A weakly universal cellular automaton with 2 states on the tiling {11,3}
aa r X i v : . [ n li n . C G ] J a n A weakly universal cellular automaton with 2 stateson the tiling { , } Maurice
Margenstern
Laboratoire d’Informatique Th´eorique et Appliqu´ee, EA 3097,Universit´e de Lorraine,Campus du Saulcy,57045 Metz Cedex, France, email: [email protected] email: [email protected]
September 12, 2018
Abstract
In this paper, we construct a weakly universal cellular automatonwith two states only on the tiling { , } . The cellular automaton isrotation invariant and it is a true planar one. The paper makes use of the improvements introduced in the paper [7]. Asmost of the author’s papers in this topic, this paper makes use of the rail-way model, see [10, 5]. We just remind the reader that the circuit simulatesa register machine instead of a Turing machine as in [10]. The Euclideancircuit consists in tracks which are either straight segments or quarters ofa circle. The circuit allows crossings and it makes use of three kinds ofswitches: the fixed one, the flip-flop and the memory switch. A uniquelocomotive runs over the circuit and the evolution of the positions of theswitches in time allows us to simulate a computation. Although initiallydevised for the Euclidean plane, this model can be implemented in the tes-sellations of the hyperbolic plane, especially in the tessellations { p, } and { p +2 , } which are spanned by a tree which, in each case allows a rathereasy way to implement horizontals and verticals.1he first implementation, in 2002, was performed in the pentagrid, thetiling { , } of the hyperbolic plane, see [1]. It required 22 states. Thenumber of states was lowered down to 9 in the same tiling in 2008, see [8].At the same time, the circuit was implemented in the heptagrid, the tiling { , } of the hyperbolic plane, see [9], requiring 6 states. Both papers of 2008gave the same implementation. The smaller number of states is compensatedby a higher number of neighbours. A bit later, I reduced the number of statesdown to 4 in the heptagrid by a slight change in the locomotive: replacingthe previously green front cell by a blue one, the same colour as that ofthe milestones. This smaller number of states in the heptagrid comparedto the pentagrid is not surprising. It was again observed a bit later in 2012with the implementation in the tiling { , } of a weakly universal rotationinvariant and planar cellular automaton with two states only, see [6, 5]. Thislatter implementation introduces many features which allowed to lower thenumber of states. The first idea was to mark the tracks by milestones insteadof assigning a specific colour to the tracks. Then, the idea was to allow one-way tracks only. This implies a strong change in the switches. The fixedone is simplified to a passive switch only: the switch is only needed whenthe locomotive comes from one of the two tracks which join into a singleone at the switch. The flip-flop was already a one-way switch in its originalimplementations as it must be crossed actively only. The constraint of one-way tracks implies to split the memory switch into two switches: an activeone and a passive one. The working of the switch makes it necessary toconnect the active switch to the passive one: when the locomotive comesfrom the non-selected track in the passive switch, the selection changes thereand it also must change in the active switch.These new features where enough to reduce the number of states downto two of them in the dodecagrid, the tiling { , , } of the hyperbolic 3 D -space. The space allows us to get rid of the crossings, which is not at all thecase in the plane. Moreover, the introduction of one-way tracks makes thenumber of crossings to seriously increase: a previous two-way crossing has tobe implemented by four one-way crossings. The implementation in { , } introduced a new feature coming from roadway traffic, the round-about. Italso reduced the locomotive to a single cell, making it closer to a particle.Trying to implement these new ideas in the heptagrid, I recently obtaineda weakly universal cellular automaton with three states only, see [7]. Now,in this paper, I had to improve the implementation of the one-way switches.In particular, introducing patterns used in asynchronous cellular automata,I could organize the implementation of the flip-flop and the active-memoryswitch in a similar way, using two new and simpler components, the fork and2he selector. The difference in the switches is now given by the assemblingof these new components, a bit different in these two switches. I thought itcould be useful to implement these new elements in order to obtain a cellularautomaton with two states in a grid { p, } with p <
13. I could do that for p = 11 by replacing the passive memory switch by a combination of the forkwith a new structure, the controller, a bit simpler than the passive memoryswitch itself.We make all of this more precise in Section 2.Before turning to Section 2, let us explain how we shall illustrate ourimplementation. The tiling { , } in the Poincar´e’s disc model is illustratedby the leftmost picture of Figure 1. Figure 1
To left: the tiling { , } in the Poincar´e’s model. To right: anotherrepresentation of a cell in the tiling { , } and the same one together with itsneighbours. We do not see very much in this representation, so that we shall replaceit by the two other illustrations given in Figure 1. As will be seen in thefurther illustrations, this representation allows us to give it some flexibilitywhich will improve the readability of the figures.
In this section, we precisely describe the implementation of the railway cir-cuit. Here, we admit that it is possible to devise such a circuit that themotion of the locomotive simulates the computation of a register machinethanks to the positions of all switches of the circuit. This is explained withall details in [5].Let us remind the reader that the systematic one-way track organizationof the circuit leads to a more complex representation of the memory switch.On the left-hand side of Figure 2 we have a sketchy representation of the3wo-way memory switch. We remind the reader that the two-way switchmay be crossed either passively or actively. On the right-hand side, we havethe one-way switch. It consists in two one-way half-switches: an active half-switch on the left-hand side and a passive one on the right-hand side. Aconnection goes from the passive half-switch to the active half-switch: this isneeded when the selection has to be changed. It first changes at the passivepart which detects that the locomotive came from the non-selected track,and the necessity to change the selection is passed to the active part throughthe orange path of Figure 2. two way switch one way switch
Figure 2
Comparison between the two-way and the one-way representation of thememory switch.
In this section, we precisely explain the new features in Sub-section 2.1and how they are implemented in Sub-section 2.2.
Let us first list the elements we shall study in Sub-section 2.2. With eachname of an element, we sketchy describe what the element is expected toperform.First, the elements of the tracks : each cell of the track is marked withappropriate milestones. Such an element may be crossed either by a singlelocomotive or by two locomotives running together, contiguously. We shallsee the elements in Sub-section 2.2.Then, the second structure we need is the fixed switch . As mentionedin the introduction, for one-way tracks it is a passive structure only. Itgathers two tracks which are melted into a single one after the cell at whichthe two tracks arrive.Next, we have two patterns involved by the round-about: the duplicator and the selector . 4he duplicator has two points of connection with the tracks: an en-trance and an exit . The locomotive arrives through the entrance. Twolocomotives leave the duplicator through the exit .The selector has three points of connection with the tracks: an entrance and two exits . Each exit correspond to the number of locomotives enteringthe selector. If a single locomotive enters, it leaves the selector through exit 1 which is connected with a track leaving the round-about. If twocontiguous locomotives enter the selector, one of them leaves the selectorthrough exit 2 . Exit 2 is attached to a piece of tracks leading to anotherselector.
A B f Figure 3
The round-about. The arrows indicate the path used by the locomotive.
This explains how a round-about works. This is illustrated by Figure 3.In the figure, the duplicator is illustrated by a green rhomboid pattern. Theselectors are illustrated by a disc. A round-bout assembles two duplicators,three selectors and a fixed switch, denoted by f in the figure. The locomotivecrosses one disc and leaves the round-about at the second one. When itarrives through A , B , the locomotive is transformed into two contiguouslocomotives after crossing the duplicator. When it meets the first selector,one locomotive is killed while the second one goes on its way on the round-about, towards the second selector. At the second selector, 3, 2 respectively,it leaves the round-about to continue its way on the same tracks.Next, we consider the controller , a structure which is used by boththe flip-flop and the active memory switch. The controller has two states : accept or reject . If on accept , the locomotive entering the controller isaccepted and it is allowed to cross it in order to go on its way on the tracks.If the controller is on reject , then the locomotive is not accepted: it does5ot leave the pattern, it is killed there.Next we consider the fork . It is different from the duplicator: here too,one locomotive enters and two ones exit, but the two exiting locomotivesare on different tracks. The fork has an entrance and two exits . Whenthe locomotive has crossed the fork, there is one locomotive on each trackleaving the pattern. L RC A S
Figure 4
Association of forks and controllers in order to constitute a flip-flop.
Both structures, the controller and the fork are used to implement a flip-flop and an active memory switch. Figure 4 shows how to combine threeforks and two controllers in order to obtain a flip-flop. functional goal scheme of implementation
Figure 5
To left: the functional goal of the new implementation of the memoryswitch with one-way tracks. To right: the scheme of implementation.
Note that this is connected with what we have indicated in Figure 2.Now it is time to make this idea more precise. Figure 5 gives the generalscheme of implementation of the memory switch under the constraint of6ne-way tracks.Now we turn to a more exact implementation of the scheme illustrated bythe right-hand side part of Figure 5. First, Figure 6 shows how to combinethe same elements in order to obtain an active memory switch, following thescheme of Figure 5.
L RC S
Figure 6
Association of forks and controllers in order to constitute an activememory switch.
L RF F S S Figure 7
Association of forks and new controllers in order to constitute a passivememory switch.
In order to implement the scheme proposed by Figure 5, we have tointroduce a a new element. Note that the idea is to align the the presentationof the passive memory switch with that of the active one by splitting a uniquecontrol at the switch into two independent controls on each branch of the7racks going to the switch. Now, due to the basic difference of working ofthe switches, we cannot use the same controller as in the active memoryswitch. The reason is that here, the controller must not stop the locomotivewhich crosses its structure. It has only to react to such a passage in case itis not the selected one. This means that the controller performs a doublefunction, but not at the same time. When the controller is on the presentlynon-selected track, it changes the selection and, at the same time, it sendsa signal to the active memory switch in order to change the selection theretoo.Figure 7 illustrates how the fixed switch, the fork and the new controllerhave to be associated in order to implement a passive memory switch, fol-lowing the scheme of implementation given in Figure 5. Note that the trackleaving the switch F2 in Figure 7 is the track arriving to the fork S in Fig-ure 6. Now, let us turn to the exact implementation of the patterns described inSub-section 2.1. We shall intensively use the representation introduced byFigure 1. Now, as can be noticed already from Figure 9, we shall not alwaysrepresent the neighbours of the same cell by circles or coloured discs of thesame size. The size of a neighbour will mainly be dictated by its role inthe considered configuration. In particular, the necessity to represent theneighbourhood of this neighbour will also play a role.We shall successively study the elements of the tracks, the duplicatingstructure, the selector, then the fork, the controller for the active switchesand the controlling structure of the passive memory switch. We shall study idle configurations only, i.e. configurations when the locomotive is notin the neighbourhood of the structure. We shall illustrate the motion ofthe locomotive through the structures when we shall establish the rules, seeSection 3.
The tracks
The tracks are the first object we have to implement. We must not neglectthis point. First of all, without tracks, the information of what happensat some switch will for ever remain in the switch, which is of no use forthe computation. Second, the tracks certainly constitute the biggest part ofthe circuit in term of quantity of involved elements. Transported into thehyperbolic plane, the horizontal parts of the tracks represented in Figures 48o 7 require a huge amount of cells.Using Figure 8 of the present sub-subsection, we can see on Figure 9how we can implement a track going from a cell to any other one. Notethat in this figure, we make use of the elements of the lower row in Figure 8.In principle, we could make use of these elements only. Indeed, consideringtwo fixed non-neighbouring cells P and Q . Consider a shortest path fromthe tile supporting P to that which supports Q . Figure 8 how to organizea track joining P to Q , using these elements only. Now, we use also theelements of the first row of Figure 8 in order to make the implementation ofthe other structures easier. In particular this will be used in the case of theround-about. IO IOOI OI
Figure 8
The four possible elementary elements of the tracks.
A B C P Q
Figure 9
The use of elements of the tracks in order to define a track going fromthe cell P to the cell Q . pivot of the cell. But, as we cansee on Figure 9, it is needed to consider the case when the tracks go fromone pivot to another one. This case is dealt with by the second column ofFigure 8. For such cells, we say that it is in between two pivots.Next, Figure 10 illustrates the implentation of the passive fixed switch.Note the particular configuration of the entrances to the switch. They areelements of the track but their working is a bit different as will appear inSection 3. I O
Figure 10
The idle configuration of the fixed switch.
The round about
In this sub-subsection, we successively study the duplicator and the selector.The idle configuration of the duplicator is illustrated by Figure 11. FromSub-section 2.1, we know that a single locomotive enters the structure andthat two contiguous ones leave it. I O
Figure 11
The idle configuration of the duplicator in a round-about.
The single locomotive enters through the cell marked by I in the figure,10hile the two locomotives successively leave through the cell marked by O .The presence of the locomotive makes cell 8 flash by turning to white andthen, at the next time, by turning back to black. When cell 8 is white, themain cell remains to be black which creates the needed second locomotivewhile the first one leaves the main cell. After that, cell 8 r eturns to blackso that the second locomotive also leave the duplicator.After the duplicator, we now look at the selector. It is a more complexstructure. It has to count how many locomotives arrive at the device andthen, depending on whether it is the case of one or two locomotives, it reactsin different ways. A BC DE Figure 12
The idle configuration of the selector used by a round-about.
The locomotive arrive to the structure by a the common neighbour of B and C at the bottom of B in Figure 12. Then, the locomotive arrives to A .There, the locomotive is duplicated on the two exits from A : the exit whichgoes along E and the one which goes along D , see the figure. Two cells cansee both A and B : the cells C and D . This allows the structure to counthow many locomotives arrive at it. We know that this number is one or twoso that, writing the state of A and then that of B , the configuration seen of AB seen from C and D is BW in the case of one locomotive and BB in the caseof two of them. If two locomotive arrive, one is killed, this means that A isblack for one time only and D turns to white while C remains black. Theeffect of this action is that the locomotive which arrives close to D is killedwhile that which was created at the common neighbour between C and E goes on along the track which will lead it to the next selector of the round-about. When a single locomotive arrives, so that the configuration seenfrom C and D is BW , C turns to white and D remains black. Accordingly,11he locomotive created at the neighbour of D goes on its way, leaving theround-about while that which was created close to C and E is killed. A DB
C 1 2
23 A B
Figure 13
Zooming at the idle configuration of the selector.
Figure 13 zooms at D and C which allows us to see more clearly howthe crossing is processed. Note that E plays a role: the locomotive createdat 1, right-hand side of the figure arrives at 2 when C becomes white. Sothat E has to become white when C is flashing. This allows us to kill thelocomotive sent in this direction. The active switches
From the structure of the selector, we can easily derive a structure whichwe call the fork which receives one locomotive and dispatches a copy of ittwo different directions.For the convenience of the reader, we reproduce the picture in the left-hand side part of Figure 14. We remind the reader that this structure isused in the flip-flop, the active memory switch and in the passive memoryswitch too, see Figures 4, 6 and 7. In the right-hand side part of the samefigure, we illustrate the controlling device used by both the flip-flop and theactive memory switch, again look at Figures 4 and 6.In the right-hand side part of Figure 14, we zoom at the cell c whichlooks at the passage of the locomotive. If the cell is white, the locomotive isallowed to pass and, necessarily, it passes: in this case, the cell t which is onthe track followed by the locomotive has the neighbourhood of an element ofthe track. If the cell is black, then it prevents the locomotive from enteringthe cell t so that the locomotive is killed.12 O O ct Figure 14
The idle configuration of the fork and of the controller used in theactive switches.
The passive memory switch
From Subsection 2.1, we know that the passive memory switch requires amore complex structure than the passive one: here we have rather a sensor than a controller. The reason is that the sensor does not stop the locomotive. c S t E Figure 15
The sensor used by the passive memory switch.
However, the sensor is a more active structure: if it is black and if alocomotive passes through the cell t , the cell S of the sensor which can seeboth c and t realizes that the locomotive is passively running on the non-13elected track. So that the sensor changes the selection: c becomes white.But on the other sensor of the switch, the cell c is also white so that secondcell c must become black. This is why the cell S sends a locomotive whichreaches both the second cell c and also the fork of the active switch in orderto change the states in both its controllers.Now, the signal sent by one of the sensors to the other enters the cell E which turns the white state of the cell c to black. In this section, we give the rules used by the automaton and we prove them.We first describe the format used for the rules and then we shall providethe rules used for each configuration. We illustrate the rules by figuresdescribing the motion of the locomotive in the different situations describedin Section 2 and, especially, in Subsection 2.2. The rules are numbered,which will allow us to follow there application in the scheduling tables ofthe section. These tables provides us with the state of the key cells in thecircuit at different times together with the rule which was applied at thattime for this cell.First, we fix the format of the rules. To this purpose, we number thesides of the cell and we say that the side i is shared by the cell and byits neighbour i . We shall consider that i ∈ { .. } and that the numbersincrease while counter-clockwise turning around a cell. As here we constructa rotation invariant automaton, it is not important to fix which side is side .For instance, considering Figure 8, the rule applied to the cell of the trackfor the leftmost cell in the upper row will be denoted WBBWBWBWWWWWW . In thisformat, in the non underlined part of the word, the i th letter from the leftindicates the state of the neighbour i : this can easily be checked on Figure 8.This format will be that of the rules which will be displayed from now on.In the word denoting a rule, the non-underlined part is called the context of the rule: it is the list of the sates of the neighbours, from 1 to 11.Second: before listing the rules we have to note that the role of manyrules consists in keeping a considered configuration persistent. By this wemean that after a possible modification introduced by the passage of thelocomotive, the configuration recovers a state it keeps most of the time or,at least, for a long time. As an example of the second situation we havethe controllers of the flip-flop: the indication of whether the passage by thelocomotive is allowed or not depends on the selected track which may bechanged but, once the selection is fixed, the indication remains permanent14ntil a possibly new one is fixed, much later if we consider the number ofsteps of the automaton. In many cases, the configurations are marked byblack cells, B in the rules while the background of the space is white, markedby W in the rules. In most situations, the white cells of the backgroundhave at most two contiguous neighbours. So that the first ten rules of theautomaton are: Table 1
The first rules: conservative rules for the milestones.
As Figure 8 represents the idle configuration only, we here provide the readerwith an illustration of the motion of one or two locomotives through anelement of the tracks, see Figures 16 and 17. This will allow the reader toeasier follow the checking of the rules given by Table 2. It is not superfluousto remind the reader that several rules of Table 1 are also applied in thissituation.
IO IO IO IOIO IO IO IOOI OI OI OIOI OI OI OI
Figure 16
The four possible motions for the elements of the tracks. Here, for asingle locomotive.
Let us look carefully at the situation. With Figure 8 in mind, fix a cellof the track. Let us consider that side is the leftmost black neighbour ofthe cell. Then, the rules of Table 2 apply to the cell of the track. Rules 11and 18, 28 and 34 apply to an idle configuration : this means that thelocomotive is far away from that cell, so that if it is white, it remains white15or the next time. For the milestones of the track, we can see that in an idleconfiguration rules 4 applies. For neighbours of a milestone, rules 3 and 5apply while rule 9 applies to the neighbours of the cell which are its entranceand its exit for the locomotive. Rule 10 applies to some milestones whenthe locomotive is in the cell. IO IO IO IO IOIO IO IO IO IOOI OI OI OI OIOI OI OI OI OI
Figure 17
The four possible motions for the elements of the tracks for two con-tiguous locomotives.
First, consider a cell with a single pivot. When a single locomotivecrosses it, the rules which apply to the cell are rules 11, 12, 13 and 14 inone direction, and rules 23, 24, 25 and 26 in the opposite direction. Whentwo contiguous locomotives cross the cell, rules 12 or 24 apply as the cell seethe first locomotive. When rules 12 and 24 are applied, the first locomotiveoccupies the cell while the second one is a neighbour of the cell, the neighbourwhich was occupied by the first locomotive at the previous time. This is whyrules 15 and 27 apply to the first locomotive. At the next time, the secondlocomotive occupies the cell while the first one already left it and occupiesa neighbour of the cell, neighbour 3 or 4, depending on the direction of themotion. Next, rules 16 and 28 apply to the second locomotive, making itto leave the cell. The cell can see the first locomotive in neighbour 3 or 4,depending on which type of cell, and it returns to white. Accordingly, atthe next step, the cell is white and the second locomotive is in neighbour 3or 4. As the cell has to remain white, rules 16 and 28 apply. At the nexttime, the configuration is idle again so that rules 11 and 23 apply to the cell.Note that a whole train of consecutive locomotives could cross the cellusing the rules for the case of two consecutive locomotives.For a cell in between two pivots, the principle is exactly the same. The16ifference is that instead of one black neighbour between the entrance andthe exit, there are two consecutive black neighbours.
Table 2
The motion rules: for a single locomotive and when two contiguous loco-motives travel on the tracks. In each case, motion in one direction and motion inthe opposite one. from left to right from right to leftone pivot in between one pivot in betweeni o
11 WBWBWBBWWWWWW12 WBBBWBBWWWWWB13 BBWBWBBWWWWWW14 WBWBBBBWWWWWW2 locomotives15 BBBBWBBWWWWWB16 BBWBBBBWWWWWW i o
17 WBWBBWBBWWWWW18 WBBBBWBBWWWWB19 BBWBBWBBWWWWW20 WBWBBBBBWWWWW2 locomotives21 BBBBBWBBWWWWB22 BBWBBBBBWWWWW o i
23 WBBWBWBWWWWWW24 WBBWBBBWWWWWB25 BBBWBWBWWWWWW26 WBBBBWBWWWWWW2 locomotives27 BBBWBBBWWWWWB28 BBBBBWBWWWWWW o i
29 WBBWBBWBWWWWW30 WBBWBBBBWWWWB31 BBBWBBWBWWWWW32 WBBBBBWBWWWWW2 locomotives33 BBBWBBBBWWWWB34 BBBBBBWBWWWWW
Table 3
Motion of a single locomotive. To left: motion from left to right. Toright: motion from right to left.
12 11 17 17 11 11 142 14
12 17 17 11 11 113 11 14
18 17 11 11 114 11 11 14
18 11 11 115 11 11 11 20
12 11 116 11 11 11 17 20
12 117 11 11 11 17 17 14
128 12 11 11 17 17 11 14
263 23 23 23 29 30
26 234 23 23 23 30
26 23 235 23 23 24
32 23 23 236 23 24
32 29 23 23 237 24
26 29 29 23 23 238
26 23 29 29 23 23 24
Table 4
Motion of two contiguous locomotives. To left: motion from left to right.To right: motion from right to left. As shown in the last line, cells and areneighbours on the tracks too.
16 15
12 17 17 11 11 142 14
16 15
18 17 11 11 113 11 14
16 21
18 11 11 114 11 11 14
22 21
12 11 115 11 11 11 20
22 15
12 116 11 11 11 17 20
16 15
127 12 11 11 17 17 14
16 15
12 11 17 17 11 14
27 28
27 28
263 23 23 23 30
33 28
26 234 23 23 24
33 34
26 23 235 23 24
27 34
32 23 23 236 24
27 28
32 29 23 23 237
27 28
26 29 29 23 23 248
26 23 29 29 23 24 Tables 3 and 4 illustrate the motion in the following way. Eight consec-utive cells of the track are taken in one direction in Table 3, in the otherdirection in Table 4. In both cases, cells are numbered from 1 to 8 and cells 4and 5 are in between two consecutive pivots. The tables indicate for eachcell and for each time which instruction applies. When the number of theinstruction is in red bold digits , this means that the corresponding cellis black before the rule is applied, otherwise it is white. Figure 9 allows us to17heck this. Note that Tables 3 and 4 were filled up by a computer programsimulating the automaton. Later on, such tables will be called scheduletables . The fixed switch has been described in Sub-section 2.2.Figure 18 illustrates the motion of one and two contiguous locomotivesthrough a fixed switch. The rules are given in Table 5. Basically, the motionis that of a locomotive through an element of the track. The first two linesillustrate the passive passage of a single locomotive. In the first row, thelocomotive comes from the left, in the second row, it comes from the right.The rule for keeping the idle configuration is rule 35. Rule 36 allows thelocomotive to enter the locomotive from the left. Rule 39 does the same fora locomotive coming from the right.
I O
I O
I O
I O
I O
I O
I O
I O
I O
I O
I O
I O
I O
I O
I O
I O
I O
I O
I O
I O
Figure 18
The motion of the locomotive through a fixed switch. First two rows:a single locomotive. Last two rows: two contiguous locomotives. able 5 The rules which handle a fixed switch. a single locomotive two locomotives tracks again i o i
35 WBBBWWBWBBWWW36 WBBBWBBWBBWWB37 BBBBWWBWBBWWW38 WBBBWWBBBBWWW39 WBBBWWBWBBBWB i ℓ o i r
40 BBBBWBBWBBWWB41 BBBBWWBBBBWWW42 BBBBWWBWBBBWB 43 WBWBWBBBWWWWW44 BBWBWBBBWWWWW45 BBWBWBBWWWWBW46 WBBWBWBBWWWWW
Rules 36 and 37 make the locomotive enter and then leave the cell.Rule 44 witnesses that the locomotive leaves the cell. In the case of twolocomotives, rules 40 and 42 allow the second locomotive to enter the cellwhile the first locomotive is about to leave it. There are two rules as thereare two possible entries. Rule 41 makes the second locomotive leave the cell.Tables 6 and 7 are the schedule tables illustrating the crossing of the switchby the locomotive(s). Here, the center of the switch is denoted by F , theentrances by i ℓ for the left-hand side one and by i r for the right-hand sideone. The exit is denoted by o . For each exit/entrance cell, its neighbourof the track is indicated as v ℓ , v ℓ and w , respectively. As the cells i ℓ and i r which are cells of the tracks are used with a different exit from whatis normally achieved, we need two additional rules for each cell: rules 43and 44 for the left-hand side and rules 45 and 46 for the right-hand one. Table 6
Motion of a single locomotive through the fixed switch. To left: the lo-comotive arrives from the left-hand side. To right: it arrives from the right-handside.
F o w v ℓ i ℓ v r i r
12 23 112 36 11 23 14
23 113
12 23 11 43 23 464 38
24 11 11 23 115 35 14
11 11 23 116 35 11 23 11 11 23 117 35 11 23 11 11 23 11 F o w v ℓ i ℓ v r i r
122 39 11 23 11 11 26
12 23 11 43 23 464 38
24 11 11 23 115 35 14
11 11 23 116 35 11 23 11 11 23 117 35 11 23 11 11 23 11
Table 7
Motion of two contiguous locomotives through the fixed switch. To left:the locomotives arrive from the left-hand side. To right: they arrive from the right-hand side.
F o w v ℓ i ℓ v r i r
16 15
23 112
12 23 14
23 463
41 15
24 11 43 23 464 38
16 27
11 11 23 115 35 14
11 11 23 116 35 11 23 11 11 23 117 35 11 23 11 11 23 11 F o w v ℓ i ℓ v r i r
28 15
12 23 11 43 26
41 15
24 11 43 23 464 38
16 27
11 11 23 115 35 14
11 11 23 116 35 11 23 11 11 23 117 35 11 23 11 11 23 11 .3 The round-about In this subsection, we first look at the duplicator and then at the selector.
Duplicator
The study of the duplicator is illustrated by Figure 19 and the rules aregiven by Table 8. Also, Table 9 allows us to check the application of therules and the working of the duplicator. For Table 9, note that D is thecentre of the duplicator, that i is the entrance for the locomotive, that i isthe neighbour of i on the tracks leading to D , that o is the exit through whichthe two locomotives leave D and that they go on the tracks, first through o and then through o .Note that the crossing of an element of the tracks or of a fixed switch bya single locomotive requires four steps exactly. If a locomotive is about toenter such a cell at time t , it just left the exit at time t +4. Now, when twoconsecutive locomotives cross the same elements, one more time is required.Here, we can see a similar situation: a single locomotive enters but twocontiguous ones exit, so that one more step is needed but no more, as thepattern is reduced to a specific neighbouring of a white cell. I O
I O
I O
I O
I O
I O
I O
Figure 19
The motion of the locomotive through the duplicator. One locomotiveenters and two ones leave the pattern.
Table 8
The rules for the duplicator. To left: the rules for the central cell. Toright, the rules for neighbour , n . the central cell neighbour i o n
47 WBWBBBWBBWBWW48 WBBBBBWBBWBWB49 BBWBBBWBBWBWB50 BBWBBBBBWWBWW51 WBWBBBBBBWBWW D52 BBBBBWWWBWWWB53 BBBBBWWBBWWWW54 WBBBBWWBBWWWB
The creation of the second locomotive is triggered by rule 50. It intro-duces a one step delay in the motion by keeping the central cell to be black.20ow, rule 50 might produce infinitely many locomotives. In order to reducethe creation of a new locomotive to a single one, neighbour of the centralcell flashes as the first locomotive entered D by rule 53 which makes it turnfrom black to white. Rule 54 makes neighbour return to the black stateat the following step. When rule 53 has been applied, so that neighbour iswhite, rule 50 applied to the central cell makes the second locomotive leavethe cell. Now, as shown by Tables 8 and 9, neighbour is ruled by rule 52. Table 9
Motion of the locomotive through the duplicator. D is the central cell, i , o the entrance, exit, respectively for the locomotive. i i D o o o n
24 47 23 11 11
48 23 11 11
24 11 11
50 27
12 11 545 23 23 51
28 15
16 15 52
13 52 Selector
As a consequence of the task assigned to the selector, its structure is morecomplex than what we have up to now studied. Figures 20 and 21 showus that several cells around the track used by the locomotive are involvedin the working of the structure. Tables 11 and 12 show that thirteen cellsare actively involved in the working of the selector. Also, Table 10 indicatesthat 43 new rules are needed. Tables 11 and 12 show that 12 rules of Table 2are also involved. Note that globally the automaton makes use of 115 rules,so that the selector requires almost the half of it.The thirteen cells involved in the working of the selector are i , the neigh-bour of B through which the locomotive(s) enter(s) the selector after crossing i , the neighbour of i on the tracks arriving to the selector: see Figure 20,last line, where the cells visited by the locomotive(s) are indicated in lightcolours. After B comes A from where the selection occurs, controlled by C and D which both can see A and B at the same time. From A , two locomo-tives exit, one through o r , neighboured by D , the other through o ℓ which isneighboured by C . The destruction of the superfluous locomotive requires toalso examine two additional cells of the tracks on each side: o r and o r onthe right hand side, o ℓ and o ℓ on the left hand side. Due to the number ofneighbours required for C , another neighbour of C , namely E , is involved inthis destruction process. Note that E is also a neighbour of the cells o ℓ , o ℓ o ℓ , while C is a neighbour of o ℓ only. Table 10
The rules for the selector. The table gives the rules for the cells A , D , C and E . Also see Figures and . A i CCBDo r o ℓ
55 WBWBWBBBBWBWW56 WBBBWBBBBWBWB57 BBWBWBBBBWBWW58 WWWBBBBBBWBBW2 locomotives59 BBBBWBBBBWBWW60 WBWWBBBBBWBBWfork only:61 WBBBBBBWBBBWW B i
62 WBBWBWBWBWWWW63 WBBWBWBBBWWWB64 BBBWBWBWBWWWW65 WBBBBWBWBWWWW66 WBBWWWBWBWWWW2 locomotives67 BBBWBWBBBWWWB68 BBBBBWBWBWWWW AoE B69 BWWBBBBBWWWWB70 BWWBBBBBWWBWB71 BWWBBBBBWWWBB72 WWBBBBBBWWWWB73 BWWWBBBBWWWWB2 locomotives74 BWWBBBBBWWBBB75 BBWBBBBBWWWBB76 BWBBBBBBWWWWBD E BBA 21o r
77 BWBBBBBBBWWWB78 BBWBBBBBBWWWB79 BWWBBBBBBWWBB80 BWWBBBBBBWBWB2 locomotives81 BBBBBBBBBWWWW82 WWWBBBBBBWWBW Co1283 BBWWWBBBBWBBB84 BWBWWBBBBWBBW85 WBWBWBBBBWBBB2 locomotives86 BBBWWBBBBWBBB87 BBWBWBBBBWBBB88 BBWWBBBBBWBBB around E89 WBBWWWBWWWWWW90 BBBWWWBWWWWWW91 WBWWBWBWWWWWW2 locomotives92 BBWWWBBWWWWWW93 WBBWWBBWWWWWW94 WBWWWBBWWWWWW 95 WBWWBWBBWWWWW96 BBBWBWWBWWWWW97 WBBBWBWBWWWWW
Table 11
The scheduling of the crossing of the selector by a single locomotive. i i B A o r o r o r o ℓ o ℓ o ℓ D C E1
63 23 55 11 11 17 29 23 29
76 69 83
24 55 11 11 17 29 23 29
76 70 83
56 11 11 17 29 23 29
77 71 83
12 11 17 30 23 29
78 22 83
12 17
24 29
18 97
80 73
857 11 62 23 55 11 14
29 23 29
77 69 83
76 69 83
Figures 20 and 21 can be viewed as different zooms with respect toFigure 12 of Sub-section 2.2. Tables 11 and 12 allow us to follow how therules are applied to the different cells constituting the selector.Let us make the description more precise. The arrival by i makes useof rules from Table 2 only. Now, i is not an ordinary element of the tracks.It has a specific configuration which is illustrated in Figure 20. Rules 62,63, 64 and 65 and also 67 and 68 play the role of motion rules for the cell i .Rule 66 is used when the cell C is white. Then, the locomotive(s) arrive(s)at B . This cell is alike an element of the track with this restriction that twoof its milestones may turn white at one moment exactly and only for thatinstant. So that the rules of Table 2 apply except when C or D is white.In the first case, rule 89 is applied. In the other one, it is rule 91. Thecell A is ruled by rules 55 up to 60. Rule 55 manages the idle configuration.Rule 56 make the locomotive enter the cell and rule 57 kills it at the next22ime. Rule 58 witnesses that each exit, the one close to C , the other closeto D , is occupied by a locomotive. When there are two locomotives, rules 59and 60 are applied instead of rules 57 and 58. Rule 61 is used only for thefork which has exactly the same neighbourhood as an idle cell A . A DB
C 1 2
A DB
C 1 2
A DB
C 1 2
A DB
C 1 2
A DB
C 1 2
A DB
C 1 2
A DB
C 1 2
A DB
C 1 2
A DB
C 1 2
A DB
C 1 2
A DB
C 1 2
A DB
C 1 2 i C B i Figure 20
The motion of the locomotive through the selector of a round-about.Focus on the cells A , B and D . On the last line of the figure, i and its neighbour-hood. We can notice that the cells C and D are applied very different rulesdepending on the number of locomotives arriving at the selector. Rules 76up to 80 are used when there is a single locomotive, witnessing the passagethrough the cells of the track which are in contact with D . Rule 81 detects23hat two locomotives arrived and turn the state of the cell D to white, whilerule 82 restores the black state at the next tick of the clock. Similarly,rules 69 up to 76 are used the cell C . It should be noticed that when thereis a single locomotive in A , the configuration of the neighbours is that of acell of the track in between two pivots when a locomotive is in the cell: thisis why rule 22 is used to make the cell turn to white. When there are twolocomotives, rules74 up to 76 are used. Note the rules for E : rules 83 upto 85 when there is a single locomotive and rules 86 up to 88 when there aretwo of them.
23 A B
23 A B
23 A B
23 A B
23 A B
23 A B
23 A B
23 A B
23 A B
23 A B
23 A B
23 A B
Figure 21
The motion of the locomotive through the selector of a round-about.Focus on the cells C and E . The remaining rules of the table appear in the destruction of the super-fluous locomotive: rule 89 for B when C , one of its milestones, momentarilyvanishes. Rule 90 kills the locomotive which is at o ℓ as the cell E is whiteat this instant. Rule 91 is used for B when D , also one of its milestones, iswhite for two ticks of the clock. Rule 92 is used by o r as its black but itsmilestone D is now white: and so, for the newt time when D is still white,24ule 94 applies. Rules 93 and 94 are used for o r which, in this case remainswhite but has a missing milestone: D . For a similar reason, rule 95 is usedfor o r as for it, D is at another milestone. Eventually, rule 96 is used by o ℓ when C is white and rule 97 is used at the next time, E being then white. Table 12
The scheduling of the crossing of the selector by two contiguous loco-motives. i i B A o r o r o r o ℓ o ℓ o ℓ D C E1
16 67
24 55 11 11 17 29 23 29
76 70 83
68 27
56 11 11 17 29 23 29
77 74 83
28 59
12 11 17 30 23 29
81 75 83
93 95
24 29 82
76 86
30 72
69 87
31 76 69 88
76 69 83
76 69 83
From Section 2, we know that we have two active switches: the flip-flop andthe part of the memory switch which is actively crossed by the locomotive.For these switches, we use two common structures: the fork and the con-troller. Figures 4 and 6 from Subsection 2.1 illustrate how these patternsare assembled in each case.
Fork
I O O I O O I O O I O O I O O I O O Figure 22
The motion of the locomotive through the fork in the switches.
The idle configuration of the fork is the same as that of the cell A in theselector. This can be checked on Figure 22. The centre of the fork makes useof the rules 55, 56 and 57 from those used by the cell A . As already mentionedit also makes use of rule 61. The slight differences with the rules applied tothe cell A of the selector come from the fact that in the neighbourhood ofthe centre of the fork, the black cells are invariant.25 able 13 The scheduling of the crossing of the fork by a single locomotive. i i t o o r o ℓ o ℓ
24 55 11 17 11 172 20
56 11 17 11 173 17 26
12 17 12 174 17 23 61
185 17 23 55 14 As will be seen further again, the fork will also be used in the passivememory switch as already shown in Figure 7 from Subsection 2.1.
Controller
For checking the controller of the active switches, we just need Figure 14and the explanations of Subsection 2.1. The rules are given by Table 14.
Table 14
Rules for the controller used by the active switches. the cell t the cell c C o i98 WBBBWBBBWWWWW t s99 WBBBBBWWBBBWW100 BBBBBBWWBBBWB t s101 WBBBBBWWBBBBB102 BBBBBBWWBBBBW
From the right-hand side of Figure 14, when the cell C of the controller iswhite, the cell t looks like a cell of the tracks: its neighbourhood is exactlyone of the neighbourhoods for cells which are elements of the tracks. Notethe use of rule 61 for the cell C when the controller is white, leaving thelocomotive cross the controller, see the lower half of Table 15. Table 15
The scheduling of the passage of the locomotive through the cell t , de-pending on whether the controller is black, upper table, or white, lower table. i i t o o r o r C s s
12 97 11 11 11
26 112 14
98 11 11 11
26 113 11 11 97 11 11 11
26 114 11 11 97 11 11 11
26 11i i t o o r o r C s s
12 23 11 11 11 99 23 112 14
24 11 11 11 99 23 113 11 14
12 11 11 61 23 114 11 11 26
12 11 99 23 115 11 11 23 14
12 99 23 116 11 11 23 11 14
99 23 117 11 11 23 11 11 11 99 23 11
When the cell C is black, rule 97 manages the idle configuration of t andrule 98 prevents a locomotive to enter t when C is black. The two right-hand26ide columns of Table 14 handle the behaviour of the cell C . Table 16
The scheduling of the change in the controller. Upper table: from un-locked to locked. Lower table: from locked to unlocked. i i t o o r o r C s s
143 11 11 97 11 11 11
26 114 11 11 97 11 11 11
26 11i i t o o r o r C s s
102 28
143 11 11 23 11 11 11 99 23 114 11 11 23 11 11 11 99 23 11
Rules 99 and 100 manage the idle configuration in which C may be eitherwhite or black but permanently in the same state. Rules 101 and 102 handlethe change of state of C which is triggered by the arrival of a locomotivethrough its neighbour 11. Note that both rules have exactly the same contextas in both cases, the neighbourhood of C is the same. From Subsection 2.1, we know that the new pattern involved in the pas-sive memory switch is the structure we called the sensor. The structure isillustrated by Figure 15.
Sensor
As in the case of the controller, if the cell C is white, the cell t behaves as acell of the track as it has one of the defined neighbourhoods for the elementsof the tracks. However, and it is one of the differences with what happens inthe controller, when C is black, the locomotive must nevertheless cross thecell t . So that in this case, we have new motion rules given by rules 103 upto 106, that one included.This time, the cell C has two auxiliary cells S and E . Rules 114 and 115handle the case of the idle configuration for S , no matter which is the stateof C . When the cell C is white, if a locomotive runs through t , C remains whiteas this is the case of a passage through the selected track: this is checkedby rule 107. When C is black, this means that the track is not selected. Thepassage of the locomotive requires that C changes to white and that it willkeep the state white until a signal of a change comes through the cell E . Thechange from black to white for C is controlled by rule 108. But at the same27ime, S must flash from white to black and back to white through rules 56and 68: look at Table 18. When S has flashed, note the locomotive movingfrom v S and v S : this locomotive will reach the other sensor of the switch tothere make the cell E flash in order to turn the cell C from white to black. Table 17
Rules for the sensor used by the passive memory switch. cell t cell c cell e cell s C o i103 WBBBWBBWBWWWW104 WBBBWBBBBWWWB105 BBBBWBBWBWWWW106 WWBBBBBWBWWBW tS E107 WBBBBBBBWBWWW108 BBBBBBBBWBWWW109 WBBBBBBWWBBWB C i110 WWBWBBWBBBWWW111 WBBWBBWBBBWWW112 WWBWBBBBBBWWB113 BWBWBBWBBBWWW Ct o114 WBWWBBBBWBWBW115 WWWWBBBBWBBBW
Let us look at that: the auxiliary locomotive arrives to v E and thento v E . As can be seen in Table 18, rule 112 makes E turn from white toblack and then rule 113 makes in turn back to white. But when C can seethat E is black, it turns from white to black thanks to rule 109. Note thatfor E , rules 110 and 111 manage its idle configuration whatever the stateof C . Table 18
The scheduling of the passage of the locomotive controlled by the sensor. i i t o o r C E S v E v E v S v E
12 103 11 11
111 114 11 11 11 112 14
104 11 11
111 114 11 11 11 113 11 14
12 11
111 56 11 11 11 114 11 11 106
12 58 110
11 11 12 115 11 11 29 14
82 110 115 11 11
126 11 11 29 11 11 82 110 65 11 11 14 i i t o o r C E S v E v E v S v E
12 29 11 11 82 110 65 11 11 11 112 14
30 11 11 82 110 65 11 11 11 113 11 14
12 11 107 110 55 11 11 11 114 11 11 32
12 82 110 65 11 11 11 115 11 11 29 14
82 110 65 11 11 11 116 11 11 29 11 11 82 110 65 11 11 11 11i i t o o r C E S v E v E v S v E
11 112 11 11 29 11 11 82 112 65
14 11 113 11 11 29 11 11 109
65 14 11 11 114 11 11 103 11 11
111 114 11 11 11 115 11 11 103 11 11
111 114 11 11 11 116 11 11 103 11 11
111 114 11 11 11 11
With the help of the figures, of the tables for the rules and of the scheduletables we have proved the following result:
Theorem 1
There is a weakly universal cellular automaton on the tiling { , } which is planar and rotation invariant.
28e remind the reader that with planar , we mean that the set of cellswhich are crossed by the locomotive(s) contains infinitely many cycles.
It can be asked whether it is possible or not to lower the number of neigh-bours in order to get a planar rotation invariant weakly universal cellularautomaton in a tiling { p, } or { p, } .A first remark on the number of rules.The tables we have displayed for the rules indicate 115 of them. However,there might be more rules: this depends on what we call the program of thecellular automaton. We should notice that these 115 rules are rotationallyindependent: none of them can be obtained from another one by a circularpermutation on the neighbour’s states. Accordingly, if we consider thatthe program of a rotation invariant cellular automaton should contain allrotated forms of the same rule, there should be much more rules: roughly115 ×
11. However, this is not exactly true. If we consider the motion rulesfor the tracks, then it is likely that any rotated form will be met in theconstruction of the tracks needed for the simulation of a register machine.However, for the cells concerning a structure, this may be not the case. Itmay be arranged that all configurations make use of the same rules whichare the ones used in the tables of the paper, or that we have only to taketwo or three rotated forms of the same rules but not all of them.Table 19 gives us the list of the rules which occur in two different set-tings. As can be checked, for the time and the cell to which they apply, theconfigurations of the neighbours and the state of the cell is the same.It should be noticed that the same cell may appear several times in thistable. As an example, the cell t of the fork appears four times in the table.It appears twice when the fork is idle, also when the locomotive is about toenter t and when two locomotives leave the fork. The neighbourhood is thatof the cell A of the selector when it is idle, when the locomotive is in i in theselector, when the locomotive is in A in the selector and for the cell C of thecontroller when the locomotive there is in t . In rules 57 and 61 which bothapply to the fork and something else, the context differ by the fact that o r is white in rule 57 but black in rule 61. We can see that the neighbourhoodof the cell is rotated in one rule with respect to what it is in the other rule.Note also that rules 79 and 82 have the same context, but the state ofthe cell is different. In this case each rule also applies to two different cells.However, there are several cases of different rules having the same context.This is in particular the case for motion rules applied to the same cell and29lso for rules managing the cell C in the controller and the cell C in the sensor.It is possible that the rather big number of rules applied in differentcontexts indicates that it must be difficult to lower the number of sides of atile in a tiling { p, } in order to obtain a weakly universal cellular automatonwhich would be rotation invariant and which would be planar. This seemsdifficult at least using the railway model. I say it might be difficult, I cannotsay it is impossible. Table 19
Table of rules which are applied at list twice, each time in a differentcontext. For each rule, identified by its number, the table gives the cell to which itis applied by its name in the structure. The time is also indicated and, possibly asan index, the case where it appears in its schedule table. Also, the context of thecell is displayed above the rule.
F i o46 WBBWBWBBWWWWWo Ei fix: i r , 3 sel: o ℓ , 6 CBDo r o ℓ
55 WBWBWBBBBWBWWi o r o ℓ Ct v S sel: A , 0 for: t , 0 sen: S , 3 CBDo r o ℓ
56 WBBBWBBBBWBWBi o r o ℓ Ct v S sel: A , 2 , 3 for t , 2 sen: S , 3 CBDo r o ℓ
57 BBWBWBBBBWBWWi o r o ℓ sel: A , 4 for: t , 3 CBDo r o ℓ
58 WWWBBBBBBWBBWtS E sel: A , 5 sen: C , 0 o r o ℓ i61 WBBBBBBWBBBWWt s for: t , 4 con: C , 3 B i
65 WBBBBWBWBWWWWv S Ct sel: i , 3; A , 5 sen: S , 0 B i
68 BBBBBWBWBWWWWv S Ct sel: i , 2 sen: S , 4 AoE B72 WWBBBBBBWWWWBC s sel: C , 5 ; D , 5 con: s , 0 BA 21o r
79 BWWBBBBBBWWBBs t sel: D , 5 sen: C , 0 BA 21o r
82 WWWBBBBBBWWBWs t sel: D , 4 sen: C , 0 vECA97 WBBBWBWBWWWWWC o i sel: o ℓ , 6 con: t , 030 eferences [1] F. Herrmann, M. Margenstern, A universal cellular automaton in thehyperbolic plane, Theoretical Computer Science , , (2003), 327-364.[2] Cellular Automata and Combinatoric Tilings in Hyperbolic Spaces, asurvey, Lecture Notes in Computer Sciences , Editors: Cristian Calude,Michael J. Dinneen, V. Vajnovszki, , (2003), 48-72, doi: 10.1007/3-540-45066-1 4 .[3] M. Margenstern,
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