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Universal Cellular Automata in the Hyperbolic Spaces
MAURICE MARGENSTERNUniversite Paul Verlaine− Metz,
LITA, EA 3097, UFR-MIMCampus du Saulcy, 57045 METZ Cedex
Abstract: This paper introduces several cellular automata in hyperbolic spaces which are able to simulate anycomputational device, as they are universal. We sketchily present five universal cellular automata, four of them ontwo grids of the hyperbolic plane and one in a grid of the hyperbolic3D space.
Key–Words:Theory of Computation, Cellular Automata, universality, hyperbolic geometry.
1 Introduction
This paper belongs to a long series of papers initiatedby the author with his contribution to a combinatorialapproach to hyperbolic geometry. Full material aboutthis approach can be found in the two-volume book ofthe author, see [3, 4], where many references can befound.
The basic idea of the approach followed by theauthor is a location technique for the tiles of regu-lar tilings of the hyperbolic plane. It is well knownthat there are infinitely many tilings in the hyperbolicplane, a consequence of Poincare’s theorem character-izing tessellations defined by a triangle, see [13]. Aninfinite family of these tilings can be defined as Cay-ley graphs of a group of displacements which leavethe tiling globally invariant. However, this classi-cal method does not work for infinitely many regulartilings of the hyperbolic plane. Moreover, it does notnecessarily solve the word problem, even in the caseswhen it can be applied as, in general, the word prob-lem for groups is undecidable, see [12].
The technique presented and developed in [3, 4]is simple and very powerful. It works on all regulartilings of the hyperbolic plane as well on the tiling{5, 3, 4} of the hyperbolic3D space and the tiling{5, 3, 3, 4} of the hyperbolic4D space. The locationtechnique presented in these books has many appli-cations. Among them, it was used by the author toimplement cellular automata in the hyperbolic planeand in the hyperbolic3D space.
The goal of this paper is to present five univer-sal cellular automata in this setting, four of them inthe hyperbolic plane and one of them in the hyper-bolic 3D space. All of these automata simulate a reg-ister machine, using the model of a railway circuit de-
scribed in [14]. We sketchily remind the reader thismodel in Section 2.
Consequently, as register machines are computa-tionally universal, these cellular automata are also uni-versal. Now, as their initial configuration is infinite,they are saidweakly universal. Note that this def-inition also contains the following constraint on theinitial configuration which, although infinite, shouldnot be arbitrary. It is required that the initial configu-ration could be defined by a primitive recursive func-tion. This will be the case: outside a finite domain,the configuration can be split into two parts, each onebeing globally invariant under an appropriate shift ofthe tiling.
The other common property of these cellular au-tomata is that they have a small number of states and,for four of them, a very small number of states. Thecellular automata of the hyperbolic plane which willbe presented work on two grids of this space whichwe present in Section 3. One of them is thepenta-grid , the tiling{5, 4} of the hyperbolic plane, and theother is theheptagrid, the tiling{7, 3}. In Section 4,we present two universal cellular automata on the pen-tagrid. The first one with 22 states, obtained in 2002,see [1] and the other with 9 states, a rather recent re-sult, see [10]. In Section 5, we present two universalcellular automata on the heptagrid, one with 6 states,see [11], also a recent result, and one with 4 states, anew result.
In Section 6, we present the implementation ofthe same model in the hyperbolic3D, giving rise to auniversal cellular automaton with 5 states, see [7].
For all the cellular automata of this paper, simu-lation programs were written to simulate them. Fourprograms were written to this aim. The first one waswritten in C, to simulate the cellular automaton on
Proceedings of the 13th WSEAS International Conference on COMPUTERS
ISSN: 1790-5109 83 ISBN: 978-960-474-099-4
the pentagrid with 22 states. The other programs werewritten in ADA: one for the3D case, another pro-gram for the pentagrid, and the last one for the hepta-grid. They were used to establish all the rules neededby the cellular automata and also to check the globalcompatibility of the rules between each other.
2 Railway simulationAs initially devised in [14] already mentioned in [2,1], the railway simulation is based on a single loco-motive running along a circuit defined by tracks andswitches. At each time, the configuration of the circuitis defined by the states of all switches. There are threekinds of switch: thefixed, thememory and theflip-flop switches. They are represented by the schemesgiven in Figure 1, using straight segments and arcs ofcircles.
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Figure 1 The three kinds of switches. From left to right:fixed, flip-flop and memory switches.
A switch is an oriented structure: on one side, ithas a single tracku and, on the the other side, it hastwo tracksa andb. This defines two ways of crossinga switch. Call the way fromu to a or b active. Callthe other way, froma or b to u passive. The namescome from the fact that in a passive way, the switchplays no role on the trajectory of the locomotive. Onthe contrary, in an active crossing, the switch indicateswhich track betweena andb will be followed by thelocomotive after running onu: the new track is calledtheselectedtrack.
As indicated by its name, thefixed switch is leftunchanged by the passage of the locomotive. It alwaysremains in the same position, always sending the loco-motive onto the same track during an active crossing.The flip-flop switch is assumed to be crossed only ac-tively. Now, after each crossing by the locomotive, itchanges the selected track. The memory switch canbe crossed by the locomotive actively and passively.In an active passage, the locomotive is sent on the se-lected track. Now, the selected track is defined by thetrack of the last passive crossing by the locomotive.Of course, at initial time, the selected track is fixed.
With the help of these three kind of switches, wedefine anelementary circuit as in [14], which exactlycontains one bit of information. The circuit is illus-trated by Figure 2, below, the indications0 and1 in-dicates the position of the switch representing the cor-responding bit value. Following the circuit, the readercan check that when the locomotive enters the circuit
throughE, it performs aread operation. When it en-ters the circuit throughU , it performs awrite opera-tion. It is assumed that the write operation is triggeredif and only if the bit is to be changed.
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Figure 2 The elementary circuit.
From this element, it is easy to devise circuitswhich represent different parts of a register machine.As an example, Figure 3 illustrates an implementationof a unit of a register.
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Figure 3 Here, we have two consecutive units of a reg-ister. A register contains infinitely many copies of units.Note the tracksi, d, r, j1 and j2. For incrementing, thelocomotive arrives at a unit throughi and it leaves the unitthrough r. For decrementing, it arrives thoughd and itleaves also throughr if decrementing the register was pos-sible, otherwise, it leaves throughj1 or j2.
Other parts of the needed circuitry are describedin [2, 1]. The main idea in these different parts is to or-ganize the circuit in possibly visiting several elemen-tary circuits which represent the bits of a configura-tion which allow the whole system to remember thelast visit of the locomotive. The use of this techniqueis needed for the following two operations.
When the locomotive arrives to a registerR, itarrives either to incrementR or to decrement it. Ascan be seen on Figure 3, when the instruction is per-formed, the locomotive goes back from the register bythe same track. Accordingly, we need somewhere tokeep track of the fact whether the locomotive incre-mented or decrementedR. This is one type of con-trol. The other type comes from the fact that severalinstructions usually apply to the same register. Again,when the locomotive goes back fromR, in general it
Proceedings of the 13th WSEAS International Conference on COMPUTERS
ISSN: 1790-5109 84 ISBN: 978-960-474-099-4
goes back to perform a new instruction which dependson the one it has just performed onR. Again this canbe controlled by what we called theselectorin [2, 1].
At last, the dispatching of the locomotive on theright track for the next instruction is performed by thesequencer, a circuit whose main structure looks likeits implementation in the classical models of cellularautomata such as the game of life or the billiard ballmodel. The reader is referred to the already quoted pa-pers for full details on the circuit. Remember that thisimplementation is performed in the Euclidean plane,as clear from Figure 4 which illustrates the case of afew lines of a program of a register machine in self-explaining notations.
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jmp 1inc Winc Z
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inc Wjmp 8dec W
inc Yjmp 12
Figure 4 An example of the implementation of a smallprogram of a register machine.
What we have shortly described is implementedin the Euclidean plane. In fact, it can be imple-mented in the hyperbolic plane and in the hyperbolic3D space.
We shall use the Poincare’s disc model of the hy-perbolic plane and its natural generalisation to3D forthe hyperbolic3D space.
We shall successively examine the three tilings inwhich our implementations will take place, to whichwe turn in our next section.
3 Grids in hyperbolic spacesWe shall consider three grids: two planar grids, thepentagrid and the heptagrid, the tilings{5, 4} and{7, 3} respectively, and a3D-one, the dodecagridwhich is the tiling {5, 3, 4}. The notation{p, q}means that each tile is a regular polygon of the hyper-bolic plane withp sides and that exactlyq of them canbe dispatched around a vertex, covering a neighbour-hood of the vertex with no overlapping. The notation{p, q, r} means that the faces are copies of a regularpolygon withp sides, thatq faces meet at a vertex on
the polyhedron and thatr polyhedra share an edge,covering a cylindrical neighbourhood around the edgewith no overlapping. This completely characterizesthe polygon of the faces which are copies of the pen-tagons of the grid{5, 4} and the fact that faces whichshare an edge are orthogonal.
3.1 The pentagrid and the heptagridAn important property is that both tilings are spannedby the same generating tree, see Figures 5 and 6.
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Figure 5 On the left: five quarters around a central tile;on the right: seven sectors around a central tile.
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Figure 6 The tree generating both the pentagrid and theheptagrid. See the representations of the numbers attachedto the nodes of the tree.
Figure 5 shows how to exactly split the hyperbolicplane into a central tile andα sectors, withα = 5 forthe pentagrid andα = 7 for the heptagrid. Number-ing the tiles on the right-hand side part of Figure 5allows to make the correspondence with the tree ofFigure 6. In the latter figure, numbers are also repre-sented in the basis defined by the Fibonacci sequencewith f1 = 1, f2 = 2, taking the maximal representa-tion, see[5, 3, 4]. The colours in both figures of Fig-ure 5 allow to show that the same tree generates a sec-tor in both tilings.
The main reason of this system of coordinates isthat from any cell, we can find out the coordinates ofits neighbours in linear time with respect to the coor-dinate of the cell. Also in linear time from the coor-dinate of a cell, we can compute the path which goes
Proceedings of the 13th WSEAS International Conference on COMPUTERS
ISSN: 1790-5109 85 ISBN: 978-960-474-099-4
from the central cell to this cell. These properties areestablished in [6, 3]. In the paper, we shall not use co-ordinates explicitly, but the programs which checkedthe correctness of the rules do use them intensively.
3.2 The dodecagrid
The method mentioned in section 3 can also be ex-tended to the dodecagrid. It gives rise to a spanningtree of spatial sectors which are corners of the space:they are the intersections of three pairwise perpendic-ular planes meeting at a point. From the propertiesof the tree, we can devise a coordinate system, as inthe planar cases of this section. However, due to alge-braic properties of the polynomial attached to the tree,we have cubic algorithms only to compute the coordi-nates of the neighbours of a cell from its coordinate.We refer the reader to [3, 9].
We shall use a standard central projection on aplane to represent the dodecagrid. Again see moreinformation in [3, 9].
4 Two universal cellular automata inthe pentagrid
In this section we remind the construction of two uni-versal cellular automata in the pentagrid. They bothimplement the circuit described in Section 2 into thepentagrid in the same way. The global setting is repre-sented by the left-hand side picture of Figure 7 wherewe give a sketchy view of the example of Figure 4 inSection 2.
Figure 7 Left-hand side: the global configuration of thecircuit in the pentagrid. Right-hand side: the same in theheptagrid. In both cases, note that instructions arrive tothe control of a register through tracks in the shape of anarc of a circle. Also note the return from the controller ofthe register when decrementing a register fails, because itscontent was zero.
4.1 The universal cellular automaton with22 states
The first universal automaton in the hyperbolic plane,see [1], was defined in the pentagrid and it has22 states.
The motion of the locomotive is performed alonga predefined path marked by a selected state denotedby B. This structure of the locomotive is the samein all the automata of the paper. On the path, thelocomotive consists of two consecutive cells, one ingreen,G, the front, the other in red,R, the rear. Theorganization of the motion in this case is very easy andstraightforward through the following rules which werefer to as themotion rules:
BBG → G, BGR → R, GRB → B, RBB → B
RGB → R, BRG → B, BBR → B
These rules are observed in three of the next fouruniversal cellular automata of the paper in the hyper-bolic plane. As indicated in [1], the crossings and theswitches are marked by a special cell at the tile whichstands on the intersection of the tracks defining thesemeetings of tracks. This special cell follows a cycleof specific states as it is crossed by the locomotive.
4.2 The universal cellular automaton with9 states
In [10], the previous result is significantly improved asa weakly universal cellular automaton is constructedin the pentagrid with 9 states only.
The main difference with the previous automa-ton lies in the fact that here, the crossing of the var-ious meeting points is as much as possible defined bythe motion rules. The actual rules may slightly devi-ate from this pattern at such a crossing: in this case,the front of the locomotive temporarily changes fromblue, B, to green,G. This deviation occurs for thecrossing of two paths and for the fixed switch.
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Figure 8 The idle configurations at a meeting of tracks.From left to right: crossing, fixed, memory and flip-flopswitches.
We have not enough room to show the motion ofthe locomotive in the different cases. Just to allow thereader to better understand the situation, Figure 8 in-dicates the configurations of the meeting points whenthe locomotive is not near the central tile. We callthem theidle configurations.
Figure 9 describes the crossing of a flip-flopswitch by the locomotive.
This crossing is calledactive as the locomotiveenters the switch from the unique track abutting thecentral tile at one isolated side. Then, the locomotiveis sent onto one of the two tracks which abut the cen-tral tile at two other consecutive sides.
Proceedings of the 13th WSEAS International Conference on COMPUTERS
ISSN: 1790-5109 86 ISBN: 978-960-474-099-4
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Figure 9 The locomotive crosses a flip-flop switch, neces-sarily actively, here from sector3. Note the change of theselected track when the rear of the locomotive leaves theswitch.
5 Two universal cellular automata inthe heptagrid
First, we have to remark that the result of the penta-grid cannot be immediately transferred to the hepta-grid. Indeed, around a vertex, there are four tiles inthe pentagrid while there are only three of them in theheptagrid. But, this is not a bad point for the hep-tagrid. In fact, this reduced number of tiles arounda vertex allows to better organize the transfer of in-formation when the locomotive approaches a switch.Also, the higher number of neighbours of a cell allowsto obtain a differentiation of the tiles with a smallernumber of states.
This was performed in [11] where the same au-thors as in [10] constructed a weakly universal cellularautomaton on the heptagrid with 6 states only.
5.1 The universal cellular automaton with6 states
The implementation is the same as in the case of thepentagrid, with this difference that we have 7 sectorsinstead of 5 ones, see the right-hand side of Figure 7,also implementing the same example as in Figure 4.
As in the case of the pentagrid where we do notgive details about the solution with 22 states, we donot go into the details of the solution with 6 states. Theparticularities of this solution can be seen as neededfrom the result with 9 states in the pentagrid although,as already mentioned we cannot go from one result tothe other by a straightforward transformation.
5.2 The universal cellular automaton with4 states
As a usual phenomenon in the construction of smalluniversal devices, when we try to the reduce the num-ber of states, the configurations of the main figures be-come much more complex. This is the case here with
the new automaton constructed to prove the followingresult:
Theorem 1 (Margenstern, see [8])There is a cellularautomaton on the heptagrid which is weakly univer-sal and which has4 states. Outside a large enoughdisc, the initial configuration consists of two infinitepatterns, each one being globally invariant under anappropriate shift.
To better understand the implementation of therailway circuit, we have to note that in the Euclideansetting, we only need to define horizontal and verticallines. In the hyperbolic plane this cannot be done butwe can implement a partial distorted grid, with hor-izontal andhalf-verticals which will be enough forour purpose. Figure 10 indicates how such a grid canbe implemented in the heptagrid.
From Figure 10, the horizontals look like horocy-cles although they are not exactly such curves. Thehalf-verticals follow rays which are defined by se-quences of consecutive yellow nodes. Such sequencesare infinite and we can manage the implementation insuch a way that the sequence always has a first tile.
Figure 10 The illustration of the implementation of a gridin the hyperbolic plane. The horizontals are the curves withmore or less alternating convex and concave arcs. The half-verticals are lines of yellow nodes.
With these indications, it is enough to implementa track which follows a half-vertical, a track whichfollows a horizontal and a piece of track which goesfrom a vertical to a horizontal and conversely. Thisthird kind of track will be called aslip road.
Figure 11 The locomotive goes along a path which maybe along a line.
Proceedings of the 13th WSEAS International Conference on COMPUTERS
ISSN: 1790-5109 87 ISBN: 978-960-474-099-4
For the automaton which is constructed to proveTheorem 1, we already redefine the implementationof the very tracks. Figure 11 shows a path which canbe used both to follow a half-vertical or to follow aslip road joining a half-vertical to a horizontal or con-versely.
In Figure 11 and all figures illustrating the be-haviour of the automaton proving Theorem 1, thefront of the locomotive has a darker colour than whatit has in the definition of the automaton: for the conve-nience of the reader this distinguishes the colour blueof the front with the same colour used in the definitionof the circuit.
Figure 12 The locomotive goes along a path which fol-lows a horizontal.
Figure 12 shows the implementation of the trackalong a horizontal. It can be noticed that this solutioncan also be used to implement a track which follows ahorizontal.
Figure 13 The idle configurations. From left to right:crossing, fixed, memory and flip-flop switches.
Figure 14 The locomotive arrives to a flip-flop switch. Itis necessarily an active passage and, after the rear of thelocomotive passed the signal, the selected track is changed.
And so, combining verticals, horizontals and sliproads which connect the two kinds of path, we getclosed curves which are distorted images of rectan-gles which cannot be similar as similarity does not ex-ist in the hyperbolic plane. But similarity is actuallynot needed: the fact that the closed curves get expo-nentionally bigger does not matter for our simulation.
Accordingly, it is possible to implement the variouselements of the circuit, following the Euclidean mapsgiven in Section 2.
Now, we can turn to the idle configurations andthen to the motion of the locomotive through thecrossings and the switches.
The idle configurations are given in Figure 13.We may notice that the differences are parallel to
those which exist in the case of the automaton with6 states, see [11]. We have two groups of meetingpoint. On one hand, the crossing and the fixed switchare similar. On the other hand, the memory and theflip-flop switch are also similar. The difference is im-portant between the two groups. Also, the mechanismof the change of selection in the memory and the flip-flop switch looks like an obstacle on the track in Fig-ure 13 which is not the case for the automaton with6 states.
As in Section 4, we only indicate the crossing ofa flip-flop switch by the locomotive, see Figure 14.Remember that the crossing of a flip-flop switch is al-ways active.
Other figures should be given in order to com-pletely illustrate all the cases of passage through thecrossings and through the switches. We cannot givethem due to the small amount of room we have forthat.
Let us indicate that the circuit can use a singletype of fixed switch, always sending the locomotiveon the same side. The other possibility can be ob-tained by appending a crossing after the selection.On the contrary, for the memory and the flip-flopswitches, we also have to consider the ’symmetrical’situation when the selected track is the other track ofthe switch.
6 A universal cellular automaton onthe dodecagrid
It is also possible to implement the same railwaymodel to simulate a register machine in the dodeca-grid. Thanks to the third dimension, crossings canbe avoided: they are replaced bybridges. So thatwe remain with the implementation of the switchesas the implementation of the tracks is straightforward:it works as in the cellular automata of Section 4.
In Figure 15, we can see the idle configurationof the switches. The pictures of the figure repre-sent the immediate neighbours of the central cell ofthe switch. They are central projections of the cen-tral cell from an appropriate point on a convenientplane. The central cell itself is the dodecahedronwhose sides are coloured in the picture. Note that
Proceedings of the 13th WSEAS International Conference on COMPUTERS
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around an edge, there are four dodecahedra. As anexample, the facesA, 2 andE of the pictures share anedge with the central cell. On each face we can put adodecahedron which is represented by a letter. Now,face 2 has a symmetric image which also shares thiscommon edge. Due to the rule of four dodecahedraaround an edge, we can conclude that the dodecahe-dron on face 2 also touches the dodecahedron on thefaceE: the common face is represented by the sym-metric image of face 2.
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Figure 15 The idle configurations for the switches in thehyperbolic3D space. From left to right: fixed, memory andflip-flop switches.
Below, Figure 16 shows the crossing of a mem-ory switch by the locomotive in the most complicatesituation, see the caption of the figure.
Figure 16 From left to right and from top to bottom: thecrossing of a memory switch by the locomotive when it ar-rives from the non-selected track. Note the change of theselected track which is completed on the last picture.
We refer the reader to [7] for further informationon this automaton which has 5 states only and whichis weakly universal, including the condition requiredon the initial configuration.
7 Conclusion
With the universal cellular automata with 4 states, weare close to the best possible result which would be2 states. Note that in the Euclidean plane there is auniversal cellular automaton with 2 states: the well-known game of lifedevised by Conway. But there isstill some work to do. Moreover, in the pentagrid weare farther from this minimal bound. Also, the ques-tion to improve the result is open in the hyperbolic3D space. The result with 4 states cannot be directlytransferred to the dodecagrid: it is obtained for the
heptagrid while planar intersections with the dodeca-grid give a pentagrid in the best case. The additionalpossibilities given by the third dimension are probablynot fully exploited.
Acknowledgements: The author is much in debtto Professor Nikos E. Mastorakis for inviting him topresent this paper to CSCC2009.
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[2] M. Margenstern, Two railway circuits: a uni-versal circuit and an NP-difficult one, ComputerScience Journal of Moldova,9, 1–35, (2001).
[3] M. Margenstern, Cellular Automata in Hyper-bolic Spaces, Volume 1, Theory,OCP, Philadel-phia, (2007), 422p.
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[5] M. Margenstern, New Tools for Cellular Au-tomata of the Hyperbolic Plane,Journal of Uni-versal Computer Science6(12), (2000), 1226–1252.
[6] M. Margenstern, Implementing Cellular Au-tomata on the Triangular Grids of the HyperbolicPlane for New Simulation Tools,ASTC’2003,(2003).
[7] M. Margenstern A Universal Cellular Automa-ton with Five States in the 3D Hyperbolic SpaceJournal of Cellular Automata1(4), (2006), 317-351.
[8] M. Margenstern, A new universal cellu-lar automaton on the ternary heptagrid,arXiv:0903.2108[cs.FL], (2009), 35pp.
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[10] M. Margenstern, Y. Song, A new universal cel-lular automaton on the pentagrid,Parallel Pro-cessing Letters, to appear.
[11] M. Margenstern, Y. Song, A universal cellularautomaton on the ternary heptagrid,Electroni-cal Notes in Theoretical Computer Science, 223,(2008), 167-185.
[12] P. S. Novikov,On the Algorithmic Unsolvabil-ity of the Word Problem in Group Theory, TrudyMath. Inst. Steklova44 (1955) (English trans-lation in Amer. Math. Soc. Translations (2),9(1958)).
[13] H. Poincare, Theorie des groupes fuchsiens,Acta Mathematicae,1 (1882), 1-62.
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ISSN: 1790-5109 89 ISBN: 978-960-474-099-4
