and (3) h s ( Q α , Q α , Q α ) = h s ( Q α , Q α , (cid:63) ) = Q β . Proposition 3.
Let α be a FSSP solution CA, β a FSSP-candidate CA and h alocal simulation from α to β . If β is a FSSP solution, then h is FSSP-compliant. The following proposition is at the same time not difficult once noted, butextremely surprising and useful: the simple constraints above are also “totallycharacterizing” and the previous implication is in fact an equivalence. This meansin particular that it is not necessary to generate space-time diagrams to check ifa constructed CA is an FSSP solution, which saves lot of computations.
Proposition 4.
Let α be an FSSP solution, β a FSSP-candidate CA and h belocal simulation from α to β . If h is FSSP-compliant, then β is a FSSP solution. illions of 6-State FSSP Solutions through Local CA Simulations 7 In our actual algorithm, we take as input an existing FSSP solution α andfix a set of state S of size | Σ α | . The search space consists of all FSSP-compliant local mappings from α to S , the neighbors N ( h ) of a local map-ping h being all h (cid:48) that differs from h on exactly one entry, i.e. ∃ !( l − , l , l ) ∈ dom ( δ α ) s.t. h s ( l − , l , l ) (cid:54) = h (cid:48) s ( l − , l , l ) . More precisely, the mappings are con-sidered modulo bijections of S . Indeed, two mappings h and h (cid:48) are consideredequivalent if there is some bijection r : S → S such that h z = r ◦ h (cid:48) z and h s = r ◦ h (cid:48) s . So the search space is, in a sense, made of equivalence classes, eachclass being represented by a particular element. This element is chosen to be theonly mapping h in the class such that h s is monotonic according to arbitrarytotal orders on dom ( δ α ) and S fixed for the entire run of the algorithm.Considering 6-states solutions, let us denote Σ α = { (cid:63) α , G α , Q α , F α , A α , B α , C α } and S = { (cid:63), G , Q , F , A , B , C } . In each of these sets, four of the states are the specialFSSP solution states (Def. 8 and Def. 9). Only the three states A , B , C come withno constraints. We can thus evaluate the size of the search space by looking atthe degrees of freedom of FSSP-compliant local mappings (Def 11).Indeed, all FSSP-compliant local mappings h from α to S have the samepartial function h z , and the same value h s ( l − , l , l ) for those entries ( l − , l , l ) ∈ dom ( δ α ) forced to (cid:63) , Q or F . For all other entries ( l − , l , l ) , h s ( l − , l , l ) cannottake the values (cid:63) nor F , leaving 5 values available. So given an initial solution α , the number of local mappings is x where x is the size of dom ( δ α ) withoutthose entries constrained in Def. 11. To give an idea, for the solution 668 of the718 solutions, x = 86 to the size of the search has 61 digits, and for Mazoyer’ssolution, x = 112 leading to a number with 79 digits. As described in Section 3.1, the local mappings are evaluated from the superlocal transition table. To build this table, we generate, for each size n from 2to 5000, the space-time diagram D α (¯ n ) and collect all super local transitionsoccurring from time to n − and from position to n . Note that for allknown minimum-time 6-state solutions, no additional super local transitionsappear after n = 250 .The starting point of the exploration is the local mapping h α correspondingto the local transition function δ α itself, i.e. ( h α ) z = q (cid:22) { (cid:63) α , G α , Q α } and ( h α ) s = q ◦ δ α for some bijection q : Σ α → S . This local mapping is obviously FSSP-compliant since it is local simulation from α to α . To explain the algorithm, let us first consider the last parameter k to be , sothat line 7 of the explore algorithm can be considered to be simply S ← N ( h ) . Recall that we do not count the (cid:63) states. T.T. Nguyen and L. Maignan
Algorithm 1: explore( ∆ α , h α , k ) H ← { h α } H current ← { h α } while | H current | > do H next ← {} for h ∈ H current do S , H ← pertN (∆ α , H , h , k ) for h (cid:48) ∈ ( S \ H ) do if isSimul ( h (cid:48) , ∆ α ) then H next ← H next ∪ { h (cid:48) } H ← H ∪ { h (cid:48) } end end end H current ← H next end return H Algorithm 2: pertN (∆ α , H , h , k ) S ← N ( h ) h (cid:48) ← perturbation ( h , k ) if h (cid:48) (cid:54)∈ H then S ← S ∪ N ( h (cid:48) ) if isSimul ( h (cid:48) , ∆ α ) then H ← H ∪ { h (cid:48) } end end return S , H In this case, the algorithm starts with h α , and explores its neighbors to collect alllocal simulations, then the neighbors of those local simulations to collect morelocal simulations, and so on so forth until the whole connected components ofthe sub-graph consisting only of the local simulations in collected.More precisely, the variable H collects all local simulations, H current containsthe simulation discovered in the previous round and whose neighbors should beexamined in current round, and the newly discovered local simulations are putin H next for the next round. The function isSimul uses the super local transitiontable to construct the local transition relation of Φ h and check if it is functional, i.e. if it is a local transition function of a valid CA Γ Φ h . By our construction, avalid CA is necessarily an FSSP solution making this operation really cheap.When k > , the neighborhood operation is altered to add more neighbors.A local mapping obtained by k modifications is considered and its neighborhoodis added to the original the normal neighborhood, in the hope of discoveringanother connected component of the local simulation subgraph. As mentioned in the introduction, this study began by the desire to analyse the718 solutions found in [2]. These solutions, numbered from 0 to 717, are freelyavailable online. We tried to search local simulation relation between them asdone in [9]. Firstly, we found a slight mistake since there are 12 pairs of equivalentsolutions up to renaming of states: (105, 676), (127, 659), (243, 599), (562, 626),(588, 619), (601, 609), (603, 689), (611, 651), (629, 714), (663, 684), (590, 596) illions of 6-State FSSP Solutions through Local CA Simulations 9 and (679, 707). This means that there are really 706 solutions, but we still referto them as the 718 solutions with their original numbering.Once local simulation relation established between the 718 solutions, we an-alyzed in the number of connected components and found 193 while expectingonly a few. When there is a local simulation h from a CA α and a CA β , thenumber of differences between h α and h varies a lot, but the median value is 3. To find more FSSP solutions, we implemented many algorithms, gradually sim-plifying them into the one presented in this paper. It has been run on an UbuntuMarvin machine with 32 cores of 2.00GHz speed and 126Gb of memory. However,the implementation being sequential, only two cores was used by the program.The original plan was to generate as many solutions as possible but we hadsome problems with the management of quotas in the shared machine. So weonly expose the some selected data to show the relevance of the approach.When running the program with the solution 355 and k = 0 , the programused 14Gb of memory and stopped after 27.5 hours and found 9,584,134 localsimulations! A second run of the program for this solution with k = 3 found11,506,263 local simulations after 80.5 hours. This indicates that perturbationsare useful but the second run find only 1922129 additional local simulations butits computing time is three times more than the first run. Testing whether alocal simulation belongs to set H obviously takes more and more times as morelocal mappings are discovered but there might be some understanding to gainabout the proper mapping landscape too in order to improve the situation.The transition table for the original Mazoyer’s solution can be found in [8],but also in [13] together with other minimal-time solution transition table. Whenrunning the program of the original Mazoyer’s solution with different values of k with obtained the following number of new solutions for different runs. Thebehavior with k = 1 seems to be pretty robust, but the bigger number of resultsis obtained with k = 2 .k number of solutions found by 10 different runs0 , 9538, 20626, 20682, 20054, 204902 9451, 9451, 20595, 8241, , 3817, 17421, 8241, 17317, 198953 644, 644, 644, 644, 644, 644, 644, 731, 8241, n − , illustrating how local simulationrearrange locally the information. The identical part in represented with lightercolors to highlight the differences. Proposition 5.
There are at least many millions of minimum-time 6-state FSSPsolutions.
This paper presents only a small part of many ongoing experimentations. Thenotion of local simulation presented here is just a particular case of the notion ofcellular field that can be used more broadly to investigate these questions. Forexample, we relate here only the small cones { d ( t − dt , p + dp ) | dt ∈ { , } , dp ∈ [[ − dt , + dt ]] } of any space-time diagram d in local mappings. If we increase therange of dt in this definition to be [[0 , h ]] for some h > , we allow CA to betransformed to a bigger extent.Another justification for this extension is that the composition of two localsimulations is not a local simulation. In fact, composing an h -local simulationwith an h (cid:48) -local simulation produces an ( h + h (cid:48) ) -local simulation in general. A0-local simulation is just a (possibly non-injective) renaming of the states.Note that since local mappings of local mappings are not local mappings,running the above algorithm on new found solutions should a priori generatemore solutions! Of course, a more exhaustive study is required.Our guess is that, with a properly large notion of such simulations, it shouldbe possible to classify the 718 solutions into only a few equivalence classes,more or less in two groups: the “mid-way division” solutions and the ”two-third division” solutions. This results also represents an important step in theunderstanding of automatic optimization of CA.Finally, the content of Section 3.2 about the preservation of correctness byFSSP-compliant local simulation is really interesting because of the simplicityof checking FSSP-compliance. It implies that a proof of correctness of a smallFSSP solution can indeed be made on some huge, possibly infinite, simulatingCA where everything is explicit as considered in [6,7]. This can be applied toease the formal proof of correctness of Mazoyer’s solution. Up to our knowledge,it is known to be long and hard but also to be the only proof to be precise enoughto actually be implemented in the Coq Proof Assistant [3].We would like to give special thanks to Jean-Baptiste Yunès who pointed usthe 718 solutions paper. If we are right, he also partly inspired the work wholead to the 718 solutions by a discussion during a conference. References
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Appl.Soft Comput. , 66:449–461, 2018.3. Jean Duprat. Proof of correctness of the Mazoyer’s solution of the firing squadproblem in Coq. Research Report LIP RR-2002-14, Laboratoire de l’informatiquedu parallélisme, March 2002.illions of 6-State FSSP Solutions through Local CA Simulations 11 ? G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A C G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A C G C G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B B B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A C G C C C G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B G Q B B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A C G C C C C C G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B G Q Q Q B B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A C G C G Q Q C C C G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B B B G C B Q B B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A C G C G Q A C Q C C C G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B B B Q A B C B Q B B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A C G C G C G C C C Q C C C G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B B B B B G Q B C B Q B B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A C G C C C G C C C C C Q C C C G C G Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B G Q B B G Q Q Q B C B Q B B B B G Q Q Q Q Q Q Q Q Q Q Q ? ? A C G C C C G C G Q Q C C C Q C C C G C G Q Q Q Q Q Q Q Q Q Q ? ? G A B G Q B B B B G C B Q B C B Q B B B B G Q Q Q Q Q Q Q Q Q ? ? A C G C C C C C G Q A C Q C C C Q C C C G C G Q Q Q Q Q Q Q Q ? ? G A B G Q Q Q B B Q A B C B Q B C B Q B B B B G Q Q Q Q Q Q Q ? ? A C G C G Q Q C G C G C C C Q C C C Q C C C G C G Q Q Q Q Q Q ? ? G A B B B G C Q B B B G Q B C B Q B C B Q B B B B G Q Q Q Q Q ? ? A C G C G Q Q C C C G C C C C C Q C C C Q C C C G C G Q Q Q Q ? ? G A B B B G C B Q B B G Q Q Q B C B Q B C B Q B B B B G Q Q Q ? ? A C G C G Q A C Q C G C G Q Q C C C Q C C C Q C C C G C G Q Q ? ? G A B B B Q A B C Q B B B G C B Q B C B Q B C B Q B B B B G Q ? ? A C G C G C G C Q C C C G Q A C Q C C C Q C C C Q C C C G C A ? ? G A B B B B B Q C B Q B B Q A B C B Q B C B Q B C B Q B B A G ? ? A C G C C C G C C C Q C G C G C C C Q C C C Q C C C Q C B C A ? ? G A B G Q B B G Q B C Q B B B G Q B C B Q B C B Q B C C B A G ? ? A C G C C C G C C C Q C C C G C C C C C Q C C C Q C B G B C A ? ? G A B G Q B B G Q B C B Q B B G Q Q Q B C B Q B C C B G B A G ? ? A C G C C C G C C C C C Q C G C G Q Q C C C Q C B G B G B C A ? ? G A B G Q B B G Q Q Q B C Q B B B G C B Q B C C B G B G B A G ? ? A C G C C C G C G Q Q C Q C C C G Q A C Q C B G B G B G B C A ? ? G A B G Q B B B B G C Q C B Q B B Q A B C C B G B G B G B A G ? ? A C G C C C C C G Q Q C C C Q C G C G C B G B G B G B G B C A ? ? G A B G Q Q Q B B G C B Q B C Q B B B A B G B G B G B G B A G ? ? A C G C G Q Q C G Q A C Q C Q C C C B C G G B G B G B G B C A ? ? G A B B B G C Q B Q A B C Q C B Q G B A B B B G B G B G B A G ? ? A C G C G Q Q C G C G C Q C C C B G B C G C G G B G B G B C A ? ? G A B B B G C Q B B B Q C B Q G B G B A B B B B B G B G B A G ? ? A C G C G Q Q C C C G C C C B G B G B C G C C C G G B G B C A ? ? G A B B B G C B Q B B G Q G B G B G B A B G Q B B B B G B A G ? ? A C G C G Q A C Q C G C A G B G B G B C G C C C C C G G B C A ? ? G A B B B Q A B C Q B A G A B G B G B A B G Q Q Q B B B B A G ? ? A C G C G C G C Q C B C A C G G B G B C G C G Q Q C C C B C A ? ? G A B B B B B Q C C B A G A B B B G B A B B B G C B Q G B A G ? ? A C G C C C G C B G B C A C G C G G B C G C G Q A C B G B C A ? ? G A B G Q B B A B G B A G A B B B B B A B B B Q A Q B G B A G ? ? A C G C C C B C G G B C A C G C C C B C G C G C B C G G B C A ? ? G A B G Q G B A B B B A G A B G Q G B A B B B A B A B B B A G ? ? A C G C A G B C G C B C A C G C A G B C G C B C A C G C B C A ? ? G A B A G A B A B A B A G A B A G A B A B A B A G A B A B A G ? ? A C A C A C A C A C A C A C A C A C A C A C A C A C A C A C A ? ? G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G ? ? F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F ? (a) original solution 668: 93 rules ? G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A A G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A A G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A A G C G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B B B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A A G C C C G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B G Q B B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A A G C C C C C G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B G Q Q Q B B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A A G C G Q Q C C C G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B B B G C B Q B B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A A G C G Q A C Q C C C G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B B B G G Q C B Q B B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A A G C G C G C C C Q C C C G C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B B B B B G Q B C B Q B B B B G Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A A G C C C G C C C C C Q C C C G C G Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A B G Q B B G Q Q Q B C B Q B B B B G Q Q Q Q Q Q Q Q Q Q Q ? ? A A G C C C G C G Q Q C C C Q C C C G C G Q Q Q Q Q Q Q Q Q Q ? ? G A B G Q B B B B G C B Q B C B Q B B B B G Q Q Q Q Q Q Q Q Q ? ? A A G C C C C C G Q A C Q C C C Q C C C G C G Q Q Q Q Q Q Q Q ? ? G A B G Q Q Q B B G G Q C B Q B C B Q B B B B G Q Q Q Q Q Q Q ? ? A A G C G Q Q C G C G C C C Q C C C Q C C C G C G Q Q Q Q Q Q ? ? G A B B B G C Q B B B G Q B C B Q B C B Q B B B B G Q Q Q Q Q ? ? A A G C G Q Q C C C G C C C C C Q C C C Q C C C G C G Q Q Q Q ? ? G A B B B G C B Q B B G Q Q Q B C B Q B C B Q B B B B G Q Q Q ? ? A A G C G Q A C Q C G C G Q Q C C C Q C C C Q C C C G C G Q Q ? ? G A B B B G G Q C Q B B B G C B Q B C B Q B C B Q B B B B G Q ? ? A A G C G C G C Q C C C G Q A C Q C C C Q C C C Q C C C G C A ? ? G A B B B B B Q C B Q B B G G Q C B Q B C B Q B C B Q B B A A ? ? A A G C C C G C C C Q C G C G C C C Q C C C Q C C C Q C B A A ? ? G A B G Q B B G Q B C Q B B B G Q B C B Q B C B Q B C C B A A ? ? A A G C C C G C C C Q C C C G C C C C C Q C C C Q C B G B A A ? ? G A B G Q B B G Q B C B Q B B G Q Q Q B C B Q B C C B G B A A ? ? A A G C C C G C C C C C Q C G C G Q Q C C C Q C B G B G B A A ? ? G A B G Q B B G Q Q Q B C Q B B B G C B Q B C C B G B G B A A ? ? A A G C C C G C G Q Q C Q C C C G Q A C Q C B G B G B G B A A ? ? G A B G Q B B B B G C Q C B Q B B G G Q C C B G B G B G B A A ? ? A A G C C C C C G Q Q C C C Q C G C G C B G B G B G B G B A A ? ? G A B G Q Q Q B B G C B Q B C Q B B B A B G B G B G B G B A A ? ? A A G C G Q Q C G Q A C Q C Q C C C B A G G B G B G B G B A A ? ? G A B B B G C Q B G G Q C Q C B Q G B A B B B G B G B G B A A ? ? A A G C G Q Q C G C G C Q C C C B G B A G C G G B G B G B A A ? ? G A B B B G C Q B B B Q C B Q G B G B A B B B B B G B G B A A ? ? A A G C G Q Q C C C G C C C B G B G B A G C C C G G B G B A A ? ? G A B B B G C B Q B B G Q G B G B G B A B G Q B B B B G B A A ? ? A A G C G Q A C Q C G C A G B G B G B A G C C C C C G G B A A ? ? G A B B B G G Q C Q B A G A B G B G B A B G Q Q Q B B B B A A ? ? A A G C G C G C Q C B A A A G G B G B A G C G Q Q C C C B A A ? ? G A B B B B B Q C C B A G A B B B G B A B B B G C B Q G B A A ? ? A A G C C C G C B G B A A A G C G G B A G C G Q A C B G B A A ? ? G A B G Q B B A B G B A G A B B B B B A B B B G G B B G B A A ? ? A A G C C C B A G G B A A A G C C C B A G C G C B A G G B A A ? ? G A B G Q G B A B B B A G A B G Q G B A B B B A B A B B B A A ? ? A A G C A G B A G C B A A A G C A G B A G C B A A A G C B A A ? ? G A B A G A B A B A B A G A B A G A B A B A B A G A B A B A A ? ? A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A ? ? G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G A ? ? F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F ? (b) a local simulation of 668: 90 rules ? G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A C C C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A C G Q C C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q Q Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A C G Q Q Q C C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q G Q C Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A C C C Q Q Q Q C C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q G Q Q Q C Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A C C C G Q C Q Q Q C C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q G Q Q Q Q Q C Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A C C C G Q Q Q C Q Q Q C C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q G Q G Q C Q Q Q C Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q ? ? G A C C C C C Q Q Q Q C Q Q Q C C C C G Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q Q Q G Q Q Q C Q Q Q C Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q ? ? G A C G Q C C G Q C Q Q Q C Q Q Q C C C C G Q Q Q Q Q Q Q Q Q ? ? A Q G Q Q Q G Q Q Q Q Q C Q Q Q C Q Q Q G Q G Q Q Q Q Q Q Q Q ? ? G A C G Q C C G Q Q Q C Q Q Q C Q Q Q C C C C G Q Q Q Q Q Q Q ? ? A Q G Q Q Q G Q G Q C Q Q Q C Q Q Q C Q Q Q G Q G Q Q Q Q Q Q ? ? G A C G Q C C C C Q Q Q Q C Q Q Q C Q Q Q C C C C G Q Q Q Q Q ? ? A Q G Q Q Q Q Q G Q Q Q C Q Q Q C Q Q Q C Q Q Q G Q G Q Q Q Q ? ? G A C G Q Q Q C C G Q C Q Q Q C Q Q Q C Q Q Q C C C C G Q Q Q ? ? A Q G Q G Q C Q G Q Q Q Q Q C Q Q Q C Q Q Q C Q Q Q G Q G Q Q ? ? G A C C C Q Q C C G Q Q Q C Q Q Q C Q Q Q C Q Q Q C C C C G Q ? ? A Q G Q G Q C Q G Q G Q C Q Q Q C Q Q Q C Q Q Q C Q Q Q G Q A ? ? G A C C C Q Q C C C C Q Q Q Q C Q Q Q C Q Q Q C Q Q Q C C A G ? ? A Q G Q G Q C Q Q Q G Q Q Q C Q Q Q C Q Q Q C Q Q Q C Q B Q C ? ? G A C C C Q Q Q Q C C G Q C Q Q Q C Q Q Q C Q Q Q C Q B C A G ? ? A Q G Q G Q Q Q C Q G Q Q Q Q Q C Q Q Q C Q Q Q C Q B B B Q C ? ? G A C C C G Q C Q C C G Q Q Q C Q Q Q C Q Q Q C Q B C C C A G ? ? A Q G Q G Q Q Q C Q G Q G Q C Q Q Q C Q Q Q C Q B B B Q B Q C ? ? G A C C C G Q C Q C C C C Q Q Q Q C Q Q Q C Q B C C C C C A G ? ? A Q G Q G Q Q Q C Q Q Q G Q Q Q C Q Q Q C Q B B B Q Q Q B Q C ? ? G A C C C G Q C Q Q Q C C G Q C Q Q Q C Q B C C C B Q B C A G ? ? A Q G Q G Q Q Q Q Q C Q G Q Q Q Q Q C Q B B B Q Q C C B B Q C ? ? G A C C C G Q Q Q C Q C C G Q Q Q C Q B C C C B C Q Q B C A G ? ? A Q G Q G Q G Q C Q C Q G Q G Q C Q B B B Q Q C A Q B B B Q C ? ? G A C C C C C Q Q C Q C C C C Q Q B C C C B C C G B C C C A G ? ? A Q G Q Q Q G Q C Q C Q Q Q G Q B B B Q Q C B G A A B Q B Q C ? ? G A C G Q C C Q Q C Q Q Q C C A C C C B C C C B Q B C C C A G ? ? A Q G Q Q Q G Q C Q Q Q C Q B Q G Q Q C B Q Q C C B B Q B Q C ? ? G A C G Q C C Q Q Q Q C Q B C A C G C C C B C Q Q B C C C A G ? ? A Q G Q Q Q G Q Q Q C Q B B B Q G Q G Q Q C A Q B B B Q B Q C ? ? G A C G Q C C G Q C Q B C C C A C C C G C C G B C C C C C A G ? ? A Q G Q Q Q G Q Q Q B B B Q B Q G Q G Q G G A A B Q Q Q B Q C ? ? G A C G Q C C G Q B C C C C C A C C C C C G Q B C B Q B C A G ? ? A Q G Q Q Q G Q A B B Q Q Q B Q G Q Q Q G Q A B B C C B B Q C ? ? G A C G Q C C A G A C B Q B C A C G Q C C A G A C B Q B C A G ? ? A Q G Q Q Q B Q A Q G C C B B Q G Q Q Q B Q A Q G C C B B Q C ? ? G A C G Q B C A G A C G Q B C A C G Q B C A G A C G Q B C A G ? ? A Q G Q A B B Q A Q G Q A B B Q G Q A B B Q A Q G Q A B B Q C ? ? G A C A G A C A G A C A G A C A C A G A C A G A C A G A C A G ? ? A Q A Q A Q A Q A Q A Q A Q A Q A Q A Q A Q A Q A Q A Q A Q C ? ? G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G ? ? F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F ? (c) original solution 355 ? G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? G G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? Q A C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? Q A C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? Q A C C C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? Q A C G Q C C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q Q Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? Q A C G Q Q Q C C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q G Q C Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? Q A C C C Q Q Q Q C C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q G Q Q Q C Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? Q A C C C G Q C Q Q Q C C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q G Q Q Q Q Q C Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? Q A C C C G Q Q Q C Q Q Q C C C C G Q Q Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q G Q G Q C Q Q Q C Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q Q Q ? ? Q A C C C C C Q Q Q Q C Q Q Q C C C C G Q Q Q Q Q Q Q Q Q Q Q ? ? A Q G Q Q Q G Q Q Q C Q Q Q C Q Q Q G Q G Q Q Q Q Q Q Q Q Q Q ? ? Q A C G Q C C G Q C Q Q Q C Q Q Q C C C C G Q Q Q Q Q Q Q Q Q ? ? A Q G Q Q Q G Q Q Q Q Q C Q Q Q C Q Q Q G Q G Q Q Q Q Q Q Q Q ? ? Q A C G Q C C G Q Q Q C Q Q Q C Q Q Q C C C C G Q Q Q Q Q Q Q ? ? A Q G Q Q Q G Q G Q C Q Q Q C Q Q Q C Q Q Q G Q G Q Q Q Q Q Q ? ? Q A C G Q C C C C Q Q Q Q C Q Q Q C Q Q Q C C C C G Q Q Q Q Q ? ? A Q G Q Q Q Q Q G Q Q Q C Q Q Q C Q Q Q C Q Q Q G Q G Q Q Q Q ? ? Q A C G Q Q Q C C G Q C Q Q Q C Q Q Q C Q Q Q C C C C G Q Q Q ? ? A Q G Q G Q C Q G Q Q Q Q Q C Q Q Q C Q Q Q C Q Q Q G Q G Q Q ? ? Q A C C C Q Q C C G Q Q Q C Q Q Q C Q Q Q C Q Q Q C C C C G Q ? ? A Q G Q G Q C Q G Q G Q C Q Q Q C Q Q Q C Q Q Q C Q Q Q G Q A ? ? Q A C C C Q Q C C C C Q Q Q Q C Q Q Q C Q Q Q C Q Q Q C C A B ? ? A Q G Q G Q C Q Q Q G Q Q Q C Q Q Q C Q Q Q C Q Q Q C Q B Q A ? ? Q A C C C Q Q Q Q C C G Q C Q Q Q C Q Q Q C Q Q Q C Q B C A B ? ? A Q G Q G Q Q Q C Q G Q Q Q Q Q C Q Q Q C Q Q Q C Q B B B Q A ? ? Q A C C C G Q C Q C C G Q Q Q C Q Q Q C Q Q Q C Q B C C C A B ? ? A Q G Q G Q Q Q C Q G Q G Q C Q Q Q C Q Q Q C Q B B B Q B Q A ? ? Q A C C C G Q C Q C C C C Q Q Q Q C Q Q Q C Q B C C C C C A B ? ? A Q G Q G Q Q Q C Q Q Q G Q Q Q C Q Q Q C Q B B B Q Q Q B Q A ? ? Q A C C C G Q C Q Q Q C C G Q C Q Q Q C Q B C C C B Q B C A B ? ? A Q G Q G Q Q Q Q Q C Q G Q Q Q Q Q C Q B B B Q Q C C B B Q A ? ? Q A C C C G Q Q Q C Q C C G Q Q Q C Q B C C C B C Q Q B C A B ? ? A Q G Q G Q G Q C Q C Q G Q G Q C Q B B B Q Q C A Q B B B Q A ? ? Q A C C C C C Q Q C Q C C C C Q Q B C C C B C C G B C C C A B ? ? A Q G Q Q Q G Q C Q C Q Q Q G Q B B B Q Q C B G B A B Q B Q A ? ? Q A C G Q C C Q Q C Q Q Q C C A C C C B C C C B Q B C C C A B ? ? A Q G Q Q Q G Q C Q Q Q C Q B Q G Q Q C B Q Q C C B B Q B Q A ? ? Q A C G Q C C Q Q Q Q C Q B C A C G C C C B C Q Q B C C C A B ? ? A Q G Q Q Q G Q Q Q C Q B B B Q G Q G Q Q C A Q B B B Q B Q A ? ? Q A C G Q C C G Q C Q B C C C A C C C G C C G B C C C C C A B ? ? A Q G Q Q Q G Q Q Q B B B Q B Q G Q G Q G G B A B Q Q Q B Q A ? ? Q A C G Q C C G Q B C C C C C A C C C C C G Q B C B Q B C A B ? ? A Q G Q Q Q G Q A B B Q Q Q B Q G Q Q Q G Q A B B C C B B Q A ? ? Q A C G Q C C A A A C B Q B C A C G Q C C A A A C B Q B C A B ? ? A Q G Q Q Q B Q A Q G C C B B Q G Q Q Q B Q A Q G C C B B Q A ? ? Q A C G Q B C A A A C G Q B C A C G Q B C A A A C G Q B C A B ? ? A Q G Q A B B Q A Q G Q A B B Q G Q A B B Q A Q G Q A B B Q A ? ? Q A C A A A C A A A C A A A C A C A A A C A A A C A A A C A B ? ? A Q A Q A Q A Q A Q A Q A Q A Q A Q A Q A Q A Q A Q A Q A Q A ? ? Q G A G A G A G A G A G A G A G A G A G A G A G A G A G A G B ? ? F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F ? (d) a local simulation of 355 Fig. 1: Some FSSP space-time diagrams of size 31
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