A weakly universal cellular automaton in the pentagrid with five states
aa r X i v : . [ n li n . C G ] M a r A weakly universal cellular automaton in thepentagrid with five states
Maurice
Margenstern
Universit´e de LorraineLITA, EA 3097,Campus du Saulcy,57045 Metz, C´edex 1, France e-mail : [email protected], [email protected] Abstract
In this paper, we construct a cellular automaton on the pentagrid whichis planar, weakly universal and which have five states only. This result much improvesthe best result which was with nine states.
Keywords cellular automata, universality, tilings, hyperbolic geometry.
In this paper, we construct a weakly universal cellular automaton on the pen-tagrid, see Theorem 2 at the end of the paper. Two papers, [1, 6] alreadyconstructed such a cellular automaton, the first one with 22 states, the secondone with 9 states. In this paper, the cellular automaton we construct has fivestates only. It uses the same principle of simulating a register machine througha railway circuit, but the implementation takes advantage of new ingredients in-troduced by the author in his quest to lower down the number of states, see [5].The reader is referred to [3, 4, 5] for an introduction to hyperbolic geometryturned to the implementation of cellular automata in this context. A short in-troduction can also be found in [2]. However, it is not required to be an expertin hyperbolic geometry in order to read this paper.Section 2 reminds the definition we take for weak universality. Section 3 isdevoted to the proof of Theorem 2. Section 3 of the paper. In that section,Subsection 3.1 reminds the basic model used in the paper, Subsection 3.2 ex-plains its implementation in the pentagrid, the tiling { , } of the hyperbolicplane, Subsection 3.2.3 explains the scenario of the simulation performed by theautomaton proving Theorem 2 and Subsection 3.3 gives the rules of the cellularautomaton. Universality is a well know notion in computer science. However, the single word’universality’ is understood in different ways, sometimes somehow divergent.Let us go back to the definition. 1 efinition Let K be a class of processes. Say that K possesses a universalelement U if, a finite alphabet A being fixed once and for all, there is anencoding c of K elements into the words on A and of the data for a K -elementinto the words on A such that for all element χ of K and for all data d of χ , U applied to ( c ( χ ) , c ( d )) ends its computation if and only if χ ends its own onewhen it is applied to d and, in that case, if U ( c ( χ ) , c ( d )) = c ( χ ( d )) . We also say that U simulates χ or that χ is smulated by U . If K possesses auniversal element, we also say that K possesses the property of being universal.We get again the standard definition of universal Turing machine,In the above definition, therei are four elements. Data χ and d , the encodingof c and the universal element U . From the definition itself, the notion ofencoding is an essential feature. Indeed, among the elements of K , it is notdifficult to construct some of them whose encoding is bigger than that of U .Indeed, it may be assumed that the encoding is an increasing function in thissense that if χ and ξ are elements of K transforming words on A onto wordson A , then c ( ξ ◦ χ ) > c ( χ ) , c ( ξ ). Consequently, encoding the elements of K intoa fixed alphabet is an essential feature. It allows U to simulate objects whichare bigger than itself. Of course, changing the encoding may result in a changeon the computation of U which may then be either faster or slower. At last,when U stops its computation, the result is an encoding of the result of theelement of K simulated by U .Now, since a few decades, these three items: the data, the encoding andthe result are not always considered in the same way. From the definition, d isfinite, as a word on A ; when the computation of χ on d stops, that of U on c ( χ )and c ( d ) also stops. Now, whether this latter condition is observed or not, ithappens that when U is applied to c ( χ ) and c ( d ), it does not yield c ( d ) when χ completes its computation on d , but something else, call it e ( d ), where e can beconsidered as another encoding of d , e being also fixed once and for all. Indeed, U ( c ( χ ) , c ( d )) = c ( χ ( d )) can also be rewritten c − ( U ( c ( χ ) , c ( d ))) = χ ( d ), so that χ ( d ) is restored by decoding U ( c ( χ ) , c ( d ))). Introducing e consists in acceptingthat the decoding funcion can be independent from the encoding one.Now, the conditions of finiteness on d and on the computation of U whenthat of chi on the considered data stops are not always observed. When allconditions of Definition 1 are observed we say that U is strongly universal .In this definition of strong universality, it is not required that the decoding bethe inverse function of the encoding. However, it is required that e and c belongto comparable classes of complexity. Specifically, it is required that there is aprimitive recursive function u such that for any d , c ( d ) , e ( d ) ≤ u ( | d | ), where | d | is the size of c ( d ), i.e. the number of symbols in c ( d ).When the conditions of strong universality are not observed, we say that U is weakly universal . In case d is in some sense infinite, it is required that c ( d ),which is an infinite word be of the form u ∗ wv ∗ , where u , v and w are wordson A . The repeated words u and v are called the periodic patterns and it is not2equired that u = v . Note that during the computation, we may consider thateach step t of U works on something of the form u ∗ w t v ∗ . It is this situationthat we shall consider, with this difference that we work in a 2 D -space and so,accordingly, we require a condition of periodicity restricted to the outside ofa big enough disc, this condition being able to involve two different periodicpatterns. Most of the cellular automata in hyperbolic spaces I constructed, myself or witha co-author, apply the same model of computation. We implement a railwaycircuit devised by Ian Stewart, see [7] which we rework in order to simulate aregister machine. This general scenario is described in detail in [4, 5]. Here, wesimply give the guidelines in order to introduce the changes which are specificto this implementation.
The railway circuits consists of tracks, crossings and switches, and a singlelocomotive runs over the circuit. The tracks are pieces of straight line or arcs ofa circle. In the hyperbolic context, we shall replace these features by assumingthat the tracks travel either on verticals or horizontals and we shall make itclear a bit later what we call by these words. The crossing is an intersection oftwo tracks, and the locomotive which arrives at an intersection by following atrack goes on by the track which naturally continues the track through whichit arrived. Again, later we shall make it clear what this natural continuation is.Below, Figure 1 illustrates the switches and Figure 2 illustrates the use of theswitches in order to implement a memory element which exactly contains onebit of information.The three kinds of switches are the fixed switch , the flip-flop and the memory switch . In order to understand how the switches work, notice thatin all cases, three tracks abut the same point, the centre of the switch. Onone side switch, there is one track, say a , and on the other side, there are twotracks, call them b and c . When the locomotive arrive through a , we say thatit is an active crossing of the switch. When it arrives either through b or c , wesay that it is a passive crossing.In the fixed switch, in an active passage, the locomotive is sent either alwaysto b or always to c , we say that the selected track is always b or it is always c .In the passive crossing, the switch does nothing, the locomotive leaves the switchthrough a . In the flip-flop, passive crossings are prohibited: the circuit mustbe managed in such a way that a passive crossing never occurs at any flip-flop.During an active passage, the selected track is changed just after the passageof the locomotive: if it was b , c before the crossing, it becomes c , b respectivelyafter it. In the memory switch, both active and passive crossing are allowed.The selected tracks also may change and the change is dictated by the following3ule: after the first crossing, only in case it is active, the selected track is alwaysdefined as the track taken by the locomotive during its last passive crossing ofthe switch. The selected track at a given switch defines its position . (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) Figure Les aiguillages du circuit ferroviaire. De gauche `a droiite : le fixe, la bascule etle m´emorisant.
The current configuration of the circuit is the position of all the switches ofthe circuit. Note that it may be coded in a finite word, even if the circuit isinfinite, as at each time, only finitely many switches have been visited by thelocomotive.Figure 2 illustrates how a flip-flop and a memory switch can be coupled inorder to make a one bit memory element. SS L E SS L E SS L E SS L E SS L E
Figure L’´el´ement de base du circuit. Deuxi`eme ligne :Deux premiers dessins : lecture de l’´el´ement. Deux derniers dessins : ´ecriture de l’´el´element.
As mentionned in the introduction, the first weakly universal cellular automatonon the pentagrid was done by the author and a co-author, see [1]. The paperimplements the solution sketchily mentioned in Subsection 3.1 with 22 states.In the next paper about a weakly universal cellular automaton on the pentagrid,see [6], the same model is implemented with 9 states. The difference with theformer paper is that in the second paper, the cell which is at the centre of theswitch has the same colour as another cell of the track. The centre of the switchis signalized by the neighbouring of the centre.4 .2.1 Former implementations
Figure 4 illustrates the implementation of the crossing and of the switches per-formed in [6], showing in particular, the feature at which we just pointed. Fig-ure 3 shows the implementation of the verticals, second row in the figure, andof the horizontal, first row. Both these figures show how to implement the basicelement of Figure 2 in Subsection 3.1. Figure 5 show a global view of how thetree structure of the tiling can be used to implement a basic element in thepentagrid.
Figure Implantation des voies dans la pentagrille.En haut, voie horizontale parcourue de gauche `a droite.En bas, voie verticale parcourue de haut en bas.
Figure Implantation du croisement et des aiguillages dans la pentagrille.De gauche `a droite : le croisement, le fixe, le m´emorisant et la bascule.
In this implementation, as well as in that of [1], the locomotive is imple-mented as two contiguous cell with different colours, which allows the imple-mented vehicle to find the direction of its motion along the tracks. The colourswere chosen as green and red, green pointing at the front and red at the rear.The direction is then obvious. These are the elements which allowed us to provethe following result.
Theorem (Margenstern, Song), cf. [6] − There is a planar cellular automatonon the pentagrid with -states which is weakly universal and rotation invariant. By planar, we mean that the trajectory of the cells of the cellular automa-5on which at some point change their state is a planar structure which containsinfinitely many cycles which cannot reduced to a 1 D -structure. This is in partic-ular the case of the units which constitute the registers of the register machineimplemented by the railway circuit. In this paper, we take benefit of various improvements which I brought in theconstruction of weakly universal cellular automaton constructed in other con-texts: in the heptagrid, another grid of the hyperbolic plane, in the hyperbolic3 D -space and in the tiling { , } of the hyperbolic plane, that latter automatonhaving two states only, see [5] for details.Our implementation follows the same general simulation as the one describedin Subsection 3.1. In particular, Figure 5 is still meaningful in this new setting. Figure Implantation de l’´el´ement de base dans la pentagrille.Les disques de la ligne du bas symbolisent, de gauche `a droite : le croisement, le fixe, lem´emorisant et la bascule.
However, here, new features are introduced.The first change is that the tracks are one-way. In some sense this is closerto what we can see for railways in real life, in particular for highspeed ones.This change entails a big change in the switches and in the crossings. There isno change for the flip-flop which was already a one-way structure from the verybeginning as passive crossings are ruled out for this kind of switches. For fixedswitches it introduces a very small change:we keep the structure for a passivecrossing and for the active one, as the selected track is the same, it is enoughto continue the active way without branching at the centre of the switch, seeFigure 6. In the same picture, we can see that the situation is different for thememory switch. This time, as there are two possible crossings of the switch andas the selected track may change, we have two one-way switches: an active one6nd a passive one. At first glance, the active switch looks like a flip-flop andthe passive switch looks like a one-way fixed one. However, due to the workingof the memory switch, we could say that the active memory switch is passivewhile the passive memory switch is active. Indeed, during an active crossing,the selected track is not changed contrary to what happens in the case of theflip-flop. Now, during a passive crossing, the switch looks at which track iscrossed: the selected or the non-selected one. If it is the non-selected one, thenthe selection is changed and this change is also transferred to the active switch.Accordingly, there is a connection between the active and the passive one-waymemory switches. fixed switch memory switch
Figure The new switches for a one-way structured circuit: the fixed and the memoryones. Note that the flip-flop remains the same as in Figure . Now, if we wish to significantly reduce the number of states, we also have tochange the tracks themselves, as it appears that giving them the same colour asthe background is better than assigning a special colour to identify the tracks.The consequence is that we have to place milestones in order to do so. Thiswas performed in previous works, see [5]. But this is not enough: we also haveto change the crossings. Contrary to what happens in the 3 D -space wherecrossings can be replaced by bridges, which makes the situation significantlyeasier, crossings cannot be avoided in the plane.In [5] we indicate a solution which allowed me to build a weakly universalcellular automaton in the hyperbolic plane with 2 states only. However, this wasnot performed in the pentagrid nor in the heptagrid, but in the tiling { , } .This solution can be implemented here and this allowed me to reduce the numberof states from 9 down to 5. There is a slight improvement in the present solutionwhich might allow us to reduce the number of neighbours for a two-state weaklyuniversal cellular automaton in the hyperbolic plane.First, we look at a crossing of two one-way tracks. The main idea is that weorganize the crossing in view of a round-about : an interference of road traficin our railway circuit. We may notice that the locomotive arriving either from A or B in Figure 7 has to turn right at the second pattern it meets on its way. Sothat it is enough to devise a pattern which allows to count from 1 up to 2 insome way. In the two-states world of the weakly universal cellular automatonon { , } described in [5], there were four patterns: a first one appears whenthe locomotive arrives at the round-about. At this point, a second locomotiveis appended to the arriving one. At the second pattern, one locomotive is7emoved and so, as a single locomotive arrives through the round-about at thethird pattern, then it knows that it has to turn right. Here, this scheme isslightly changed as follows. At the first pattern it meets the locomotive, which,arriving in a state S , is changed into T . At the second pattern, as a locomotivein the state T arrives, it is sent on the track which leaves the round-about. Thisis why three patterns are needed in Figure 7. A B f Figure The new crossing: the one-way tracks from A and B intersect. We have a three-quarters round-about. The small disc at f represents a fixed switch. Discs , and representthe pattern which dispatches the motion of the locomotive on the appropriate way. Patterns and are needed as explained in the description of the scenario. BA f A B f B A f A B
12 3 f Figure The new crossing: four possible one-way track. Assembling them allows toperform a two-way crossing. The notations are those of Figure . A , B go opposite to A , B , respectively. Figure 8 shows us how to assemble four one-way round-abouts in order toperform a true crossing fro two intersection two-ways tracks.8ow that we have seen the scenario to implement the circuit, we have toprecisely look at how to implement it with five states only. We turn now to thisquestion.
As already announced, we need to use five states only. The first state is thequiescent state which we denote by W . Remember it is defined by the followingrule: if a cell c and all its neighbours are in the state W , then at the next top ofthe clock, the cell c remains in the state W . We shall also call W the blank and wealso shall say that a cell in W is white. The other states are B , G , R and Y . Thecells in these states are said to be blue, green, red or yellow, respectively. Thestate B is mainly used for the milestones which delimit the tracks. The state G is the basic state of the locomotive. In crossings, the locomotive turns to R . Thestate Y is used for special markings in the passive memory switch. The states B , G and R are also used for marking in the crossings and in the other switches.The state G also appears in the milestones for the tracks.Checking the new scenario requires to first study the implementation of thetracks. We have to define verticals and horizontals.Vertical and horizontal tracksThe verticals are easy to define, they correspond to branches of the Fibonaccitree. In fact they are finite sequences of cells for which a side lies on a fixed line.The line can be changed inside the sequence as illustrated by Figure 9 where agreen locomotive goes on two verticals, one in a bottom-up run, the other in atop-down one. Figure A vertical with a green locomotive, First row: top-down traversal; second row:bottom-up traversal.
The structure of an element of the track is simple. Assume that the loco-motive leaves the cell through its side 1. Then, the milestones are always onsides 2 and 5. It enters either through sides 3 or 4. Both cases occur as illus-trated in Figures 9 and 10. Now, if we consider the standard numbering of thesides, then the place of the milestones depends on the direction of the motion.Assume that in the central cell, side 3 is the horizontal side. Then, for the cell1(1), cell 1 of sector 1, the milestones are on the sides 3 and 5 in a bottom-up9otion while they are on the sides 2 and 5 in the top-down one. The leftmostpicture of Figure 11 illustrates the milestones as just defined.
Figure A vertical with a red locomotive, First row: top-down traversal; second row:bottom-up traversal.
Figure The elements of the tracks.Leftmost picture: the standard element. Second and third pictures from left: the elementwhich allows to perform sharp turns. Fourth and fifth pictures, illustration for a sharp turn.
The other pictures of Figure 11 illustrate two other patterns for the tracksand their importance. The second and third picture of the figure illustrates anelement of the track which allows the locomotive, either green or red, to performa sharp turn: this means that the locomotive enters through a side and exitsthrough a contigous one. Such a turn is absolutely needed in the pentagrid inorder to have cycles in the trajectory of the locomotive. If such a possibilitywould not be allowed, the locomotive would run to infinity without returningto any tile it had already visited.The patterns illustrated by the second pictures allow us to perform a sharpturns as it is illustrated by the last two pictures. A blue milestone is replaced bya green one. There are two possiblities and each of them is used for a direction ofthe motion. Rules are devised in such a way that the green milestone preventsa backward motion of the locomotive. Indeed, if in the fourth picture, thegreen milestone is replaced by a blue one, then a locomotive going from sector 3into sector 4 would go on its way in sector 4 but a copy of it would return totile 1 of sector 3. This backward motion is prevented by the green milestonewhen it is in tile 1 of sector 5. Now, the sharp turn is possible as the exit ofthe locomotive does not occur through the expected side 1 but through a sidewhich is contiguous to the one through which the locomotive entered the cell.10 igure A horizontal with a green locomotive, first four rows and then, a red one: thelast four rows. For each locomotive, right-left and left-right runs.
Figure 13 allows us to establish that any horizontal can be run by the lo-comotive with the elements indicated in Figure 11. The horizontals follow thelevels which are defined in [3]: they belong to bigger and bigger Fibonacci treeswhose roots follow a vertical as defined above. Weprove that the track cango from a black node on a level to the next one. There are four situations,depending on the father of the black node and the arrival to the black node,either from the upper level or from its son, on the lower level. On Figure 13three horizontals are indicated by a thin coloured line which crosses the tiles, amove line, a green one and a red one. Note that contiguous tiles on the same11orizontal do not share an edge, they only share a vertex. In Figure 13 and inthe next figures in this subsection, we shall use the following numbering of thesectors: the sectors are counterclockwise numbered from 1 to 5 and sector 3 isdefined by the pentagon which is below the horizontal side of the central tile.In each sector s the tile is given by its number n , we write n ( s ) if it is neededto indicate the sector.Figure 13 indicates the motion from a black node β on a given level, thegreen one in the figure, to the next one γ on the same level and in the considereddirection. Both directions are illustrated in the figure in order to facilitate thechecking. Now, a black node may have either a white or a black father andthe track may arrive at β either from the upper level or from the lower one.This explains the four cases illustrated on each row of the figure. Note that inthe motion from left to right, the track goes through 2(2), 0 and 2(5) when thefather of β is black and it goes through 2(2), 0 and 2(1) when the father is white.In the motion from right to left, the visited cells occur in the opposite order.In the figure, the central cell is the father of β in the motion from right to left.From Figure 12, we can check that a horizontal can be run by the locomotive,whether it is green or red. Figure Closer look on a horizontal. The central cell is the father of β in the motionfrom left to right. Note the indication of the horizontals through the coloured lines. In whitenodes the line has a ⌣ shape, in black nodes it is a ⌢ -shape.First row: from left to right; second row: from right to left. Figure 12 shows that the connection with a vertical, either up or down orto the left or to the right raises no problem. The figure as well as Figures 9and 10 also shows that such tracks can be run indifferently by a green or a redlocomotive.It is the place to remark that as horizontal tracks run on three consecutivelevels, a two-way protion of the tracks require at least seven consecutive levelsas we need at least one level to separate the tracks run on each direction. Asimilar remark also holds for vertical lines. A two-way section must be separatedby several nodes on the same level at the level where the distance between thesupporting line is minimal: this distance must be positive, which guarantees12hat the lines are not secant. These constraints require much space for theimplementation, but in the hyperbolic plane, we are never short of space.Implementing the crossingsAs indicated in Subsubsection 3.2.3, the tracks are organized according towhat is depicted in Figures 7 and 8. From the latter figure, it is enough to focuson the implementation of Figure 7. From the implementation of the tracks, weonly have to look at the implementatation of the patterns symbolically denotedas , , and f in the figure. As f is a fixed switch whose implementation isindicated a bit further, we simply implement as the other patterns are strictcopies of this one. Figure 14 illustrates this implementation and the behaviourof the locomotive when it crosses the pattern. We can see that the differencestrongly depends on the colour of the locomotive.From Figure 7, the locomotive always arrives to the pattern from the samecell, namely 4(4). Then the locomotive goes to the centre, cell 0. Figure The key pattern of the crossings. Notice the difference of behaviour dependingon the colour of the locomotive.
When the locomotive is green, it goes to cell 1(1) arriving there as a red cell,and then it goes out onto the round about towards the next pattern, still as ared locomotive. When the locomotive which arrives at cell 4(4) is red, it alsogoes to 0 but from there, it goes to 1(5) where it arrives as a green cell. Indeed,cell 1(5) remains white until it sees a red cell through its side 1. Note thatside 1 is clearly indentified thanks to the pattern of the neighbours of cell 1(5).It is the pattern of an element of the tracks with a red milestone in place of thegreen one.Implementation of a fixed switchThe first two rows of Figure 15 illustrate the passive crossing of a fixed switchfor a green locomotive while the last two rows of the same figure does the samefor the red locomotive.With one-way tracks, we need a single kind of passive fixed switch. Indeed,there is no need of an active fixed switch, as the locomotive never goes in thenon-selected direction. The active switch is reduced to the track which goes inthe selected direction, as illustrated by Figure 6.13 igure The fixed switch: passive crossing by the green locomotive and then the redone.
Implementation of a flip-flopYp to now, we used four states only, those which were introduced withthe tracks. It was not difficult to implement the tracks and it was possible toimplement the crossings using still these states. With the flip-flop, we introducethe fifth state, Y . Figure 16 illustrates the configuration of a flip-flop and itsactive crossing by a locomotive. Figure The flip-flop.
We can imagine that the locomotive crosses a second time the flip-flop,14urning it back to the position it had before the first crossing. We remarkthat the change of the selected track occurs some time after the locomotive leftthe switch. In fact, the white cell at 2(2), it has three red neighbours, detectsthe passage of the locomotive just before the latter leaves the switch. Then, itpasses the information to the cell 2(1) through 1(1). This makes 2(1) to flash,turning from Y to R and then turning back to Y . This flash makes both 1(1)and 1(5) to change their states in a way which triggers 2(2) and 2(5) to changetheir states. When the cell 2(2) is red, a symmetric process occurs.Implementation of a memory switchNow, we arrive to the most difficult situation. We have to implementtwo switches with a connection between them. As already noticed in Sub-section 3.2.3, the active switch has a passive behaviour when crossed by thelocomotive and the passive switch has an active behaviour when the locomo-tive takes the non-selected track. This action of the passive switch triggers thechange of selection in the active switch: hence we have to organize the connec-tion from the passive witch to the active one. The patternsof these switches,when the locomotive is not present, is illustrated by Figure 17. Figure The stable configuration of the active and passive memory switches. To left,the switches selected the right-hand side track. To right, they selected the left-hand side track.
Figure 18 illustrates the crossing of the switch by the locomotive. Due to itsway of working, the memory switch has two basic positions according to whichis the selected track. We say that the left-, right-hand side switch selects theleft-, right-hand side track respectively. We can check on the figure that at theswitch remains unchanged adter the traversal of the locomotive. Note that theswitch remains unchanged after the locomotive crossed the switch.
Figure The active memory switch. The two versions of the switch: above, left-handside switch, below, right-hand side one.
15e can see that the pattern of the active memory switch looks like that ofthe flip-flop. The difference is restricted to the cell 2(1) and its neighbours.In the flip-flop, the cell 2(1) is in Y and three consecutive neighbours ar in B :cells 5, 6 and 7 of sector 1. Now, in the active memory switch, the cell 2(1) isin B and the cell 6(1) is white. Moreover, the cells 15(1) and 18(1) are in G . Weshall see the role of these green cells in a while.Before, we look at the crossing of the passive memory switch. We havefour situations as the switch has two positions and as for each position, thelocomotive may arrive either through the selected track or through the non-selected one. Figure The passive memory switch. The two versions of the switch and the two pos-sible crossings: above, first two rows, left-hand side switch; below, last two rows, right-handside one.
As is clear on Figure 17, the pattern of the passive memory switch is verydifferent from the active one. Although a part of it is taken from a fixed switchwhose centre would be at the cell 2(3), the surrounding of this cell and thepassive tracks is very specific.Figure 19 illustrates all the four cases of crossing the switch. In the fig-ure, it can be checked that when the locomotive arrives through the selectedtrack, nothing is changed outside the tracks themselves. It can also be checkedthat when the locomotive crosses the non-selected track, this changes the selec-tion according to the definition of the memory switch: the non-selected trackbecomes the new sekected track.Now, this information has to be transferred to the active memory switch.This is performed by the pattern of the passive memory switch. The crossingthrough the non-selected track is detected by the cell 1(5)i which is in contactwith the central cell. That latter one has a B -neighbour on the side of theselected track and a Y -one on the side of the non-selected track. When the16ocomotive becomes a neighbour of the central cell, it abuts the cell on the sideof the Y -neighbour or of the B -one. This allows the central to know whetherthe locomotive has run through the selected track or through the non-selectedone. Accordingly, if the run goes through the non-selected one, the central cellflashes: it turns from B to R and then turns back to B . Now, this flash makes thecells 1(1) and 1(4) to take the opposite colour, from B to Y or from Y to B : thischanges the signalization of the non-selected track. But the cell 1(5) also cansee the flash of the central cell. This makes the cell 1(5) to also flash: it turnsto G and then turns back to Y . Now, the cells 5(1) and 11(5) are milestonesfor the cell 4(5) which, accordingly, appears to be a possible element of a track.Consequently, the flash of 1(5) creates a second locomotive which can go alongthe tarck whose starting point is cell 4(5). It is enough to define a track goingto the active memory switch to make the needed connection between the twoparts of the memory switch. Figure The organisation of the memory switch.
Figure 20 illustrates the global setting for implementing the memory switchwith its tow parts, the passive and the active one and the connection betweenthem. Now, the path whose starting point is the cell 4(5) in the passive switch,see Figure 17, goes to the active switch as indicated in Figure 20 and it arrivesat the cell 6(1) of the active switch, see Figure 17.
Figure The change of selection in the active memory switch triggered by the arrival ofthe second locomotive at the cell . To left the case when the left-hand side track is selected,to right, when it is the case for the right-hand side track. The presence of the locomotivein can be seen in the first picture of each series.
The cells 15(1) and 17(1) allow to make the second locomotive sent from thepassive switch go to the cell 16(1) from where it is driven to the cell 6(1). Now,17hen the cell 2(1) can see the second locomotive, it flashes, turning to R andthen back to B , which makes the cells 1(1) and 1(5) trigger the signal to thecells 2(2) and 2(5) which then take the opposite colour.Figure 21 illustrates the situation when the second locomotive arriving atthe cell 6(1) triggers the change of selection. As can be seen in the figure, thesecond locmotive vanishes just after it arrived at 6(1). This arrival makes thecell 2(1) flash and then the same mechanism as seen for the flip-flop apply: thesituation for cells 2(2) and 2(5) is exactly the same.And so, outside the locomotive which yields the simulation of the computa-tion in this model, call it the main locomotive , from time to time a secondlocomotive appears for a while in order to transmit the appropriate signal to anactive memory switch. It is important to notice that the motion of this secondlocomotive does not interfer with the motion of the main one. Indeed, althoughthe track from a passive memory cell to its corresponding active one is verylong, the distance between whole switches is much larger. In any case it canbe made much larger: this can easily be seen on Figure 5. Also note that thesecond locomotive may sometimes be red. Indeed, as indicated by Figure 20,the second locomotive travels through two crossings. In order to prove the existence of the cellular automaton whose working wasdescrived in the previous sections, we have to implement its rules. Here, wedisplay all of them, but we list them in several groups according to the presen-tation of the previous implementation. We present the rules according to thesame order as we presented the implementation in Subsection 3.2.3
First, we have the rules for the motion of the locomotive on the tracks whichare displayed in Table 1. The rules are rotation invariant .This means that if we perform a circular permutation on the neighbours ofa cell, this does not change the new state. Accordingly, in the tables we give,the rules are rotationally invariant: a given rule ρ has four over rules with thesame current state and the same new state, the states of the neighbours beinga circular permutation of those indicated by ρ .The rules are presented in the same order as established while checking themduring the construction by a computer program.As explained in the caption of the table, the rules are divided into conser-vative ones and motion ones. This is a general feature: at each top of theclock, the locomotive is in one cell. There are at most two locomotives at agiven time so that, at most 12 cells may change from one time to the next one.Accordingly, the rest of the configuration must not change. This is the role ofthe conservative rules: they keep the configuration invariant, the locomotivesbeing supposed to be absent. The motion rules control the whole simulation.They define the changes of states but also they sometime do not change the18tate: either they simply witness the passage of a locomotive, this is typicallythe case for milestones, either this lack of change is part of the process: this iswhat we observe in the management of the switches. Table The rules for the tracks. The first two columns are conservative rules. The lastcolumn contains the motion rules.
The motion rules controlling the motion of the locomotive, whatever itscolour, define the change of state which translates in terms of the cellular au-tomaton the fact that the locomotive goes from one cell of the track to the nextone, see rules 31 up to 51. But the locomotive must not go to a white neighbourof the track which is not an element of the track. This property is shared bythe motion rules which are conservative: this is the case for rules 43-43, it cor-responds to the exit of the locomotive from the considered element of the track.It is also the case for rule 37 which prevents the entrance into an element whereone of the two milestones is green: the entrance is allowed from the side of theblue milestone, rule 35, and forbidden from the side of the green milestone, itis the role for rule 37.
Table 2 provides additional rules for the crossing and for the fixed switch.
Table The rules for the crossings and for the fixed switch. The first two columns areconservative rules. The last two columns contains the motion rules.
52 WWWWWRW53 WWWWRRW54 WWWWRBW55 WWBWRRW56 RWWWWRR57 BWWWWRB58 WWWWBRW59 RWWWWBR60 RWWWWWR61 WWWRWBW 62 RGBWWWR63 BGRWWWB64 BRRWWWB65 BGWWWRB66 WGWRWBW67 BRWBWWB68 BRBWWWB69 RBRWWWR70 RRBWWWR 71 WWBBGWG72 WGBWRRR73 GWBBWWW74 WGBBWWW75 WRBBWWW76 RWBWRRW77 BRWWWBB78 RGWWWWR 79 WWBBRWR80 RWBBWWW81 WRBWRRW82 WRWRWBG83 WWBBWGW84 GWWRWBW85 BGWBWWB86 BBRBWWB
There is slightly less rules strictly required by the crossings than those used19or the tracks themselves, taking into account the above mentioned rules for thered locomotive. The rules controlling the difference between a green locomotiveand a red one are rules 72 and 76 for the cell 1(1) and rules 82 and 84 for thecell 1(5). After these cells, the cell 3(1) for the red locomotive, and the cell 4(5)for the new green one, motion rules apply: rule 39 for the cell 3(5) and rule 36for the cell 4(5).
Table 3 gives the rules for the flip-flop, of course in its two versions. Theconservative rules are mainly required by the new state Y which was introducedfor the simulation of this switch. They are listed in the first column.Note that rule 105, which does not change the current state is typically acell which contributes to the role of the switch. This rule applies to the cell 1(4)when the cell sees the locomotive in the central cell. As the cell has a redneighbour a single white neighbour, which is not the case of the cell 1(2) which,at the same time has two white neighbours. This is why rule 104 prevents thelocomotive to enter the first element of the non selected track while rule 38makes the locomotive enter the first element of the selected track. Table The rules for the flip-flop.
87 GWYGGWG88 YGGBBBY89 GGBWWWG90 GGWWWRG91 BGYWWWB92 WWGRRRW93 GWRWWWG94 WWBWGRW95 GWRGGYG96 RGWRRRR97 GGRWWWG98 GGWWWBG99 RRGWWWR100 RGRWWWR 101 WGWGWGG102 BWGBWBB103 GGWWWGW104 GGYGGWG105 WGBWGRW106 GGRGGYG107 WGGWWGW108 WGGRRRG109 GWYGGGW110 WWGGGBW111 GWGRRRG112 RGGWWWR113 WWYGGGW114 YWGBBBR115 GWBWWWG 116 GWWRRRG117 WWRGGGB118 RWGBBBY119 GWRGGRW120 BWYGGGG121 YBWBBBY122 GBWWWRG123 GWBRRRR124 WWRGGYG125 RWWRRRW126 GWYGGRG127 WWRGWBW128 RWGRRRR129 GWWGGYG 130 WGWRRRW131 GGYGGRG132 WGRGWBW133 GGWGGYG134 BGWBWBB135 WWBGGGW136 GWGGGYW137 GGWRRRG138 YGWBBBR139 WWGGGYW140 RGWBBBY141 WWGGGRB142 WWYGGRG143 BWGGGYG144 GBWRRRR
When the flip-flop selects the track starting from 1(4), rule 35 applies to thecell. Note that it is a rule already used for a simple motion of the locomotiveon the tracks. Now, at the same time, rule 132 applies to 1(2) which preventsthe locomotive to enter the non-selected track. Many rules are used to managethe change of the selection. Their action is symbolized by Table 4 which showsin both positions of the switch the action of the rules in the cells concerned bythe motion of the locomotive and the action of the switch. As an example, therules acting on the cell 2(1) when the selected track is the left-hand side oneare, successively: 88, 114, 118 and 121, the flash being controlled by rules 114and 118. In the motion when the selected tracl is the right-hand side one,rules 88 and 121 again apply but the flash is now controlled by rules 138 and 140.20 able Execution trace of the crossing of the flip-flop by the locomotive when the selectedtrack is: to left, the left-hand side track; to right, the right-hand side track.
Here, the rules will be divided into two tables: Table 5 and Table 7 for theactive and the passive part respectively.We know that the action of the locomotive crossing actively the active partof the memory switch makes no change in the configuration of the switch. Thisis why the in first two columns of Table 5 all rules keep the current state. Inthe last two columns, we can see the rules involved by the flash of the switchtriggered from the passive part of the memory switch.
Table The rules for the active memory switch. First to columns to left: no change ofthe current state. Last two columns: managing the flash of the switch.
145 GWBGGWG146 GWRGGBG147 WWGGWBW148 WWWGGBW149 BGGBWBB150 BBGWWGB151 WBGWWGW152 WWGRBRW153 RGWRBRR 154 GGBGGWG155 GGRGGBG156 WGGRBRW157 GWBGGRG158 RWGRBRR159 GWWGGBG160 WGWRBRW161 GGBGGRG162 GGWGGBG 163 BGGBGBR164 GBGWWGW165 GWRGGWR166 RGGBWBB167 BRGWWGB168 WRGWWGW169 RWBGGWG170 BRWBWBB171 GRWWWRG 172 WWRRBRR173 WWRGGBG174 RWWRBRW175 GWWGGRR176 WWBGGRG177 BWRBWBB178 RWWGGBG179 WRWRBRR180 GRWWWBG
Table Execution trace of the flash in the active memory switch in both possible situa-tions.
Table 6 produces the trace of execution when the flash issued from the passivememory switch reaches the active memory one. The two kinds of actions areindicated. The rules acting on the cell 2(1) are successively: 163, 166 and 170when the selected track was the left-hand side one. Rules 163 and 166 managethe flash and rule 170 witnesses the transfer of the signal to the left-hand side.When the selected track is the rght-hand side one, rules 163 and 166 againmanage the flash and this time, rule 177 witnesses that the signal goes to the21ight-hand side.We arrive to the passive memory switch which also requires a lot of rules,almost as many of them as in the flip-flop, see Table 7. The reason is theimportance of the detection of the passage along the non-selected track. Thistriggers the change in the cell 0, which entails a change in both sides of the cell.In the memory switch we apply the principle of changing the states when it isneeded to do so: a straightforward application of the definition of the memoryswitch could consist in re-defining the selection even if the new selection is thesame as the previous one. Here, the change is triggered only if the locomotivepassed through the non-selected track. This allows us to implement a shortestprocess requiring less states.
Table The rules for the passive memory switch. First column and upper part of thesecond one: no change of the current state. Motion rules appear already in the lower part ofthe second column.
181 BBWWYYB182 BBYYBWB183 YBYBWWY184 YBWWWWY185 BYWWWWB186 WYWWWWW187 WYYWWWW188 WBYWWWW189 YBWBYYY190 WYWBWWW191 BYWBWWB192 YYWWWWY193 WYBWWWW194 YBYRWYY195 YYYWWWY 196 RYWWWWR197 WYWBWBW198 WRYWWWW199 BBYYBGB200 BBGWYYB201 BGGWWWB202 WYWBWGG203 BYGBWWB204 YBGBYYY205 GYWBWWW206 BBWGYYR207 WYGBWWW208 RBWWYYB 209 BRYYBWY210 WRWWGGW211 WRGBWWW212 YRWBYYB213 YRYRWYG214 BYWWBGB215 YYGBWWY216 BYWWBWB217 WWYWWBW218 GBYRWYY219 YGBWWWY220 BYWWBYB221 YYYBWWY222 BYGWWWB 223 YBYRGYY224 YYBWWWY225 GYWBWBW226 WGRWWWW227 BYWGBYB228 WWYGWBG229 GWYWWBW230 BYGWBYR231 WGYWWBW232 RYWWBYB233 YRYYBWB234 BRWBYYY235 YBGBWWY236 YGYWWWY
Table Execution trace of the flash in the active memory switch in both possible situa-tions.
Table 8 allows us to see the process of changing the selected track when thecell 0 can see the locomotive on the non-selected track. The rules which applyto this cell are rules 181, 206, 208, 214 and 220 when the selected track is theleft-hand side one. The flash is managed by rules 206 and 208, and rule 214witnesses the change of selection. Rule 220 is the conservative rule for the otherposition. Namely, when the selected track is the right-hand side one, the rulesare 220, 230, 232, 203 and 181. New rules manage the flash as the cell 0(0)can directly see from where the locomotive comes. Rule 215 again witnesses the22hange and the conservative rule is again rule 181.With these rules and with this study illustrated bu the figures, we completedthe proof of the following result:
Theorem There is a rotation invariant cellular automaton on the pentagridwith states which is planar and weakly universal. References [1] F. Herrmann, M. Margenstern, A universal cellular automaton in the hy-perbolic plane,
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Cellular Automata in Hyperbolic Spaces, vol. , Theory ,Collection: Advances in Unconventional Computing and Cellular Automata ,Editor: Andrew Adamatzky, Old City Publishing, Philadelphia, (2007),422p.[4] M. Margenstern,
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