A Language for Particle Interactions in One-dimensional Cellular Automata
AA Language for Particle Interactions inRule 54 and Other Cellular Automata
Markus Redeker
Hamburg, [email protected]
This is a study of localised structures in one-dimensional cellular au-tomata, with the elementary cellular automaton Rule 54 as a guidingexample.A formalism for particles on a periodic background is derived, ap-plicable to all one-dimensional cellular automata. One can computewhich particles collide and in how many ways. One can also computethe fate of a particle after an unlimited number of collisions – whetherthey only produce other particles, or the result is a growing structurethat destroys the background pattern.For Rule 54, formulas for the four most common particles are givenand all two-particle collisions are found. We show that no other parti-cles arise, which particles are stable and which can be created, providedthat only two particles interact at a time. More complex behaviour ofRule 54 requires therefore multi-particle collisions.
1. Introduction
This article is part of a project to develop a higher-level language forthe dynamical behaviour of cellular automata. In the current investi-gation we search for an intermediate-level description of the elementarycellular automaton Rule 54, in order to learn how to handle periodicbackground structures and simple particle interactions. The investi-gation leads to further streamlining and an extension of the existingformalism. The formalism is called
Flexible Time . It was introduced in [18] andfurther developed in [20]. Flexible Time makes it possible to “calculate”with the localised structures in a cellular automaton and to determinetheir development over time. The structures in Flexible Time resemblethe way in which a human observer views an evolution diagram of acellular automaton (like Figure 1): by grouping the states of cells fromdifferent times and places to a single pattern in space-time.Rule 54 is an elementary cellular automaton that was first inves- This article started as an extension of [19], but has now grown considerablyand is completely rewritten.
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Figure 1.
Development of a random initial configuration under Rule 54. Timeruns from bottom to top. tigated in detail by Boccara et al. [1]. When evolving from randominitial configurations, it develops a simple background pattern with asmall number of interacting particles that move on this background.While it has not been shown to be computationally universal, it can atleast evaluate Boolean expressions [10]. So it is a rather simple system(but not too simple) and therefore an ideal test object for a formalismthat is still under development.The right methods to handle large complex structures must still befound. I ask here new questions about the behaviour of Rule 54, andFlexible Time must “learn” how to handle them. As a result, there aredifferences and extensions of the formalism in this article that were notpresent in [20]. I will point them out and review them at the end.
Context
Researchers on cellular automata have developed a numberof concepts to describe the localised structures that arise in a cellularautomaton.The oldest named structures must be the particles (also called glid-ers or signals) and their collisions. This goes back at least to Zuse[23], whose cellular automaton simulates idealised physical particles.Particle-based research has continued since then, with Cook’s con-struction of a universal computer in Rule 110 as its most spectacularresult [2].This rule, and Rule 54, belonged also to those rules in which a stableperiodic background pattern occured; it was called “ether” by Cook.For Rule 54, the starting point was the work by Boccara et al. [1]; they identified the most common particles that arise from ran-dom initial configurations, described their interactions and gave them
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Language for Particle Interactions the names that are still used. This research was later continued by thegroup around McIntosh [10, 11, 13], who found more complex particlesand interactions.The descriptions of these particles were mostly given by pictures andby a simple symbolism that showed which particle collides with which.But, especially to find general theorems about cellular automata, moreabstract representations were developed too.There is a more detailed investigation of particles and what they canachieve [3, 14]. For Rule 110 there is an approach for the systematicspecification of initial configurations with interacting gliders [13], andto express the behaviour of the cellular automaton through a blocksubstitution system [21].There are also the approaches by Hordijk et al. [6] and by Martin[9], who use properties of the background and draw conclusions aboutthe particles and particle interactions that are possible. More gen-eral, the cellular automaton is subdivided in “regular” regions and theboundaries between them [4, 5, 7, 8, 17]; the boundaries move, often inan almost random fashion, and are thus a generalisation of the morestraight-moving particles.Other approaches view the evolution of the cellular automaton astwo-dimensional, with one space and one time dimension. The cellularspace-time is then subdivided into finite patches that represent e. g. apiece of the background or a collision between particles. The theoryof cellular automata then becomes a special tiling problem. One cando this in a more informal way, like McIntosh and Martínez [12], ordevelop a complex formal theory around it, as Ollinger and Richard[15, 16] do it. (This approach is closest to the work described here.) Overview
After an introductory section about cellular automata andRule 54, Section 3 recapitulates the work in [20], as far as it is relevantfor the present work. At its end, a representation of Rule 54 as a“reaction system” (defined below) is shown, the same that was derivedin [20]. In Section 4 we then find a way to compress this and similarsystems, and we use the compressed reaction system to understandthe local behaviour of Rule 54 better. Section 5 then turns to largerpatterns and describes the triangular structures in Rule 54 and thestable background pattern that is formed by them. Then, in Section 6,the four kinds of particles found by Boccara et al. [1] are representedin Flexible Time, together with the collision between the particles. Asummary follows in Section 7.
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2. Cellular automata and Rule 54
Rule 54 is an one-dimensional cellular automaton, more specificallyan elementary cellular automaton . This kind of cellular automata wasmade popular by Stephen Wolfram [22], who also introduced the systemof code numbers from which Rule 54 got its name.One-dimensional cellular automata are dynamical systems with dis-crete time. The state of such an automaton is called a configuration .It consists of an infinite sequence of simpler objects, the cells . Thestate of each cell is an element of a finite set Σ ; the configuration attime t is therefore a function c t : Z → Σ . We write Σ Z for the set ofconfigurations; c t ( x ) is then the state of the cell at position x at time t .The evolution of the automaton is then a sequence ( c , c , c , . . . ) of configurations that follow a common rule that is described belowin (2). While the sequence here starts at time 0, we will also acceptother starting times.)An elementary cellular automaton is a one-dimensional cellular au-tomaton with two states and a three-cell neighbourhood. The set ofstates is Σ = { , } , and its behaviour is given by its local transitionrule ϕ : Σ → Σ . (1)This is the function with which the configuration c t +1 is computedfrom its predecessor c t . To do this, we apply ϕ to every three-cellneighbourhood of c t , and the result is the next state of the middle cell: c t +1 ( x ) = ϕ ( c t ( x − , c t ( x ) , c t ( x + 1)) for all t , x ∈ Z . (2)The function ϕ defines then a global transition rule ϕ global : It is thefunction that maps the configuration c t to its successor c t +1 accordingto (2).The transition rule (2) is also called a rule of radius
1, because onlythe c t ( y ) with | x − y | ≤ contribute to c t +1 ( x ) . Rules with other radiiare defined similarly. Rule 54 has a left-right symmetric transition rule, ϕ ( s ) = (cid:26) for s ∈ { (0 , , , (1 , , , (0 , , , (1 , , } , otherwise. (3)The rule is easier to remember in form of the following slogan [20], “ ϕ ( s ) = 1 if s contains at least one 1, except if the cells in state1 touch.” Complex Systems , Volume (year) 1–1+
Language for Particle Interactions Here we say that two cells “touch” if they are direct neighbours. Thusthe two cells in state 1 touch in the neighbourhood (1 , , , but not inthe neighbourhood (1 , , . Figure 2.
Rule icon for Rule 54.
Figure 2 shows how the neighbourhoods influence the next state ofthe central cell. White squares are in state 0, black squares are in state1, and the time runs upwards. This is also our convention in the otherdiagrams, even if white and black may also become dark and brightgrey in the parts of the diagram that are less emphasised.
3. Flexible Time
We need a means to describe and understand the interactions ofgliders and other patterns under Rule 54. Flexible Time was developedin [20] for this task. The motivation was that it is easier to find patternsin the evolution of cellular automata if one works with structures thatinvolve the states of cells at different times. These structures are calledhere situations .They generalise the finite sequences of cells that are part of the con-figurations c t described above. In order to express e. g. that c t (0) = c t (1) = 0 and c t (2) = 1 , one would often write that the subsequence of c t that begins at cell position 0 is 001. Situations generalise this nota-tion. They may extend not only over space but also over time. To writethem, we use additional symbols that express a jump in spacetime.Under Rule 54, situations are written as sequences of the symbols0, 1, (cid:9) i and ⊕ i , for i ∈ { , } . The intended interpretation can mosteasily be described in terms of instructions to write symbols on squaresin a grid. The squares are labelled by pairs ( t, x ) ∈ Z ; x is the positionof a cell and t a time step in its evolution. The writing rules are: Start reading at the first symbol. For writing, place the cursor atsquare (0 , of the grid.If the cursor is at ( t, x ) and the current symbol is – an element of Σ , write it down and move the cursor one squareto the right, to ( t, x + 1) , – (cid:9) i , move the cursor to ( t − , x − i ) , – ⊕ i , move the cursor to ( t + 1 , x − i ) . Complex Systems , Volume (year) 1–1+
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Then continue with the next symbol.
No overwriting : One cannot write different symbols on the samesquare.
To get an example for such a writing process, let us set for a moment
Σ = { , , , } and look at the situation ⊕ . First, the cell states0 and 1 are written to the squares (0 , and (0 , . The cursor is thenat square (0 , . Now the symbol ⊕ moves the cursor to (1 , . Thefollowing symbols 2 and 3 are then written to the squares (1 , and (1 , , leaving the cursor at (1 , . The result is then the following grid: t = 1 t = 0 x = -2 -1 0 1 2 3 4The horizontal rules mark the beginning and end of the symbol se-quence, or, more exactly, the squares left of the starting point andright of the end point of the state sequence. Similar lines will laterappear in the illustrations.Now we need to express this construction in a mathematical form.We will use two-dimensional coordinates and call a coordinate pair ( t, x ) ∈ Z a space-time point . A pair ( p, σ ) ∈ Z × Σ is a cellularevent . The event (( t, x ) , σ ) provides the information “At time t , the cellat position x is in state σ ”. We will usually write them [ t, x ] σ or [ p ] σ forbetter readability. A situation is then a sequence of cellular events, to-gether with the final cursor position: s = (([ p ] σ , . . . , [ p n − ] σ n − ) , p n ) .For the final cursor position of s we write δ ( s ) , the size of s . This meansthat we have in our example δ ( s ) = p n .In a situation, the sequence of the cellular events is significant, andthe size too, since they make algebraic operations possible. In manycases, we want to ignore however this information: Then we will usethe cellular process that belongs to a situation; it is simply the set ofits cellular events. The cellular process of a situation s is written pr( s ) .In our example, with s = 01 ⊕ , we have therefore s = (([0 , , [0 , , [1 , , [1 , , (1 , . This means that δ ( s ) = (1 , and pr( s ) = { [0 , , [0 , , [1 , , [1 , } .Usually we will not need this explicit form, since situations are meantto make this unnecessary. It helps us however to explain the “no over-writing” rule above. This rule concerns expressions like ⊕ (cid:9) ,where the cursor reaches the same point twice. If it were a situation,its cellular process would be { [0 , , [0 , , [1 , , [0 , } . This wouldprovide contradicting information about the space-time point (0 , :At time 0, is the cell at position 1 in state 0 or 3? The overwriting ruleprevents this problem. Complex Systems , Volume (year) 1–1+
Language for Particle Interactions The most important algebraic property of situations is that they canbe multiplied. The product of s and s is found by first writing s and then, with the cursor at δ ( s ) , writing s . The resulting productis written s s , but due to the overwrite rule, it may not always exist.More complex terms of situations are defined in the usual way: s is the result of writing s twice, and so on. The Kleene closure of asituation s is the set s ∗ = { s k : k ≥ } . (4)The Kleene closure always contains the empty situation , which is writ-ten [0] .In Flexible Time, situations are used to express the evolution ofa cellular automaton. But in order to understand how this is done,we first have to look at the way in which the evolution of a cellularautomaton is expressed by cellular processes. In a similar way to that in which a configuration c ∈ Σ Z can be thestarting point of an evolution ( c , c , c , . . . ) , a cellular process π can beextended to a larger process cl π , its closure . Figure 3 shows how this is Figure 3.
A process and its closure. meant for the initial configuration π = pr(10 . The cellular eventsof the original process π are displayed in black and white; together withthe the events in grey they form the process cl π . Each horizontal rowin the diagram contains the events that belong to a specific time step.We see that the diagram becomes smaller at the top; this means that astime progresses, fewer cell states can be deduced from the informationgiven by the initial process π .To motivate the exact definition of the closure, we first express theconfigurations of the cellular automaton and their evolution in termsof cellular processes. This will then allow us to generalise the definitionof evolution to processes that do not correspond to configurations.Let now c be the configuration of a cellular automaton. We definethe embedding of c at time t to be the process η t ( c ) = { [ t, x ] c ( x ) : x ∈ Z } . (5)A kind of inverse of the function η t is the concept of time slices . Thetime slice at time t of a process π is the process π ( t ) = { [ t, x ] σ : x ∈ Z } . (6) Complex Systems , Volume (year) 1–1+
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The time slice is a process and not a configuration because π ( t ) mustexist for all processes, not just for embeddings of configurations.With these concepts, the cellular process that belongs to the evolu-tion sequence ( c o , c , c . . . ) is γ = (cid:83) t ≥ η t ( c t ) . It has the time slices γ ( t ) = η t ( c t ) , which represent the configurations c t . The process γ mustthen be the closure of η ( c ) .A time slice π ( t ) of an arbitrary process is then understood as partialknowledge about the state of a cellular automaton at time t . In orderto determine the state of the automaton at time t + 1 , we take allconfigurations that are compatible with this knowledge, evolve themfor one time step and accept only the states of those cells about whichall configurations agree. The result is the cellular process ∆ t ( π ) = (cid:92) { η t +1 ( ϕ global ( c )) : η t ( c ) ⊇ π ( t ) } (7)of those events that are determined by π ( t ) . The cellular events ofwhich it consists all belong to time t + 1 .We can now easily check that the process γ has the property that γ ( t ) = ∆ t ( γ ) for all t > . Every time slice, except the first, can becomputed from the previous one. Only γ (0) , which represents the initialconfiguration, must still be handled separately.This inconvenience in resolved in the full definition of the closure.In it, the initial process no longer needs to be the embedding of aconfiguration. This is possible because it is now split into time slicesand then added piece-wise to the partial results of the computation. Definition 1 (Closure [20, Def. 3.10])
Let π be a cellular process for whichthere is a time t ∈ Z such that π ( t ) = ∅ for all t < t .If there is a process γ with the property that γ ( t ) = (cid:40) ∆ t ( γ ) ∪ π ( t ) for t ≥ t , ∅ for t < t , (8)then we write γ = cl π and say that it is the closure of π .It is easy to see that the choice of t has no influence on cl π .We can now see that the set γ that was defined above satisfies (8)if we set t = 0 and π = η ( c ) : Then we have γ ( t ) = ∅ for t < , γ (0) = η ( c ) , and γ ( t ) = ∆ t ( γ ) for t > , and indeed γ = cl η ( c ) .Definition 1 is thus a generalisation of the transition rule (2) to cellularprocesses.Not to all cellular processes, however. One of the requirements ofDefinition 1 is that γ must be a process, and this can easily be broken.All we need is conflicting information in ∆ t ( γ ) and π ( t ) : If there is atime step t at which there is an event [ t, x ] σ ∈ ∆ t ( γ ) and another event Complex Systems , Volume (year) 1–1+
Language for Particle Interactions [ t, x ] τ ∈ γ ( t ) with σ (cid:54) = τ , then γ ( t ) is no cellular process, and neitheris γ .We will however introduce in the next section a class of situationswhose cellular processes all have a closure: They will then be used todescribe the evolution of cellular automata in an economical way. The evolution of a cellular automaton is represented in Flexible Timeby reactions . We will say that there is a reaction between two situations a and b if the situation b consists only of events that are determinedby of the events of a . They belong to the future of a , so to speak. → → (10 ⊕ ) (cid:9) Figure 4.
A reaction under Rule 54.
Figure 4 shows a reaction. On the left side we see the process ofthe situation a = 10 , together with its closure. As in Figure 3, theevents of pr( a ) are highlighted while the remaining cells of the closureare displayed in grey. On the right side we see the same closure, butwith different events highlighted. This time they belong to the situation b = (10 ⊕ ) (cid:9) . With these diagrams we therefore see that theevents of the process b are determined by the process a .The formal definition of reactions is then: Definition 2 (Reactions [20, Def. 4.8])
Let a and b be two situations with cl pr( a ) ⊇ pr( b ) and δ ( a ) = δ ( b ) . (9)Then the pair ( a, b ) is the reaction from a to b . It is usually written a → b .We will use the expression a → b also as a proposition, meaning thatthere is a reaction from a to b . The symbol “ → ” then specifies arelation, and as it is normal for relations, we can also write longerchains of reactions, like a → b → c . One can verify that if such a chainexists, then there is also a reaction a → c .Reactions are useful because they can be applied to situations. Itcan be shown [20, Th. 4.11] that if there are situations x , y and a forwhich cl pr( xay ) exists and if there is a reaction a → b , then there isalso a reaction xay → xby . This reaction is then called the application of a → b to xay . Complex Systems , Volume (year) 1–1+ Complex Systems
Table 1.
The local reaction system for Rule 54, long form.
Generating Slopes: (cid:9) , (cid:9) , (cid:9) , (cid:9) , ⊕ , ⊕ , ⊕ , ⊕ .Reactions: (cid:9) → (cid:9)
00 000 ⊕ → ⊕ (cid:9) → (cid:9)
01 100 ⊕ → ⊕ (cid:9) → (cid:9)
10 010 ⊕ → ⊕ (cid:9) → (cid:9)
11 110 ⊕ → ⊕ (cid:9) → (cid:9)
00 001 ⊕ → ⊕ (cid:9) → (cid:9)
01 101 ⊕ → ⊕ (cid:9) → (cid:9)
10 011 ⊕ → ⊕ (cid:9) → (cid:9)
11 111 ⊕ → ⊕ → ⊕ (cid:9) (cid:9) ⊕ → [0]01 → ⊕ (cid:9)
01 1 (cid:9) ⊕ → → ⊕ (cid:9)
10 1 (cid:9) ⊕ → → ⊕ (cid:9)
11 00 (cid:9) ⊕ → Now it is possible that there is also a reaction that can be applied to xby . We would then have a reaction b (cid:48) → c and two processes x (cid:48) and y (cid:48) such that xby = x (cid:48) b (cid:48) y (cid:48) → x (cid:48) cy (cid:48) and therefore, by transitivity, alsoa reaction xay → x (cid:48) cy (cid:48) . This way application allows one to specify alarge set of reactions by a small set of “generator reactions”, providedonly that there is a large enough set of situations to which they can beapplied.The result is a reaction system . It is the foundation of all calculationsin Flexible Time. Definition 3 (Reaction System [20, Def. 4.13])
Let D be a set of situationsand R a set of reactions between them. We say that R is a reactionsystem with domain D if the following is true:
1. If a ∈ D , then a → a is in R .2. If a → b and b → c are in R , then a → c is in R .3. R is closed under application of reactions to the situations in D . We will now define a reaction system by specifying D and a set G ⊆ R of generators; it is then extended by repeated application andconcatenation of reactions, as described above. The system describesRule 54; its derivation is described in detail in Chapters 6 and 7 of [20].The reaction system is summarised in Table 1. The top of the ta-ble, entitled “Generating Slopes”, specifies the domain D of Φ . Morespecifically, it lists the neighbourhoods that a (cid:9) or ⊕ operator mayhave if it is is part of a situation s ∈ D . The first entry, (cid:9) , specifiesthat a (cid:9) may occur in s at the left of the term 00, the second entry Complex Systems , Volume (year) 1–1+
Language for Particle Interactions (cid:9) , that it may occur between a 1 (at its left) and a 01 (at itsright). No other possibilities exist since the remaining entries refer toother operators. One can prove [20, Theorem 6.10] that all situationsin D have a closure.The bottom of Table 1 contains the generating reactions of Φ . Itsupper part (i. e. the middle of the whole table) contains the reactionsthat involve a single (cid:9) or ⊕ operator. If we had only them, no reactioncould have an element of Σ ∗ at its left side: Therefore we have atthe bottom left of the table a set of reactions that create a (cid:9) anda ⊕ operator from an element of Σ ∗ . Their converses are listed atthe bottom right: reactions that destroy a (cid:9) and a ⊕ operator. Allreactions of Φ are the results of repeated applications of these fourtypes of generators.The arrangement of the reactions in Table 1 has also another pur-pose. It allows one to read off two important subsystems of Φ . Definition 4 (Slopes)
Let R be a reaction system with domain D .The system R + (with domain D + ) of positive slopes consists of thesituations of D that only contain ⊕ operators and the reactions betweenthese situations.The system R − (with domain D − ) of negative slopes consists ofthe situations of D that only contain (cid:9) operators and the reactionsbetween these situations.In case of Rule 54, we can find the generators of Φ − if we take onlythe generating slopes at the right and the generator reactions at thetop right of the middle section in Table 1. Similarly, Φ + is representedby the slopes and reactions at the top right of the table. Details of the reaction system
We will now have a closer look atthe way in which the reaction system Φ represents Rule 54.We begin with the slopes. Figure 5 displays the generating slopes for Φ , first the negative slopes and then their mirror images, the positiveslopes. (cid:9)
00 1 (cid:9)
01 1 (cid:9)
10 00 (cid:9) ⊕ ⊕ ⊕ ⊕ Figure 5.
Generating slopes.
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In this and in later diagrams, the endpoints of the situations aremarked by horizontal lines. They represent the places where the sur-rounding events would be expected if the slopes were parts of largersituations. Or, in the interpretation of Section 3.1, the square at whichthe left horizontal line ends is always one point left of the coordinateorigin, while the right horizontal line always begins at δ ( s ) . The be-ginning of the situation is also marked by the small vertical bar, whichis located at the left boundary of the square at the coordinate origin.An important property of the generating slopes is that they tracethe boundaries of the closure. We can see in Figure 6 what this means. ⊕ (cid:9) Figure 6.
Generating slopes as boundaries of the closure.
It shows a situation, , together with two generations of itsclosure. We see at its left the slope ⊕ (the mirror image of (cid:9) in Figure 5), and at its right, the term (cid:9) , both in boldercolours. Note that the situation (cid:9) reaches over two time steps andhas its starting point directly at the right end of the second time stepof the closure. This is the way the slope terms trace the boundary ofa closure.The generator reactions of Φ − are designed with the goal that thereaction result consists of events near the right boundary of the closureof the initial situation. (For Φ + it is similar, with left and right ex-changed.) How this is done is shown in Figures 7 and 8. They containreactions of the form a → b and display pr( a ) and pr( b ) in relation tothe closure of pr( a ) . Figure 7 shows the generator reactions of Φ − . Init, we see that the process of b is always located more to the right than pr( a ) and that it touches the right boundary of cl pr( a ) . The reactionsinvolve only two time steps, and one of the (cid:9) operators must always bepresent. To get the system started from situations in Σ ∗ , we need thereactions at the right side of Figure 8. Here we see reactions in which pr( b ) completely fills the closure of pr( a ) and b is a situation with botha ⊕ and a (cid:9) operator.The converses of the reactions at the left side of Figure 8 are shownat its right side: Reactions a → b in which a contains one (cid:9) and one ⊕ ,while b contains none. We can use them for cleanup, since they removepairs of neighbouring (cid:9) and ⊕ operators. The same manoeuvre is alsopossible in all other cases where a (cid:9) is left of a ⊕ , and we get a resultthat for every situation a there is a reaction a → b + b − with b + ∈ D + and b − ∈ D − . If we start from a and continue to apply the generatorreactions as long as possible, we can even enforce that b + and b − tracethe boundaries of cl pr( a ) . Complex Systems , Volume (year) 1–1+
Language for Particle Interactions →(cid:9) → (cid:9) →(cid:9) → (cid:9) → (cid:9) → (cid:9) → (cid:9) → (cid:9) → (cid:9) → (cid:9) → (cid:9) → (cid:9) → (cid:9) → (cid:9) → (cid:9) → (cid:9) Figure 7.
Reactions of Φ − as motion towards the boundaries of the closure. This was a summary of the content of [20] as far as it concerns Rule54.
4. Understanding the Reaction System
Up to now, the representation of Rule 54 in Table 1 looks complexand does not provide much insight. This makes it difficult to do cal-culations about Rule 54 without always looking at the table. We willtherefore develop a more compact representation of the reaction sys-tem. The goal is to find “slogans” for it that are easy to remember,analogous to the slogan for ϕ on page 4. As a first simplification, we omit the indices from the ⊕ and (cid:9) operators. This is possible because the indices of the operators arealways determined by the environment. We can see from the list ofgenerating slopes in Table 1 that if (cid:9) i is followed by a , then always i = 1 , and if it is followed by a , then i = 2 . A similar law is valid for ⊕ i , and we can recover the indices of (cid:9) and ⊕ from the equations (cid:9) (cid:9) , (cid:9) (cid:9) , ⊕ = 0 ⊕ , ⊕ = 1 ⊕ . (10)This kind of abbreviation is possible in every reaction system, becausein a generating slope u (cid:9) i v , the term u (cid:9) i is completely determinedby v . Complex Systems , Volume (year) 1–1+ Complex Systems → → ⊕ (cid:9) →(cid:9) ⊕ → [0] → → ⊕ (cid:9) → (cid:9) ⊕ → → → ⊕ (cid:9) → (cid:9) ⊕ → → → ⊕ (cid:9) → (cid:9) ⊕ → Figure 8.
Reactions that generate and destroy slopes. The generator reactionsare shown at the left, the destructors, right.
For the same reason we can shorten the generator reactions by re-moving common factors from their left and right sides. The generatorreactions of Φ − all have the form u (cid:9) vσ → ux (cid:9) v (cid:48) , with a generatingslope u (cid:9) v . When such a reaction is applied to a situation s , theremust be always a factor u to the left of (cid:9) v in s . Therefore we canshorten these generator reactions to the form (cid:9) vσ → x (cid:9) v (cid:48) and do notget new reactions when the shortened reactions are applied.We then get four pairs of reactions as generators for Φ − : (cid:9) → (cid:9) , (cid:9) → (cid:9) , (cid:9) → (cid:9) , (cid:9) → (cid:9) , (11a) (cid:9) → (cid:9) , (cid:9) → (cid:9) , (cid:9) → (cid:9) , (cid:9) → (cid:9) . (11b)They can be compressed further with the help of a new notation. Fora cell state σ ∈ Σ we will write ¯ σ for the complementary state , suchthat ¯0 = 1 and ¯1 = 0 . Then we can write the following reactions, valid Complex Systems , Volume (year) 1–1+
Language for Particle Interactions
The local reaction system for Rule 54, short form.
Generating Slopes G − = {(cid:9) , (cid:9) , (cid:9) , (cid:9) } G + = { ⊕ , ⊕ , ⊕ , ⊕ } Reactions (cid:9) σ → σ (cid:9) σ σ ⊕ → σ ⊕(cid:9) → (cid:9) ⊕ → ⊕(cid:9) σ → ¯ σ ¯ σ (cid:9) σ σ ⊕ → σ ⊕ ¯ σ ¯ σ (cid:9) σ → ¯ σ (cid:9) σ σ ⊕ → σ ⊕ ¯ σu (cid:9) v ⊕ u → uv → v ⊕ u (cid:9) v for u (cid:9) v ∈ G − Abbreviations (cid:9) (cid:9) ⊕ = 0 ⊕ (cid:9) (cid:9) ⊕ = 1 ⊕ for all σ , (cid:9) σ → σ (cid:9) σ, (cid:9) σ → ¯ σ ¯ σ (cid:9) σ, (12a) (cid:9) → (cid:9) , (cid:9) σ → ¯ σ (cid:9) σ . (12b)Written in this form we will analyse the reaction system and show whatthe generator reactions actually mean. But before we can do this, wemust see how to simplify the rest of Table 1.The reactions at the bottom of the table can be brought easily to acommon form, when we define the set G − = {(cid:9) , (cid:9) , (cid:9) , (cid:9) } of negative generating slopes . With this name at hand, we can see thatthe bottom reactions have the common form v → v ⊕ u (cid:9) v u (cid:9) v ⊕ u → u (13)whenever u , v ∈ Σ ∗ and u (cid:9) v ∈ G − . This then completes the com-pression of Table 1: The result is Table 2. Relation to the Transition Rule
In order to understand this newform of the reaction system and to see how it is related to the transitionrule ϕ , we write the reactions of Φ − in the following manner: The bottom left reaction has been shortened even more, it should have been (cid:9) σ → (cid:9) σ . Complex Systems , Volume (year) 1–1+ Complex Systems τ τ τ (cid:9) σ → τ (cid:9) σ ϕ (0 , , σ ) = σ ϕ (0 , σ, · ) ↑(cid:9) → (cid:9) ϕ (0 , σ, · ) ↑(cid:9) σ → τ τ (cid:9) σ ϕ (0 , , σ ) = ¯ σ ϕ (1 , σ, · ) = ¯ σ ϕ ( σ, · , · ) ↑(cid:9) σ → τ (cid:9) σ ϕ (1 , σ, · ) = ¯ σ ϕ ( σ, · , · ) ↑ In the reactions at the leftmost column of the table, each variable τ i stands for the state of the cell at position (0 , i ) . The other columns thenshow for each τ i the computation that determines its value – or, if itcannot be computed, which application of ϕ fails to have a determinedvalue.We can see e. g. in the first row that the state of the cell at (0 , canbe computed from the information presented in the initial situation (cid:9) σ . The cellular process of this situation consists of the events [ − , − , [ − , , and [ − , σ , and therefore the state τ of the cellat (0 , must be ϕ (0 , , σ ) .In the same way we can see that in the third row, τ is ϕ (0 , , σ ) .The diagram contains however also entries for which not all argumentsof ϕ are known. The missing arguments are marked by a dot. Whenthe value of ϕ is independent of the missing argument, it is entered inthe table, otherwise the entry is marked with an arrow.We can see that the values of the τ i only depend on three equations, ϕ (0 , , σ ) = σ, ϕ (0 , , σ ) = ¯ σ, ϕ (1 , σ, · ) = ¯ σ . (14)They all can be derived from the rule that a pair of touching 1’s causea ϕ value of 0, while one or more isolated 1’s make the value equal to 1.In the case of ϕ (0 , , σ ) , a pair of touching 1’s cannot occur, thereforethe value of ϕ is one if and only if σ = 1 . In the other two cases, σ = 1 creates a touching pair and σ = 0 inhibits it, therefore the functionvalue is ¯ σ . In a similar way we can see that in the remaining entriesof the table, the value of ϕ is undefined. This is how ϕ influences thereactions in Φ .In the table, the (cid:9) have been written once again with indices—notjust to ease the translation from situations to cellular processes, butalso because with them we can see how many new events are generatedin the reactions. One can thus see that in the first reaction one newevent is generated because δ ( (cid:9) σ ) must be equal to δ ( τ (cid:9) σ ) , andso on. If the left side of a reaction has a (cid:9) i operator and the right sidea (cid:9) j , then j − i new cell states must be generated in the reaction. Slogans
These considerations may help to understand the reactionsof the system Φ a bit better. To help memorising them, I will introducetwo slogans. Both refer to the left side of the reactions of Φ − . Thisside can always be written as (cid:9) αβσ , with α , β , σ ∈ Σ . The first slogan Complex Systems , Volume (year) 1–1+
Language for Particle Interactions tells in which cases the value of αβ makes the reaction product longeror shorter than the initial term: “01 causes growth, 10 shrinking, everything else no change.” The second slogan describes the influence of βσ on the newly generatedcell states. They can be either be a copy ( σ ) or the inversion ( ¯ σ ) ofthe variable σ , and the rule is: “ σ copies and σ inverts.”
5. Triangles and Ether
In the rest of this article we will describe the behaviour of largersystems of cells under Rule 54. We want to describe the interaction ofparticles that move on a periodic background, the so-called ether . Sowe will now introduce, as a first step, reactions for the ether. Since ithas been done already to some extent in [20, Ch. 8], we will do it herein a shorter form and from a higher point of view.The first tool that we will use are reaction families , which allow torepresent many similar reactions in a single formula. Reaction familiesappeared already in[20], but here we use a more streamlined notation.
Definition 5 (Reaction Families)
If there is a reaction a k → b k for every k ≥ , we will write this as ( a k → b k ) k . (15)The notation will be extended in the usual way to expressions like ( a k → b k ) k ≥ N or ( a j,k → b j,k ) j,k . We will also speak of ( a k ) k as a situation family . We will first find general formulas for reactions that represent tri-angular structures like that in Figure 4.There are two general laws that we will use here. The first onemakes it possible to iterate a reaction of a special form. This can bedone in two ways,if ax → ya, then ( ax k → y k b ) k , (16a)if xa → ay, then ( x k a → by k ) k . (16b)The second law “iterates” a specific reaction family; in it, n is a con-stant: if ( a k + n → xa k y ) k , then ( a kn + i → x k a i y k ) i,k . (17) Complex Systems , Volume (year) 1–1+ Complex Systems
Both laws can easily be proved by induction [20, Ch. 8.1].We now search for cases in which the first law can be applied and inwhich the left side is a generator reaction. There are two candidates, (cid:9) → (cid:9) and (cid:9) → (cid:9) . The first one has a = (cid:9) and x = y = 0 and leads to ( (cid:9) k +2 → k (cid:9) k , (18a)while the second reaction has a = (cid:9) , x = 1 and y = 0 and leads to ( (cid:9) k +2 → k (cid:9) k . (18b)Family (18a) is the more interesting one. It becomes the core ofanother reaction family, (10 k +2 → ⊕ k k , (19)whose derivation I will show here in detail, as an example for calculationwith reactions: k → ⊕ (cid:9) k → ⊕ (cid:9) k → ⊕ k (cid:9) → ⊕ k (cid:9) . Parts of the situations are underlined; they are the places that changein the next reaction step. We will use this notation in later calculationwithout special notice.Reaction family (19) can now be iterated by rule (17), with a k =10 k and n = 2 . The result is (cid:0) k + i → (10 ⊕ ) k i (cid:9) k (cid:1) i,k . (20)In families like these, the cases with i < are the most important ones,since the reactions in (19) have been applied in them for the highestnumber of times. For i = 1 , we can add one more step, since we have → ⊕ (cid:9) → ⊕ (cid:9) . therefore (20) can be written astwo families, (cid:0) k → (10 ⊕ ) k (cid:9) k (cid:1) k , (21a) (cid:0) k +1 → (10 ⊕ ) k +1 (cid:9) k +1 (cid:1) k . (21b)They, and all reactions of the form ( a k + n → x k a n y k ) k , are called trian-gle reactions .Diagrams for the reactions with k = 3 are shown in Figure 9.If we try the same manoeuvre with the other reaction family, (18b),we get (01 k +2 → ⊕ k +2 (cid:9) k . This is a family to which (17) Complex Systems , Volume (year) 1–1+
Language for Particle Interactions → → → (10 ⊕ ) (cid:9) → (10 ⊕ ) (cid:9) Figure 9.
Triangle reactions for k = 3 . cannot be applied. Therefore we will now use the reaction families (21)as our base for the description of the ether. We will find now represent the ether of Rule 54 by reactions. Thereactions for Rule 54 will turn out to be a special case of a genericscheme that applies to periodic patterns in any one-dimensional cellularautomaton.In Rule 54 [1], the ether is a periodic structure whose configura-tions consist alternatingly of the two patterns . . . . . . and . . . . . . . When one of them occurs again, it is shifted hori-zontally by two cells, so that the true time period is 4.Our starting point for representing them by reactions must be theconfiguration . . . . . . , since to it we can apply one reactionof type (21b), → (10 ⊕ ) (cid:9) . (22)It would be therefore advantageous to decompose the initial configu-ration into components of the form 10001. With a small extension ofour notation, this is actually possible. Definition 6 (Overlapping Situations)
Let ax be a situation. The a (cid:104) x (cid:105) isalso a situation, and (cid:104) x (cid:105) is the overlapping part . We have pr( a (cid:104) x (cid:105) ) = pr( ax ) and δ ( a (cid:104) x (cid:105) ) = δ ( a ) . (23)A product of situations with overlap, like a (cid:104) x (cid:105) b (cid:104) y (cid:105) , is only allowed ifthe situation by begins with x ; then a (cid:104) x (cid:105) b (cid:104) y (cid:105) = ab (cid:104) y (cid:105) .A reaction that begins with a (cid:104) x (cid:105) must have the form a (cid:104) x (cid:105) → a (cid:48) (cid:104) x (cid:105) ; (24)it exists if ax → a (cid:48) x is a reaction.If we remind ourselves that the transitions of a cellular automatonare defined in terms of overlapping cell neighbourhoods, then the newextension looks quite natural.We can now write a term like (1000) k as a product (1000 (cid:104) (cid:105) ) k and apply the ether reactions in parallel to each factor, except for Complex Systems , Volume (year) 1–1+ Complex Systems the final 1. In this style, Reaction (22) is best written in the form (cid:104) (cid:105) → (10 ⊕ (cid:104) (cid:105) ) (1 (cid:9) (cid:104) (cid:105) ) .But now we should better introduce abbreviations. We will write, ε + = 10 ⊕ (cid:104) (cid:105) and ε − = 1 (cid:9) (cid:104) (cid:105) , (25)such that (22) becomes (cid:104) (cid:105) → ε ε − . (26)The terms ε + and ε − are the simplest of the higher level structures inRule 54 that we will identify.There is also a complementary reaction to (26), ε − ε → (cid:104) (cid:105) . (27)In contrast to (26), this reaction does not belong to a known family,and we will derive it by hand (see below). Together the two reactionsform a type that naturally represents the periodic patterns of one-dimensional cellular automata. Before a formal definition is given, weintroduce the abbreviations e − = ε − , e + = ε , b = 1000 (cid:104) (cid:105) . (28)Then we see that (26) and (27) are example of the following generalpattern: Definition 7 (Background Pairs)
Two situations, e − , e + , form a backgroundpair if there is a reaction e − e + → e + e − . (29a)If there is also a situation b ∈ Σ ∗ with e − e + → b → e + e − , (29b)then b is the baseline of the background pair.A background pairs represent the elementary region of a tiling ofthe two-dimensional space-time (Figure 10). If a background pair ispresent, we automatically get the reaction families ( b k → e k + e k − ) k , (30a) ( e k − e (cid:96) + → e (cid:96) + e k − ) k,(cid:96) , (30b)which represent larger patches of the background. As we can see inFigure 10, the reactions of (30a) represent the generation of a largerpiece of ether from an initial configuration, while (30b) represents thedevelopment of a background fragment at a later time. Complex Systems , Volume (year) 1–1+
Language for Particle Interactions t = 0 e − e + b e e − b e e − e e − Figure 10.
An ether, represented by a background pair e − , e + with baseline b . Derivation of the remaining ether reaction
We have not yetproved equation (27), the reaction e − e + → (cid:104) (cid:105) . This will be donenow.The computation is an example for a larger calculation with FlexibleTime. We will prove (27) via the two reactions ε − ε + → (cid:104) (cid:105) , (31a) ε − ε → (cid:104) (cid:105) (31b)and the auxiliary step → ⊕ (cid:9) . (31c)The last reaction is an element of the reaction family (01 k +2 → ⊕ k +2 (cid:9) k . (32)Its derivation uses the reaction family (18b) and is done in the followingway: k → ⊕ (cid:9) k → ⊕ (cid:9) k → ⊕ k (cid:9) → ⊕ k +2 (cid:9) . Now we can derive the other two reactions of (31): ε − ε + = 1 (cid:9) (cid:104) (cid:105) ⊕ (cid:104) (cid:105) = 1 (cid:9) ⊕ (cid:104) (cid:105)→ (cid:9) ⊕ (cid:104) (cid:105) → (cid:104) (cid:105) ,ε − ε − ε + ε + → ε − (cid:104) (cid:105) ε + = 1 (cid:9) (cid:104) (cid:105) (cid:104) (cid:105) ⊕ (cid:104) (cid:105) = 1 (cid:9) ⊕ (cid:104) (cid:105)→ (cid:9) ⊕ (cid:9) ⊕ (cid:104) (cid:105) → (cid:104) (cid:105) . Complex Systems , Volume (year) 1–1+ Complex Systems
In the second computation we have used (31a) and (31c).
6. Particles
In the ether particles move. Boccara et al. [1] have found four ofthem and called them ←− w , −→ w , g o and g e (Figure 11). We will refer tothe moving particles ←− w and −→ w sometimes as gliders , in contrast to thestatic particles g o and g e . −→ w ←− w g o g e Figure 11.
Particles under Rule 54. The diagrams show the four types ofgliders on an ether background.
Now we will represent these particles by situations and reactions.The characterisation of particles is a natural generalisation of that ofa background:
Definition 8 (Particles)
Let ( b − , b + ) be a background pair. A particle thatmoves in this background is a situation p for which there is a reaction b m − pb n + → b n + pb m − . (33)The pair ( m, n ) is the type of the particle.The type of p represents its speed relative to the background. Toconvert it to a more conventional form, we notice that in the initialsituation of the reaction (33), the left side of p is located at the space-time point mδ ( b − ) , while in its final situation, it is at nδ ( b + ) . The period vector (∆ t, ∆ x ) = nδ ( b + ) − mδ ( b − ) is therefore the displacementthat p undergoes during one cycle of its existence. After ∆ t time steps,the particle is in the same state, and it has ∆ x positions to the right.The speed of p is then ∆ x ∆ t . (Figure 12.)Often it is simpler to work with speeds relative to the background.For this we use the vectors T = δ ( b + ) − δ ( b − ) and X = δ ( b + ) + δ ( b − ) as our base, the first one pointing to the future and the second oneto the right. A particle of type ( m, n ) has then a period vector of n + m T + n − m X and we can say that its relative speed is n − mn + m . Complex Systems , Volume (year) 1–1+
Language for Particle Interactions b − b − b − b − p p (∆ t, ∆ x ) Figure 12.
A particle of type (2 , as part of a periodic background. Itsrelative speed is . The particles of Rule 54
For Rule 54 we use the following defini-tions: ←− w = ε − (cid:104) (cid:105) , g o = ε + ε − , −→ w = 1 ε + , g e = ε + ε − . (34)They have this specific form because we can then use a simple subset ofour reaction system to represent their behaviour. This subset consistsof two reaction families and one extra reaction, ( ε − k ε + → ε k +1+ ε k +1 − ) k ≥ , ε − ε + → (cid:104) (cid:105) , (35a) ( ε − k +1 ε + → ε k +1+ ε k +1 − ) k , (35b)which transform situations that consist only of ε − , ε + and 1 into eachother. They can easily be derived from the reaction families (32)and (21). With the reactions of (35a), the ether reaction ε − ε + canbe proved, as we have seen on page 21.With these reactions we can now verify that the terms in (34) areindeed particles: −→ w e + = ε − ε → ε ε − ε + → ε ε − (cid:104) (cid:105) = e + −→ w , (36a) e − g o e + = ε − ε + ε − ε → ε − ε + → ε ε − = e + g o e − , (36b) e − g e e + = ε − ε + ε − ε → ε − ε + → ε ε − = e + g e e − . (36c)The reaction e − ←− w → ←− w e − has been omitted since the reactions in (34)are left-right symmetric. We see from these reactions that the types of −→ w and ←− w are (0 , and (1 , , while g o and g e both have type (1 , .Figure 13 contains diagrams of the reactions. Collisions of two particles
With the reactions of (35) we can al-ready find out simple facts about the particles and their interactions.
Complex Systems , Volume (year) 1–1+ Complex Systems →−→ w e + → e + −→ w → → e − g o e + → −→ w ←− w → e + g o e − . → → e − g e e + → −→ w ←− w → e + g e e − . Figure 13.
Evolution of the Rule 54 particles. The particles are shown instrong colours, and the outlined squares are ether.
One fact is hidden in (36b): the reaction −→ w ←− w → e + g o e − (37)can easily be recognised once we remember that −→ w ←− w = ε − ε + .This is the reaction in which two colliding w particles create a g o . It isin fact the only reaction that is possible between the two w particles.To see this, we note that if −→ w moves towards ←− w with nothing else thanether between them, this must be represented by a situation −→ w E ←− w ,where E is a product of an arbitrary number of e − and e + terms.Then there must be a reaction E → e m + e n − , where m is the numberof e + factors in E and n the number of e − factors. This leads to areaction chain −→ w E ←− w → −→ w e m + e n − ←− w → e m + −→ w ←− w e n − (38)to which we can apply (37). We have thus seen that two w glidersalways move towards each other unchanged until they react to theposition −→ w ←− w , and that therefore (37) is their only possible collision. Complex Systems , Volume (year) 1–1+
Language for Particle Interactions The same principle can be applied to any pair of colliding particles.We have then the following theorem:
Theorem 1 (Particle Collisions)
Let p and p (cid:48) be two particles of types ( m, n ) and ( m (cid:48) , n (cid:48) ) , with p left of p (cid:48) . Then p moves toward p (cid:48) if nm (cid:48) > mn (cid:48) ,away from p (cid:48) if nm (cid:48) < mn (cid:48) , otherwise they keep the same distance.If they collide, then there are nm (cid:48) possible interactions betweenthem. Proof . If p and p (cid:48) collide, the relative speed of p must be greater thanthat of p (cid:48) . This means that n − mn + m > n (cid:48) − m (cid:48) n (cid:48) + m (cid:48) , or equivalently that nm (cid:48) >mn (cid:48) . The other two cases are similar.For the second statement we represent the relative positions of p and p (cid:48) by a situation apbp (cid:48) c with a , b , c ∈ { b − , b + } ∗ . Here a and c represent the empty space left and right of the particles. We can makethem arbitrarily large without changing the relative position of p and p (cid:48) . (A change of a changes the absolute position of p and p (cid:48) , but thathas no influence on their behaviour.) Especially we can assume that a = b m − and c = b n (cid:48) + . The situation b represents the space between p and p (cid:48) , and we can always bring it by background reactions to the form b i + b j − .So we can assume that the environment of the particles has the form b m − pb i + b j − p (cid:48) b n (cid:48) + . Since p and p (cid:48) collide, none of the reactions b m − pb n + → b n + pb m − and b m (cid:48) − p (cid:48) b n (cid:48) + → b n (cid:48) + p (cid:48) b m (cid:48) − can be applied to this situation. Thismeans that i < n and j < m (cid:48) , for which there are nm (cid:48) possibilities. (cid:3) Interaction between the static particles and the w gliders. When we start with a random initial configuration and let it evolve fora short time, we typically see some g o and g e particles on a background,with −→ w and ←− w moving between them (Figure 1). The formalism forRule 54 is now developed far enough to describe with it the behaviourof these particles in reasonable detail.Specifically, we can now describe the behaviour of isolated g o and g e particles, which never interact with each other, only with −→ w and ←− w . InFlexible Time we can express this requirement by restricting ourselvesto the reactions that start from a situation xgy with x ∈ { e − , −→ w } ∗ , g ∈ { g o , g e } and y ∈ { e + , ←− w } ∗ .The g o case is the simplest, since the collision with a w always de-stroys this particle. Up to symmetry we have only the following reac-tions, −→ w g o e + → e + −→ w e − , −→ w g o ←− w → e e − . (39)They could be verified directly, but we will now compute them in away that is also useful in the more complex case of g e . For this webegin with −→ w g o , a common factor of the two left sides in (39), and alsothe smallest situation that represents a collision of −→ w and g o . Their Complex Systems , Volume (year) 1–1+ Complex Systems reaction is wg o = ε − ε + ε − → ε ε − = e + ε − e − . The end result ishere interpreted as an ε − surrounded by two ether fragments. We canconsider it as a short-lived intermediate stage, or a resonance , if we useonce again the jargon of particle physics. In the next step we ignore theether fragments and consider only the development of the ε − . Thereare two ways in which it can interact with an ether fragment or a w particle, namely through the reactions ε − e + = ε − ε → ε + = ←− w and ε − −→ w = ε − ε + → ε ε − = e + e − . No further resonances arise fromthese reactions, so we can stop here.The result is a scheme of three reactions; they describe the behaviourof g o in the same way as (39): −→ w g o → e + ε − e − , (40a) ε − e + → −→ wε − ←− w → e + e − (40b)One can use them to derive the reactions of (40), e. g. with the reac-tion chain −→ w g o e + → e + ε − e − e + → e + ε − e + e − → e + ←− w e − for the firstreaction. But for most purposes, (40) can be interpreted directly asa two-step scheme that describes how an ε − is created (40a) and howit decays to −→ w or ether (40b). The ether particles at the right sideof (40a) can be thought as becoming part of the surrounding space,which is why they do not appear in (40b).A similar but more complex scheme describes the collision of g e withone or more w particles. Up to symmetry it has the intermediate states1, ε − and (cid:104) (cid:105) and can be written as follows: −→ w g e → e + e − ε − (41a) ε − e + → ←− w ε − ←− w → e + e − (41b) e − e + → (cid:104) (cid:105) e − ←− w → ←− w e − −→ w ←− w → e + g e e − (41c) e − e + → ←− w g e −→ we − ←− w → ←− w e e − −→ w ←− w → e g e e − . (41d)All these reactions are short and can be verified directly. They showthat an isolated g e can neither be destroyed nor does it explode toa larger structure. (See [9] for the deeper reasons behind this.) Theintermediate states can however persist for an indefinite time if theright pattern of incoming w gliders is given. One can see this e. g. Complex Systems , Volume (year) 1–1+
Language for Particle Interactions from the reaction e − ←− w → ←− w e − in (41c). It can be iterated to ( e k − ←− w k → ←− w k e k − ) k , which shows how the intermediate state 1 can bekept alive indefinitely by a sequence of incoming ←− w gliders.In summary we get a description of the behaviour not just of asingle g o and g e , but also of a whole system of particles, providedthat the g particles and their intermediate states all keep a distancefrom each other. The distance must be so large that next to each g particle or intermediate state there is always a w particle or an etherfragment. As long as this is true, the g o particles are created (37)and destroyed (40) by w gliders, while the g e persist but go throughintermediate states (41).
7. Summary
This text consists of two interleaving tracks, one with the goal ofunderstanding Rule 54 better, the other to find concepts that are validfor all cellular automata.After a recapitulation of the results derived in [20], we began withconstructing a shorter representation of the local reaction system forRule 54 (Table 2). We then described how the transition rule ϕ in-fluences the local reaction system Φ and at the end introduced twoslogans to summarise the generator reactions of the local system.With (16) and (17), we learned how to iterate reactions. This helpedto derive expressions for the triangles under Rule 54 and to find asubsystem (35) of Φ that consists only of modified triangle reactions.It also introduced the situations ε − and ε + , which, together with thesituation 1, were the building blocks of the following construction.We introduced definitions for the background and for particles andexplored particle collisions. A formula for the number of particle inter-actions was already found in [6] under a different framework, but theproof here seems more direct.Expressions for the ether and the main particles of Rule 54 werefound and the collisions of the particles computed. We could see thatan isolated g e is stable under all collisions with incoming w gliders. Thisextends in a way a result in [9], which already showed that a single g e could not be destroyed, but the current, more detailed investigationalso shows that it could not “explode” either and become a steadilygrowing perturbation in the ether.On the way to this result, we saw an efficient method to display allpossible interactions of an isolated particle with all other particles andthe background (41).The track about Rule 54 lead therefore to results about the interac-tion of its particles, while the general track lead to generic definitionsof triangles, background and particles and a theorem about glider colli-sions. Both show how Flexible Time helps to understand an automaton Complex Systems , Volume (year) 1–1+ Complex Systems like Rule 54 as a system of interacting particles.
Changes in the formalism
One of the aims of this work was toextend the capabilities of Flexible Time by applying it to the under-standing of a “naturally occuring” cellular automaton, i. e. one that wasnot constructed for a specific purpose. This resulted in the followingchanges with respect to the version in [20]:
1. The interpretation of (cid:9) and ⊕ were changed silently in (10). In [20],they were abbreviations for (cid:9) r and ⊕ r , where r was the radius of thecellular automaton. Now the horizontal offsets associated to (cid:9) and ⊕ depend on the context in which the symbols occur.2. Reaction families, which were already present in [20], got a shorternotation.3. A short notation for overlapping situations was introduced in Defini-tion 6. There was already an overlap notation in [20], but it was moreclumsy. Now overlapping situations are part of the normal formalism. The new interpretation of (cid:9) and ⊕ allowed us to write the formulasof the local reaction system completely without indices and to makethe similarities between the basic reactions more visible.With overlaps, definitions like those of a background pair (29) couldbe written in a concise way. Acknowledgement
I want to thank Nazim Fatès for reading themanuscript and giving many helpful hints.
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