# A strongly universal cellular automaton on the pentagrid with six states and
Moore neighbourhood.

Maurice Margenstern

###### Abstract

In this paper, we prove that there is a strongly universal cellular automaton
on the pentagrid with six states. The rules of the cellular automaton are
rotation invariant and the cellular automaton works with a Moore neighbourhood
for each cell. The cellular automaton requires $1072$ rules.

##  1 Introduction

In many papers, the author studied the possibility to construct universal
cellular automata in tilings of the hyperbolic plane, a few ones in the
hyperbolic $3D$ space. Most often, the constructed cellular automaton was
weakly universal. By weakly universal, we mean that the automaton is able to
simulate a universal device starting from an infinite initial configuration.
However, the initial configuration should not be arbitrary. It was the case
that it was periodic outside a large enough circle, in fact it was periodic
outside such a circle in two different directions as far as the simulated
device was a two-registered machine. In almost all papers, the considered
tiling of the hyperbolic plane was either the pentagrid or the heptagrid, i.e.
the tessellation $\\{5,4\\}$, $\\{7,3\\}$ respectively. Both tessellations
live in the hyperbolic plane only. In the pentagrid, the basic tile is a
regular convex pentagon with right angles. In the heptagrid, it is a regular
convex heptagon with the angle $\displaystyle{{2\pi}\over 3}$ between
consecutive sides. Below, Figure 1 provides us with a representation of the
pentagrid in the Poincaré’s disc model of the hyperbolic plane.

In the left-hand side picture, we can see five tiles which are counter-
clockwise numbered from 1 up to 5, those tiles being the neighbours of a tile
which we call the central tile for convenience. Indeed, there is no central
tile in the heptagrid as there is no central point in the hyperbolic plane. We
can see the disc model as a window over the hyperbolic plane, as if we were
flying over that plane in an abstract spacecraft. The centre of the circle is
the point on which our attention is focused while the circle itself is our
horizon. Accordingly, the central tile is the tile which is central with
respect to the area under our consideration. The same picture also shows us
five arcs of circle issued from a vertex of the central tile, continuing a
side of that tile. Those rays delimit a right angle $S$, and we call sector
headed by $1$ the set of tiles whose centre are inside $S$, and tile 1 is
called the head of that sector. Similarly, we can define sectors $i$ with
$i\in\\{2..5\\}$ each of them having the tile $i$ as its head.

###### Figure 1

To left: the pentagrid; to right: showing the tree structure of the sectors
around a central tile indicated by the left-hand side picture.

In the right-hand side picture, we can see the central tile and the five
sectors which surround it: they are defined by the rays shown on the left-hand
side picture of Figure 1. In each sector we can see four tiles numbered from 1
up to 4: tile 1 is the head of the sector while tiles 2, 3 and 4 are the sons
of tile 1, constituting the first level in the sector. In each sector we have
three kind of tiles: green, yellow and blue ones. More precisely, the tiles of
a sector are coloured with three colours: green, blue and yellow. Blue tiles
have two neighbours which are blue and green in that order and green and
yellow tiles have three neighbours which are blue, green and yellow in that
order, turning counter clockwise around the tile. These indications define
rules from which we can build a tree which spans the tiles of the sector. Also
the numbering of the tiles can be continued level after level and from left to
right on each level. More indication on that topic and on the way to locate
the tiles can be found in [3, 2].

Now that the global setting is given, we shall proceed as follows: Section 2
indicates the main lines of the implementation which is precisely described in
Subsection 2.2. At last, Section 3 gives us the rules followed by the
automaton. The figures illustrating the various configurations considered in
Subsection 2.2 were established from pieces of figures drawn by a computer
program which applied the rules of the automaton displayed by Table 2 to an
appropriate window in each of the above mentioned configurations. The computer
program also checked that the set of rules is coherent.

That allows us to prove the following property:

###### Theorem 1

There is a strongly universal cellular automaton on the pentagrid with six
states whose rules are rotation invariant. The cellular automaton is truly
planar and for each cell the neighbourhood is considered in Moore sense. The
automaton makes use of $1072$ rules.

The number of rules of the automaton is much higher than those for cellular
automata on the heptagrid. That number is also a consequence of the Moore
neighbourhood used by the cells. We remind the reader that the Moore
neighbourhood of a cell $c$ consists of the cells whose support shares at
least a vertex with the support of $c$. Accordingly a Moore neighbourhood
consists of 10 neighbours instead of 5 of them in an ordinary neighbourhood.
If needed we distinguish the full neighbours of $c$, those which share a side
with $c$, from the partial neighbours which share a vertex only with $c$.

##  2 Main lines of the computation

The first paper about a universal cellular automaton in the pentagrid, the
tessellation $\\{5,4\\}$ of the hyperbolic plane, was [1]. This cellular
automaton was also rotation invariant, at each step of the computation, the
set of non quiescent states had infinitely many cycles: we shall say that it
is a truly planar cellular automaton. That automaton had 22 states. That
result was improved by a cellular automaton with 9 states in [12]. Recently,
it was improved with 5 states, see [5]. A bit later, I proved that in the
heptagrid, the tessellation $\\{7,3\\}$ of the hyperbolic plane, there is a
weakly universal cellular automaton with three states which is rotation
invariant and which is truly planar, [6]. Later, I improved the result down to
two states but the rules are no more rotation invariant, see [7]. Paper [4]
constructs three cellular automata which are strongly universal and rotation
invariant: one in the pentagrid, one in the heptagrid, one in the tessellation
$\\{5,3,4\\}$ of the hyperbolic $3D$-space. By strongly universal we mean that
the initial configuration is finite, i.e. it lies within a large enough
circle.

In the present paper, we borrow the ideas of [11].

Our automaton simulates a two-registered machine which is enough to entail the
strong universality as far as a two-registered machine can simulate any Turing
machine as proved in [13]. In our simulation, each register consists of a
finite sequence $\sigma$ of tiles where each tile of $\sigma$ is a full
neighbour of two other tiles of $\sigma$, two tiles of $\sigma$ being excepted
which have a single full neighbour in $\sigma$. One of those exceptional tiles
is the beginning of the register, the other is its end. Moreover, the tiles of
$\sigma$ have a side which lie on a line $\ell$ which is specific to $\sigma$.
At each time of the computation, the register is a finite sequence of tiles.
That sequence is appended one element to its end each time it is incremented
by the application of an appropriate instruction of the register machine
program. That appended element becomes the new end of the register.
Symmetrically, the sequence is removed the ending tile each time it is
decremented, with the exception of the case when the register is empty. The
new end of $\sigma$ is then the neighbour in $\sigma$ of the removed tile.

The simulation is based on the railway model devised in [14] revisited by the
implementations given in the author’s paper [10]. Sub-section 2.1 describes
the main structures of the model. In Sub-section 2.2 we indicate the new
features used in the present simulation. Those new features, introduced in
[8], allowed us to reduce the number of auxiliary structures used to simulated
the apparatus needed by the railway model as described in [14].

###  2.1 The railway model

The railway model of [14] lives in the Euclidean plane. It consists of tracks,
crossings and switches and the configuration of all switches at time $t$
defines the configuration of the computation at that time. There are three
kinds of switches, illustrated by Figure 2. The changes of the switch
configurations are performed by a locomotive which runs over the circuit
defined by the tracks and their connections organised by the switches.

A switch gathers three tracks $a$, $b$ and $c$ at a point. In an active
crossing, the locomotive goes from $a$ to either $b$ or $c$. In a passive
crossing, it goes either from $b$ or $c$ to $a$.

###### Figure 2

The switches used in the railway circuit of the model. To left, the fixed
switch, in the middle, the flip-flop switch, to right the memory switch. In
the flip-flop switch, the bullet indicates which track has to be taken.

In the fixed switch, the locomotive goes from $a$ to always the same track:
either always to $b$ or always to $c$. In any switch, the track followed by
the locomotive in an active crossing is called the selected track. The passive
crossing of the fixed switch is possible. The flip-flop switch is always
crossed actively only. The crossing of a locomotive entails the change of its
selected track. The memory switch can be crossed actively or passively. Now,
the selected track is the track taken by the locomotive in the last passive
crossing.

As an example, we give here the circuit which stores a one-bit unit of
information, see Figure 3. The locomotive may enter the circuit either through
the gate $R$ or through the gate $W$.

If it enters through the gate $R$ where a memory switch sits, it goes either
through the track marked with 1 or through the track marked with 0. When it
crossed the switch through track 1, 0, it leaves the unit through the gate
$E_{1}$, $E_{0}$ respectively. Note that on both ways, there are fixed switch
sending the locomotive to the appropriate gate $E_{i}$. If the locomotive
enters the unit through the gate $W$, it is sent to the gate $R$, either
through track 0 or track 1 from $W$. Accordingly, the locomotive arrives to
$R$ where it crosses the switch passively, leaving the unit through the gate
$E$ thanks to a fixed switch leading to that latter gate. When the locomotive
took track 0, 1 from $W$, the switch after that selects track 1, 0
respectively and the locomotive arrives at $R$ through track 1, 0 of $R$. The
track are numbered according to the value stored in the unit. By definition,
the unit is 0, 1 when both tracks from $W$ and from $R$ are 0, 1 respectively.
So that, as seen from that study, the entry through $R$ performs a reading of
the unit while the entry through $W$, changes the unit from 0 to 1 or from 1
to 0: the entry through $W$ should be used when it is needed to change the
content of the unit and only in that case. The structure works like a memory
which can be read or rewritten. It is the reason why call it the one-bit
memory.

###### Figure 3

The basic element containing one bit of information.

We shall see how to combine one-bit memories in the next sub-section as far as
we introduce several changes to the original setting for the reasons we
indicate there.

###  2.2 Tuning the railway model

We first look at the implementation of the tracks in Sub-subsection 2.2.1 and
how it is possible to define the crossing of two tracks. In Sub-subsection
2.2.3 we see how the switches are implemented. Then, in Sub-subsection 2.2.4,
we see how the one-bit memory is implemented in the new context and then, in
Sub-section 2.2.5, how we use it in various places.

####  2.2.1 The tracks

The tracks play a key role in the computation, as important as instructions
and registers: indeed, they convey information without which any computation
is impossible.

As in [9], the tracks are one-way. But as in [1] we define the tracks as cells
with a particular colour. However, as far as the locomotive goes from the
program to the registers and from there back to the program, it should be
possible for the locomotive to move in opposite directions in some sense. But
we keep to the one way constraint as far as we impose a passive crossing for
the fixed switch. As in [8], the memory switch is split into two different
structures: one of them for an active passage, the other for the passive
passage.

Although the rules are rotation invariant a word should be said about the way
to read the states of the neighbourhood. The problem is the numbering of the
neighbours of a cell. In the Euclidean case it is assumed that the north,
south, east and west directions are defined without further consideration.
There are no such directions in the hyperbolic plane. However, the ’natural’
structure of that plane is a tree, namely, the Fibonacci tree mentioned in
Subsection 1 and illustrated by the right-hand side part of Figure 1.
Consequently, we assume that each cell knows its father in the tree spanning
the sector where it lies. By convention, we say that the father is neighbour 1
and the others are increasingly numbered while counterclockwise turning around
the cell from the father. There is some freedom for cell 0: it has no father
as far as its neighbours are the cell 1 in each of the sectors defined by
those neighbours. We assume that for each cell, the neighbourhood is read in
that way around the cell starting from neighbour 1. In Section 3 we give more
details about the format of the rules.

As far as the way of reading the neighbours is fixed we can now indicate how
the tracks are defined. The increasing and decreasing of the registers must be
along particularly convenient paths. The only ’natural’ way of following a
track is, in our setting to follow a father or, alternatively, once the father
is known, the same son. We decide to have two kinds of tracks: one kind we
call verticals are defined as follows: for each cell, its previous neighbour
of the track is its neighbour 1 and its next neighbour in the track is its
neighbour 4. The second kind of tracks are what we call horizontals. In a
sector, a horizontal is the set of cells which are at the same distance from
the head of the tree. There is a simple way to define a horizontal in a local
way. Besides its numbering, it is ’natural’ to assume that a cell in a
Fibonacci tree, knows the status of its supporting tile: either the tile is a
white node of the tree, or its a black one. In terms of Subsection 1, green
and yellow tiles are said white while blue ones are said black. In a white
tile $\tau$, the horizontal passing through $\tau$ connects its neighbours 2
and 5. In a black tile $\sigma$, the horizontal passing through $\sigma$
connects its neighbours 3 and 5.

Consider four tiles pairwise adjacent, the third one being not in contact with
the first one. The motion from the first tile to the third one can be
symbolized as follows:  FWWW RFWW WRFW WWRF WWWR

and the reverse motion is given by:  WWWF WWFR WFRW FRWW RWWW  From that
observation, we can deduce that a uniform colour for the tracks can accept a
single direction, although it could accept both of them as far as the
locomotive consists of two contiguous cells, the front F and the rear R.

Consider two tiles $U$ and $V$. A path from $U$ to $V$ is a finite sequence of
tiles $\\{T_{i}\\}_{i\in[0..n]}$, where $T_{0}=U$, $T_{n}=V$ and, for any $i$
in $[1..n$$-$$1]$, $T_{i}$ and $T_{i+1}$ share a common side. In that case,
$n$$-$1 is the length of the path. The distance from $U$ to $V$ is the
smallest length of the paths from $U$ to $V$. A circle around $T$ of radius
$n$ is the set of tiles $U$ whose distance to $T$ is $n$. An arc of circle is
a path from a tile $U$ of a circle $\mathcal{C}$ to another tile $V$ of
$\mathcal{C}$, whose all tiles also belong to $\mathcal{C}$.

###### Figure 4

To right, the tracks used for testing the implementation. To left, the sectors
in order to locate the tiles of the tracks. The single colour allows us to
realize the required circuit as far as horizontals and verticals constitute a
kind of grid. It is especially the case if the whole configuration lies in a
single sector, its closest cell to the origin of the coordinates is far enough
for implementing the configurations we describe in Subsection 2.2.

###### Figure 5

Motion of the locomotives on a track which tests verticals and horizontals.
Top two rows: the B-locomotive, first row from top to bottom; second row, from
bottom to top. Bottom two rows, the G-locomotive, first from left to bottom
and then from bottom to top.

The colour of the tracks is fixed as Y. However we shall indicate a track with
respect to the colour of the locomotive which runs on that track. A B-,
G-track is run by a locomotive whose front is B-, G\- respectively. We also
say B-, G-locomotives respectively. Paths from a part of the circuit to
another part consists of pieces of verticals and horizontals connected
together. On most parts of the circuit, the locomotive is B-. Later, we shall
say that B\- and G\- are opposite colours.

###### Figure 6

Motion of the locomotives on a track which tests verticals and horizontals.
Top two rows: the B-locomotive, first row from top to bottom; second row, from
bottom to top. Bottom two rows, the G-locomotive, first from top to bottom and
then from bottom to top.

Figure 4 represents two combinations of a vertical line with two horizontal
lines. In the left-hand side picture, the combination constitutes an ’S’ which
also allows us to test two kinds of junctions of a vertical with a horizontal:
the ’S’ allows us to test a top left corner together with a bottom right
corner. The right-hand side picture of Figure 4 represents another combination
which constitutes a ’Z’. That picture allow us to test the remaining corners:
top right one and bottom left one. Use Figure 1 in order to locate the tiles
of the tracks. For the ’S’ configuration, we have from top to bottom, the
following tiles; horizontal part in the top: (5,$i$) for $i$ ranging in
$\\{$3,10,4$\\}$ and (1,$i$) for $i$ in $\\{$5,15,6,18,7$\\}$; then the
vertical part: (1,3), (1,1), 0, (4,1) and (4,3). And last, the horizontal part
in bottom: (4,$i$) for $i$ in $\\{$7,18,6,15,5$\\}$ and (3,$i$) with $i$ in
$\\{$4,10,3,7,18,6,15,3$\\}$. For the ’z’ configuration, the tiles are,
respectively: (1,$i$) for $i$ in $\\{$4,10,3,7,18,6,6,15,5$\\}$, then (5,4),
(5,1), 0, (2,1) and (2,4) and, at last (3,$i$) with $i$ in
$\\{$5,15,6,18,73,10,4$\\}$ and (4,$i$) with $i$ in $\\{$5,15,6,18,7$\\}$.

The locomotives, both B\- and G\- can run on those tracks from (5,3) down to
(3,5) in the ’S’ and also in the reverse order, from (3,5) up to (5,3). For
the ’Z’, the run occurs from (1,4) down to (4,7) and also in the reverse
order. Those motion have been tested by a computer program and it is
illustrated by Figures 5, 6 for the ’S’-, ’Z’-configuration respectively. As
already noticed, those motions also test the various junctions of a vertical
track with a horizontal one.

We now turn to Sub-subsection 2.2.2 where we describe the auxiliary structures
and the passive fixed switch used to implement the crossings and then the
remaining switches whose implementation is discussed in Sub-subsection 2.2.3.

####  2.2.2 Auxiliary structures and crossings

The auxiliary structures we need are the fork, the changer and the filter. The
changer and the filter are implemented in two versions : a B\- and a G-one.
The B, G-changer transforms a G-, B-locomotive respectively into a B-, G-one
respectively. A filter has a colour, either B or G. It let the locomotive of
the same colour as its one and it destroys a locomotive having an opposite
colour to its own one.

The figure illustrates the idle configuration of those structure. We define an
idle configuration of a structure to be a structure contained in a disc of
radius 2 around the central tile in which disc there is no locomotive.

###### Figure 7

To left, the sectors. To right, top line, the changer from B\- to G-, the
changer from G\- to B. To right, bottom line, from left to right: the fork,
the filter for B-, the filter for G-. With all pictures, the symbol which will
represent them in further diagrams.

In the changers, a track crosses the figure through tiles (5,21), (5,8),
(5,3), (5,1), 0, (3,1), (3,3), (3,8) and (3,21). In (4,1) we have a B\- or a
G-tile and in (4,3) and (4,4) we have a G-tile which fixes the tile in (4,1).
We also have an R-tile at (3,1) with R-tiles too at (3,3) and (3,4) to fix the
tile in (3,1). In the filter, we have the same track and another one with
tiles (1,$i$) for $i$ in $\\{$3,7,18,6$\\}$. Moreover, there is a B\- or a
G-tile at (1,1) together with R-tiles at (2,5), (1,10) and (1,9) to fix the
tile in (1,1) and to allow the locomotive a motion in tile (1,3). There are
also R-tiles at (3,1), (3,3) and (3,4), the last two ones fixing the tile
(3,1).

At last, we have the fork. It gathers three tracks around a Y-tile. The
arriving track passes through (5,33), (5,12), (5,4), (5,1), 0 and (3,1); a
track leaves to right through (4,$i$) with $i$ in
$\\{$5,15,6,17,7,3,10,4$\\}$; a track leaves to left through (3,$i$) with $i$
in $\\{$7,18,6,15,5$\\}$ and (2,$i$) with $i$ in $\\{$4,10,3$\\}$. As far as
the rules of the cellular automaton are rotation invariant, the above fork is
enough to perform such a function in various configurations where the arriving
and the leaving tracks may be different.

A computer program checked the motion of the locomotive through the structure
illustrated by the configuration of Figure 7. It can be checked on Figure 8
which illustrates those various motions for a B-locomotive. Figure 8
illustrates the motion for a B\- and for a G-locomotive, to left and to right
respectively on the figures.

###### Figure 8

The locomotives through the fork. Top, the B-locomotive; bottom, the G-one.

The fixed switch is crossed passively only in the present implementation,
following [8]. Figure 9 illustrates the idle configuration of a fixed switch.
However as far as the rules are rotation invariant, the configuration of
Figure 9 may be rotated by any angle ${2\pi k}\over 5$ with $k$ in
$\\{$1..4$\\}$.

###### Figure 9

To right, the configuration of the passive fixed switch. Rightmost red disc:
representation of a fixed switch in later figures.

Note the R-tile at (3,4) in Figure 9 . The symmetry of the figure allows the
locomotive to pass along one of its sides while it prevents the front of the
locomotive to enter the branch starting from its side where the other branch
of the switch arrives.

The fork, the changers, the filters and the fixed switch can be assembled in a
structure which performs the working of a crossing.

###### Figure 10

Organisation of a crossing.

###### Figure 11

The filters for a horizontal line. To left, the motion goes from $A$ to $C$.
To right, it goes from $C$ to $A$. At the bottom of the large disc, to right,
the symbol of a G-converter.

That organisation is illustrated by Figure 10. A locomotive arrives at $E$. If
it comes from $A$, $D$ it goes to $C$, $B$ respectively. We assume that the
locomotive are B-ones. A locomotive coming from $A$ crosses the fixed switch
at $E$ and is duplicated by the fork at $F$ into two B-locomotives, one of
them going to $C$ and the other to $B$. On the track from $F$ to $B$, the
locomotive meets a G-filter which stops it. On the track from $F$ to $C$, the
B-locomotive meets a B-filter which let it go towards $C$. The B-locomotive
coming from $D$ is converted to a G-one before reaching $E$. From there, the
locomotive goes to $F$ where the fork sitting there duplicates it into two
copies. One of them goes to $C$ where it is stopped by a B-filter which sits
on the track from $F$ to $C$. The other G-locomotive goes from $F$ to $B$. It
first meets the G-filter which let it go towards $B$. A bit further towards
$B$, the G-locomotive meets a B-converter which changes it to a B-locomotive
going towards $B$. Note that if a G-locomotive runs along $D$ to $E$, there is
no need of converters before $E$ nor after $F$ towards $B$. It is assumed that
a B-locomotive runs along the horizontal line. If it is not the case,
appropriate converters should be placed before $E$ and after the B-filter
placed after $F$. We remind the reader that tracks are one way. Accordingly,
other possible crossings can be obtained from the configuration of Figure 9 by
a suitable rotation or a symmetry in a line, or a combination of both
transformations.

From Figure 7, we can see that the converters and the filters are assumed to
sit on a vertical line. Yet, we just mentioned the fact that it could be
appropriate that filters could also sit on a horizontal track. However, it is
possible to set filters on a horizontal track while the portion where a
locomotive crosses it lies on a vertical track. Figure 11 illustrates how it
can be organised. As indicated by the caption, there are two configurations
each one depending on the direction of the motion.

Before turning to the configuration of the fixed switch, we draw the attention
of the reader on Figures 12 and 13. On Figure 12, we can see the change of
colour of a locomotive: on the top row of the figure, the G-converter changes
a B-locomotive to a G-one. On the bottom row of the figure, the B-converter
changes a G-locomotive to a B-one. It is important to indicate that a
converter should never be applied to a locomotive of the same colour.

###### Figure 12

The motion of the locomotives through the converters.

On Figure 13, we can see the behaviour of a filter: the filter let go the
locomotive with the same colour while it kills the locomotive with an opposite
colour. The figure illustrates all cases: the top two lines illustrate the
case when the filter let the locomotive go. The third line illustrates the
case when the filter kills the locomotive: on that line, to left the case of
the B-filter and to right, that of the G-one. The bottom line of the figure
illustrates the change of colour of a filter: to left, the B-filter becomes G,
while to right, the G-filter becomes B. In both cases, the cell at 1(1) can
see the front of the B-locomotive arriving along the track 1($i$) with $i$ in
$\\{$6,18,7,3$\\}$ as far as the front of the locomotive eventually reaches
the cell 1(7) which can be seen from the cell 1(1) although it is a partial
neighbour of that latter cell.

###### Figure 13

The motion of the locomotives through the filters. First row: the B-filter let
the B-locomotive go. Second row, the G-filter let the G-locomotive go. Third
row, to left the B-filter kills a G-locomotive; to right, the G-filter kills a
B-locomotive. Fourth row: changing the colour of the filter: to left, from B\-
to G-; to right, from G\- to B-. We can now turn to the passive fixed switch.
As already indicated, we have four possible configurations illustrated by
Figure 9. Each configuration is defined by the track which leaves the switch:
it may be a vertical track or a horizontal one. In the first case the
locomotive may come from above, in the second one, it may come from below.
When we deal with a horizontal track, the locomotive may come from the left,
or it may come from the right.

From the rotation invariance of the rules, the configuration of Figure 9 may
be turned by an angle which is a multiple of ${2\pi}\over 5$. Accordingly it
is also true for the pictures of Figure 14.

###### Figure 14

The B\- and G-locomotives passively crossing a fixed switch: first row, a
B-locomotive arriving from the right. Second row, a B-locomotive is arriving
from the left. Third row: a G-locomotive is arriving from the right. Fourth
row: a G-locomotive is arriving from the left.

Figure 14 illustrates the crossing of a passive fixed switch by a B-locomotive
and by a G-one.

####  2.2.3 The switches

In order to better understand the schemes later presented, it is important to
indicate the global frame in which the simulating computation holds.

It holds within a grid which we define as constituted by the vertical tracks
and the horizontal ones. However, up to now, vertical tracks and horizontal
ones were not clearly defined. It is time to do that right now.

Figure 15 illustrates a part of a grid in the hyperbolic plane. The vertical
tracks are represented by mauve arcs in the figure while horizontal ones are
represented by blue arcs. However, the tracks are not lines but sequences of
tiles in the pentagrid. We can identify tracks by lines in the figure
considering that our window encloses a large area of the hyperbolic plane.

Let us describe more exactly what our tracks consist of. Figure 1 shows us
five sectors around the central tile, that which we call tile 0. Let us fix
cell 0 once for all and once for all too, let us fix the neighbour one of the
tile 0. Our computation takes place in sector 4. In that sector our vertical
tracks are segments of a branch of the tree rooted at the cell (4,1) and the
horizontal tracks are pieces of levels in that tree. We remind the reader that
a level of the tree in a sector is the set of tiles which are at the same
distance from the head of the sector. However, we do not take any branch as
support of vertical tracks: a vertical passes through white tiles and for such
a tile, its neighbours in the track are neighbour 1 and neighbour 4. A
horizontal track lies in a level. For a white tile of a horizontal track, its
neighbours in the track are neighbour 2 and neighbour 7. For a black tile of a
horizontal track, its neighbours in the track are neighbour 3 and neighbour 7.
In order to have the freedom of organisation we can deduce from Figure 15, it
is needed to assume that the closest level for a horizontal track in the
circuit we shall describe is at least 10: there are 17711 tiles on level 10 as
far as $f_{21}=17711$. Indeed, the number of nodes at level $n$ is $f_{2n+1}$
where $f$ is the Fibonacci function with initial conditions $f_{0}=f_{1}=1$,
see for instance [3, 2] for proofs of that property of the levels in a sector.
As each node produces one black node exactly among its sons, there are
$f_{2n-1}$ black nodes on the level $n$, so that there are $f_{2n}$ white
nodes on that level. Accordingly, 10946 vertical tracks can start from level
10. For the simulation of a universal register machine, it is much more than
what is required as will soon be seen. It should be noted that at level 20,
the distance between two consecutive vertical tracks raised at level 10 is
more than 17711. The previous figures show us that a distance of 10 levels is
a small distance between the extremal tiles of a vertical track.

###### Figure 15

Representation of a grid in the hyperbolic plane.

We postpone the implementation of the register machine to Subsection3.5 which
describes how registers are implemented in our simulation. Presently, we deal
with the implementation of the switches. The present sub-subsection deals with
the flip-flop switch and with the memory one.

A flip-flop changes the selection after an active crossing and there is never
a passive crossing through a flip-flop switch. In a memory switch, its passive
part defines the selection of its active part. Accordingly, we need two kinds
of structures: a structure which let the locomotive go, the other one which
kills it. In the flip-flop, the change is made by the crossing of the switch,
in the memory switch the change is triggered by the crossing of the passive
switch. Basically, it is the same constraint. Accordingly, the controlling
structure in the flip-flop and in the memory switch should be programmable, a
point which we explain further.

Figure 16 illustrates the implementation of the flip-flop. Figure 17
illustrates both parts of the memory switch: the active and the passive ones.

In the flip-flop, a locomotive arrives to $A$. There, a fork sends two
locomotives of the same colour: one towards $L$, the other towards $R$. The
locomotive which is sent to $R$ meets a filter which is sitting there. If the
filter has the colour of the locomotive, that latter goes on. Otherwise, the
locomotive is destroyed. Symmetrically, another filter is sitting at $L$. Now,
the filter at $R$ and that at $L$ have opposite colours. Accordingly, the
locomotive goes on further on the side of the switch where the filter has the
same colour. That property defines the selected track: it is the leaving track
on which the filter has the colour of the locomotive arriving at $C$. When the
locomotive goes on along the selected track, at some distance after the
filter, it meets a fork: one branch of the fork let the locomotive go on along
the selected track, the other sends the locomotive to $B$. There, a fixed
switch is sitting and any locomotive arriving there passively crosses the
switch. From $B$, the locomotive arrives to a fork sitting at $C$ which sends
a copy of it to the filter at $L$ and another copy to the filter at $R$. Those
copies change the filter to the opposite colour, so that now, the selected
track of the switch is changed which is conformal to the definition of a flip-
flop switch. Note that the change of the colour in a filter is performed by a
B-locomotive so that a changer is possibly put on the track leaving $S$ and
before the arrival to the fork leading to $R$ and to $L$.

###### Figure 16

The flip-flop and its symbolic representation.

We assume that the arriving track at the flip-flop and the leaving tracks lie
on different kind of tracks: if the arriving track is vertical, horizontal,
the leaving tracks are horizontal, vertical respectively. When a locomotive
crosses the switch, we may always assume that the change at $L$ and $R$ is
performed before a new visit of the locomotive at the switch. We may also
assume that a given flip-flop switch is always crossed by a locomotive of the
same colour: either always B-, or always G-.

The active memory looks like the flip-flop but it is simpler as far as in its
working when the locomotive arrives at $A$ only the arcs from $A$ to $L$ and
from $L$ to $R$ are concerned. We may assume that a passive crossing occurs
when the locomotive is not involved in the active switch.

###### Figure 17

Memory switch: to left, the active switch; to right the passive one. Each one
has a specific symbolic representation.

The structure of the passive switch is very different. The locomotive arrives
either at $P$ or at $Q$. Assume that it arrives at $P$. The case of an arrival
at $Q$ is symmetrically dealt with. A fork at $P$ sends a locomotive to $A$,
where a fixed switch let the locomotive leave the switch. The other locomotive
is sent to $L$. There, if the filter in the active part let the locomotive go,
the filter at $L$ stops the locomotive, so that the memory switch does not
change the selection. If the filter of the active part stops the locomotive,
the filter of the passive switch let the locomotive go. That locomotive goes
further to $S_{p}$ via $T$ where a fixed switch is sitting. At $S_{p}$, a fork
sends the locomotive to $U$ where an other fork sends a copy of the locomotive
to the fork $S_{a}$ of the active switch and according to the scheme of the
active switch that action will exchange the role of its filters. The second
locomotive sent by the fork at $S_{p}$ goes to $U$ where another fork
duplicates that locomotive sending one copy to $L$ and the other to $R$, both
copies exchanging the roles of the corresponding filters. Accordingly, both
parts of the memory switch operates on the needed way. Here too, we have to
make sure that the arrival at $L$ and $R$ of the locomotives sent from $T$
occurs later at the permissive controller than the locomotive arriving from
$P$ or from $Q$. The picture of Figure 17 ensures us that it is easy to
fulfill that condition, possibly appending tracks in order to introduce a
possibly required delay.

Note that the separation of the two parts of the memory switch allows to use
one passive part for changing several active parts in a same way, a situation
we shall soon encounter.

At last and not the least, the switches we have just implemented can be
accessed in different ways by the tracks: the possibility of free crossings
allow us to satisfy the very structure of those implementations.

####  2.2.4 The one-bit memory

We are now ready to implement the one-bit memory in our setting. The
implementation is illustrated by Figure 18 which reminds us the Euclidean
implementation in its left-hand side part. We adopt the same notations as in
the Euclidean picture in order to facilitate the comparison for the reader.

###### Figure 18

To left, the scheme of Figure 3, to right, its implementation in our setting.
In the hyperbolic picture, the colours at $W$, $R$ and $E$, namely yellow,
light green and light purple respectively represent a flip-flop switch, an
active memory switch and a passive memory switch. The yellow points with an x
letter indicate crossings.

Here again, a B-locomotive is involved. If it arrives at the disc labelled
with $R$ it exits through the gate 0 or through the gate 1 according to the
track selected by the active memory switch sitting at $E$.

If the locomotive arrives at $W$, the flip-flop sitting there sends the
locomotive to $E$ either through track 0 or track 1, depending on what is
selected by the switch at $R$. If the track is $i$, then it arrives to $E$
through its track $1$$-$$i$. Now, from what we have seen on Figure 17, the
switch at $E$ sends a locomotive to $R$ making the active memory switch
sitting there select the values which are compatible with those at $E$. But
from $E$, another B-locomotive is sent out of the switch through the gate $E$.
Note that the memory switch of the left-hand side picture is split into two
parts on the right-hand side figure. There, the active part sits at $R$ while
the passive one sits at $E$. Note that there is a path joining $E$ to $R$.

As far as we shall later use on-bit memories, we shall use the names R, E, W,
0 and 1 to label the gates as in Figure 18. Here two, we assume that the one-
bit memories used later will be implemented as indicated in Figure 18: free
crossings allow us to make that assumption.

####  2.2.5 A register and its discriminating structures

Let us illustrate the implementation of the structures used by the registers
by a toy example given by Figure 19.

###### Figure 19

A toy example. The content $x$ of $R_{1}$ is transformed into $\lfloor{x\over
2}\rfloor$ stored in $R_{1}$ too. Initially, the content of $R_{2}$ is $0$.

In Figure 19, we have the implementation of a program simulating a two-
registered machine which computes the half of the initial value contained in
$R_{1}$. At the beginning, $R_{2}$ is empty and it is used as an auxiliary
structure. The illustrated program of the figure can be written as follows:  1
dec R1,2 dec R1,2 inc R2 jmp 1 2 dec R2,3 inc R1 jmp 2 3 halt Figure 20
illustrates the implementation of a register in our setting. As can be seen on
the figure, we have a segment of line enclosed in a path surrounding it.

The value of the register is the number of W-tiles along that line from tile 0
in the left-hand side of Figure 20 up to tile 0 of the right-hand side part,
both tiles included. Denoting the register by $\mathcal{R}$, we denote the
W-cells representing its content $n$ by $\mathcal{R}$(m) with $0\leq m<n$. By
construction, for any $n$, we have that $\mathcal{R}$(n) is Y. In particular,
when the register is empty, whose value is 0 by definition, $\mathcal{R}$(0)
is Y.

The register is the occasion to remind the reader the fact we have two kinds
of locomotives. The B-locomotive is devoted to increment the register while
the G-one is devoted to decrement it. The locomotive returning from the
register has the same colour as it had when entering the register. However,
once it performed its operation, the locomotive does not remember which
instruction of the program dictated it to execute that operation. Accordingly,
we need two new structures outside a register: a structure to remember which
instruction required to increment the register, the $\mathbb{DD}$I-structure,
see 2.2.5.a; and a structure to remember which instruction required to
decrement the register, the $\mathbb{DD}$D-structure, see 2.2.5.b. Note that
the colour of the locomotive is enough to characterise which operation the
locomotive did perform. Both structures are required for each register. Those
structures too make use of one-bit memories.

###### Figure 20

To left, the beginning of a register when $n=0$; to right, its end when $n>0$.
Both ends are marked as far as they play an important role.

2.2.5.a Remembering the incrementing instruction

As can be guessed from Figure 20, when the locomotive goes back from the
register, as far as we have two types of locomotive only, the locomotive does
not remember from which point of the program it was issued. In order to go
back to that point in order to define the next instruction, the instruction
must be marked in some way near the register itself. The
$\mathbb{DD}$I-structure is devoted to play that role. There is a
$\mathbb{DD}$I-structure for each register $\mathcal{R}$. The structure
contains as many units as there are instructions in the program to
specifically increment $\mathcal{R}$. From each instruction $I$ incrementing
$\mathcal{R}$, a path goes from the place of $I$ in the program to the unit of
the $\mathbb{DD}$I attached to $\mathcal{R}$ associated by that path to that
unit. The unit itself contains a one-bit memory, see Figure 21.

At the initial configuration, in all units of each $\mathbb{DD}$I-structure,
the one-bit memory contains 0. When the locomotive starts its way to perform
an incrementing instruction $I$ over the register $\mathcal{R}$, the track run
by the locomotive arrives to the W-gate of the unit of the $\mathbb{DD}$I
attached to $\mathcal{R}$ which is associated with $I$. As far as the
locomotive arrives to a W-gate it rewrites the 0-bit of the one-bit memory to
1. Then, the locomotive exit the memory through its E-gate. It meets a flip-
flop at $A$ which, form the initial configuration, sends it towards
$\mathcal{R}$. After crossing that flip-flop switch, the selected track is now
a track which goes back to the program, to the instruction which has to be
executed after $I$ has completed its operation on $\mathcal{R}$. At that time,
there is a single unit in that $\mathbb{DD}$I whose one-bit memory contains 1.

###### Figure 21

The unit of $\mathbb{DD}$I devoted to an incrementing instruction. The figure
does not mark the crossings which are clear from the picture itself.

When the locomotive comes back from $\mathcal{R}$, it is a B-locomotive and a
fork combined with appropriate filters of opposite colours sends that
locomotive to the $\mathbb{DD}$I of $\mathcal{R}$.

The locomotive visits each unit of the $\mathbb{DD}$I-structure until it meets
the single one whose one-bit memory contains 1. To do that, the locomotive
enters the memory of the unit through its R-gate. If it reads 0, it is sent to
the next unit. When it reads 1, it knows that the right unit is reached.
Leaving the one-bit memory through the 1-gate, the locomotive is sent to the
W-gate so that it rewrites the content of the one-bit memory from 1 to 0.
Leaving the memory through theE-gate, the locomotive again meets $A$ where the
selected track of the flip-flop sends it on the track leading to the right
place of the $\mathbb{DD}$I or of the $\mathbb{DD}$D of the appropriate
register. Once the flip-flop at $A$ is crossed, the selected track again
becomes the track leading to $\mathcal{R}$. Accordingly, when the locomotive
leaves the $\mathbb{DD}$I-structure after preforming its operation, that
$\mathbb{DD}$I recovered its initial configuration.

2.2.5.b Remembering the decrementing instruction

If the locomotive has to decrement the register $\mathcal{R}$ it visits a
similar structure as $\mathbb{DD}$I, the $\mathbb{DD}$D-structure attache to
$\mathcal{R}$. The principle of that structure is similar to that of
$\mathbb{DD}$I. However a unit of $\mathbb{DD}$D is more complex than a unit
of $\mathbb{DD}$I. The reason comes from the difference between a decrementing
instruction and an incrementing one. It is always possible to increment a
register. It is not possible to decrement an empty register as far as register
machines deal with natural numbers only. When a decrementing locomotive
arrives at $\mathcal{R}$(0) when $\mathcal{R}$ is empty, it leaves
$\mathcal{R}$(0) through a different track from which it leaves that place
when it had successfully decremented it.

Accordingly, if the working of $\mathbb{DD}$D is similar to that of
$\mathbb{DD}$I for an arriving locomotive from the program, it is not the case
for a locomotive returning from the register. We leave the case of an arriving
locomotive from the program to the reader.

###### Figure 22

The unit of $\mathbb{DD}$D devoted to a decrementing instruction.

Consider the case of a locomotive coming back from $\mathcal{R}$. It comes
from the D-track after a successful operation or it comes through the Z-track
because it could not decrement an empty $\mathcal{R}$. Both tracks behave in
the same: they lead to the R-gate. If the locomotive reads 0, it goes to the
next unit through the same kind of track as the one it used to arrive at the
current unit. As can be seen, there are two possible exits for the locomotive
entering the one-bit through the R-gate when it comes from the register. But
in both cases, the exit splits again in two possibilities depending on which
track the locomotive entered the unit. In case of an exit to the next unit,
the exit track is a D-, Z-track if it was respectively the case for the
entering track. In case of an exit to the program, the exit track goes to the
next instruction or to another one defined by the jump condition if the
entering track was D\- or Z-respectively. The distinction is known at the
entry of the unit, see point $F$ in Figure 22. It is the reason why at $F$ a
passive memory switch is sitting. As far as that difference has to be known at
$B$ and at $C$, there is at each of those points an active memory switch
connected to the passive one at $F$. The connection to those active memory
switches is established by a track from $F$ to a fork before reaching $C$
which sends a copy to $C$ and another copy to $B$. Accordingly, the whole
structure performed what is expected from it in all situations.

Figure 23 illustrates the tracks followed by a locomotive returning from the
register. Its colour indicates which operation it performed. Coming from the
register it arrives at $A$ where a fork is sitting. One branch has a B-filter
at $B$, meaning that the branch leads to $\mathbb{DD}$I. The other branch has
a G-filter at $G$, so that the branch leads to $\mathbb{DD}$D. Accordingly,
the locomotive goes to the appropriate structure which, as already seen, sends
it back to the right place in the program.

###### Figure 23

The structure sending a locomotive returning from the register to the
appropriate $\mathbb{DD}$I\- or $\mathbb{DD}$D-structure. Note at $B$ and at
$G$ the symbol for the filter: the forbidden colour is circumscribed by a red
circle.

2.2.5.c Decrementing and incrementing the register

The operation performed by a locomotive inside the register is illustrated by
Figure 24. The figure illustrates a standard situation for a non trivial value
of the content: the Y-tile is at some distance from the beginning of the
structure, depending on the initial content of the register which may be
empty. On the leftmost picture of the figure, we can see the Y-cells 1(3),
1(4) and 1(5) around the Y-cell at 0 which we call the hat of the register. In
the rightmost picture of the figure we also see the hat around the Y-cell at
1(1), it consists of the Y-cells 1(2), 0 and 1(7). In the second picture from
left, illustrating the incrementing operation, the hat consists of four
Y-cells around the Y-cell at 1(4): 2(4), 3(4), 4(4) and 2(5).

Figure 24 illustrates how the operation is performed. The figure is split into
two parts. To left, the register before the operation, to right, the register
after it was performed. In that right-hand side part, two possibilities, to
left: incrementing; to right, decrementing as mentioned.

###### Figure 24

To left, the register before the operation. To right, the configuration after
the operation: incrementing and, rightmost one, decrementing.

The action illustrated by Figure 24 has been tested by a computer program and
can be checked in Figure 25 and in Table 2.

###### Figure 25

Top, execution of an incrementing instruction when $n>0$ together with the
motion of the locomotive through the beginning of the register and then the
motion after the operation is completed. Bottom, the similar executions for a
decrementing instruction when $n>2$.

Figures illustrating the particular cases are given in Section 3 to which we
now turn.

##  3 Rules

The definition of the rules will give us the opportunity to more explicitly
study the behaviour of the automaton.

We start the section by introducing a formalism for writing rules and schemes
of rules which will allow us to better present some rules. Then, in each
further sub-sections, we successively examine the cases studied in the sub-
subsections of Subsection 2.2.

###  3.1 Fixing notations

We fix the rules in the following format: c-s1s2s3s4s5s6s7s8s9s10:n where c
denotes the current state of the cell $\sigma$ to which the rule applies
provided that $s_{i}$ with $i\in\\{1..10\\}$ is the state of the neighbour $i$
of $\sigma$ and when the rule applies, the new state of $\sigma$ is n. Of
course, c, si and n take their value in W, Y, B, R, G, M, numbered in that
order. Note that neighbours $i$ with $i$ in $\\{1..5\\}$ are full ones while
neighbours $i$ when $i$ is in $\\{6..10\\}$ are partial ones. From now on, as
far as we speak about the cellular automaton, we shall say cell in place of
tiles. As an example, the central tile of a window will become the central
cell of the window. The reason is that a cell is supported by a tile but it
also contains the finite automaton, the same for all cells, which rules the
change of cell at each tip of the discrete clock. All those conditions are
just an application of the definition of a cellular automaton.

The first letter is the current state of the cell and the last one is the new
state obtained from the current state of the cell and the current states of
its neighbours: if we order the neighbours according to the letters from 1 up
to 10 from left to right in the word delimited between two commas, the $i^{\rm
th}$ letter gives the state of the corresponding $i^{\rm th}$ neighbour of the
cell. That latter word is the neighbourhood word.

An observation must be made about rotation invariance. A rotation which leaves
a tile globally unchanged operates a permutation on the sates of its full
neighbours. Clearly, the same permutation operates on the partial neighbours.
The new rule resulting after such a permutation is called a rotated form of
the rule. With that new definition of a rotated form, we can define the
minimal form of a rule: it is the lexicographically smallest one among the
five rotated forms of the rule.

###  3.2 Motion of a locomotive on the tracks

As already mentioned, we consider the tracks as combinations of segments of
vertical lines and arcs on levels of a tree, see Figure 15 for instance. In
order to establish the rules and to test them, we constructed two figures: the
’S’-track and the ’Z’-one, see Figures 5 and 6.

Take, as an example, the motion rules on a vertical track for a blue
locomotive displayed in $(vT)$. The number to right of a rule is the time
measured from the application of the rule Y-YWWYWWWWWW:Y applied to an idle
configuration. The number to left of the rule is its number in Table 2.
Similar rules hold for a G-locomotive: in the rules, each occurrence of B is
replaced by G. Of course, rule 89 also applies to a G-locomotive when the rear
R of the locomotive is seen from the Y-cell just left by the locomotive.

$\downarrow$ $\uparrow$ cell 0 5 Y-YWWYWWWWWW:Y $0,5$ 63 Y-BWWYWWWWWW:B $1$ 67
B-RWWYWWWWWW:R $2$ 76 R-YWWBWWWWWW:Y $3$ 89 Y-YWWRWWWWWW:Y $4$ cell 1(1) 6
Y-YWYWWWWYWW:Y $0,6$ 57 Y-YWYWWWWBWW:Y $1$ 60 Y-YWBWWWWRWW:B $2$ 25
B-WRWWYWYWWW:R $3$ 41 R-WYWWBWYWWW:Y $4$ 49 Y-WYWWRWYWWW:Y $5$ cell 0 6
Y-YWWYWWWWWW:Y $0,5$ 63 Y-YWWBWWWWWW:B $1$ 147 B-YWWRWWWWWW:R $2$ 152
R-BWWYWWWWWW:Y $3$ 157 Y-RWWYWWWWWW:Y $4$ cell 4(1) 6 Y-YWYWWWWYWW:Y $0,6$ 57
Y-YWYWWWWBWW:Y $1$ 60 Y-YWBWWWWRWW:B $2$ 25 B-WRWWYWYWWW:R $3$ 41
R-WYWWBWYWWW:Y $4$ 49 Y-WYWWRWYWWW:Y $5$ $(vT)$ In $(vT)$, the number(s) after
an instruction is the time at which the rule is applied, 0 being the initial
time for the table.

Note that, in $(vT)$ instead of rule 25 one would expect the rule
B-YWRWWWWYWW:R to be applied. However, that latter rule is a rotated form of
rule 25 as can easily be checked. Note that in rule 25, the track joins
neighbours 2 and 5 as required for a white tile which is the case for the
support of the cell 4(1).

Say that a rule is conservative if and only if the new state is the same as
the current one and also if the states of the neighbours are also unchanged
when the rules for the neighbours are applied. If the state of the cell only
is unchanged, we speak of a witness rule. As an example, rule 75, namely
W-WWYWWWBWWW:W is a witness rule as far as the B-cell in the neighbourhood of
the current cell to which the rule applies belong to a blue locomotive.

The rules dealing with the motion of a locomotive on tracks are given in Table
2 from rule 7 until rule 377. However, several rules among those ones also
apply to the fork, to the fixed switch, to the other auxiliary structures and
to many cells of the registers too.

Note that among the mentioned rules, many of them deal with arcs of a circle.
Together with rules applying to a segment of line, it requires additional
rules. They have the form Y-WYWWYYYWWW:Y . If F stands for B or G, the front
of a locomotive, we also have Y-WYWWYYFWWW:Y, rule 37, rule 175 for a B-,
G-locomotive respectively. As motion rules, we have the rules Y-WFWWYYRWWW:F,
F-WRWWYYYWWW:R, the rule R-WYWWFYYWWW:Y, the rule Y-WYWWRFYWWW:Y and the rule
Y-WYWWYRYWWW:Y respectively rules 45, 52, 56, 59 and 62 for a B-locomotive,
and rules , 181, 186, 188, 191 and 62 again for a G-locomotive. We also have
the rules Y-WYWWYYWWWW:Y, Y-WFWWYYWWW:F, F-WRWWYYWWWW:R, R-WYWWFYWWWW:Y,
Y-WYWWRFWWWW:Y and Y-WYWWYRWWWW:Y. We leave as an exercise to the reader to
identify them in Table 2. We refer the reader to Figure 5 illustrating those
motions where the rules can be checked. We also leave as an exercise to the
reader to identify the motion rules on a ’Z’-track.

###  3.3 Fixed switch and fork

We start with the fixed switch, considering its passive version only as
already mentioned.

We have four cases: a blue locomotive coming from the left-hand, right-hand
side branch and the similar two sub-cases with a green locomotive. We refer
the reader to Figures 9 and 7 to follow the application of the rules given by
Table 1 where the notations are those introduced in Subsection 3.1.

In the table, the rules are given for both locomotives: in the considered
rule, replace the occurrences of F, all of them by B or all of them by G.

For other configurations of the fixed switch, the reader is referred to Figure
14 and to Table 2.

Note the symmetry of the neighbourhood words for the rules dealing with the
transformation of the current cell: the position of F in that word depends
from which side of the switch the locomotive is arriving.

In the case of a fork, rule 380, namely Y-YWWYYWWYRY:Y gives the neighbourhood
of the Y-cell where all tracks meet. That cell is a cell of the track and it
is at the same time a good witness of what happens:

That Y-cell is a good witness of the action of the fork: rules 488 and 518,
namely R-YWBBWWWYRY:W and R-YWGGWWWYRY:Y indicate that the cell can see both
fronts of the locomotives being created from the arriving one and rules 495
and 525, namely Y-YWRRWWWBYB:Y and Y-YWRRWWWGYG:Y show us that the cell can
see both rears and both fronts of the leaving locomotives. And last, rule 502,
namely Y-YWYYWWWRRR:Y, show us that the cell can see the rear of the leaving
locomotive, whichever its colour.

###### Table 1

Rules for the motion of a B-locomotive through the cell $1(3)$ in a fixed
switch with an upward vertical exit track. To left, to right, when the
locomotive arrives from the left, from the right respectively. The index at
the number indicates at which time(s) the rule is applied, $0$ being the
initial time for the table.

1(3), $\bullet$ $\rightarrow$, B-, G- Y-YWYYWWWYRY:Y 3800,6 Y-YWYYWWWFRY:Y
395,4381 Y-YWFYWWWRRY:F 397,4402 F-YWRYWWWYRY:R 403,4463 R-FWYYWWWYRY:Y
412,4544 Y-RWYYWWWYRY:Y 4195 1(3), $\leftarrow$ $\bullet$, B-,G-
Y-YWYYWWWYRY:Y 3800,6 Y-YWYYWWWYRF:Y 423,4581 Y-YWYFWWWYRR:F 425,4602
F-YWYRWWWYRY:R 430,4653 R-FWYYWWWYRY:Y 412,4544 Y-RWYYWWWYRY:Y 4195

1(3): F is either B or G Y-YWYYWWWYRY:Y 3800,6 Y-FWYYWWWYRY:F 481,5121
F-RWYYWWWYRY:R 483,5142 R-YWFFWWWYRY:Y 488,5183 Y-YWRRWWWFRF:Y 495,5254
Y-YWYYWWWRRR:Y 5025 $(Fk)$ Here too, the index at a number of a rule indicates
the time of its application.

###  3.4 Converters and filters

In the case the converter, the structure contains a B\- or a G-cell. It
transforms a locomotive of the opposite colour into a locomotive of its
colour. Looking at Figure 7, it is considered that the locomotive arrives from
the right. Accordingly, the first cell of the track seeing the B\- or the
G-cell is cell 1(5). However, we shall follow the move of a locomotive from
the cell 1(4) where the colour of the changer is sitting. It is a perfect
place to witness the transformation:  1(4):$\\{$F,H$\\}$ = $\\{$B,G$\\}$
F-YWGGWYRWWW:F 536,5720,4 F-YWGGWHRWWW:F 548,5831 F-FWGGWRRWWW:F 555,5902
F-RWGGWYRWWW:F 563,5983 0:$\\{$F,H$\\}$ = $\\{$B,G$\\}$ Y-WYRFYWWWWW:Y
532,5700,5 Y-WYRFHWWWWW:F 547,5821 F-WYRFRWWWWW:R 551,5862 R-WFRFYWWWWW:Y
559,5943 Y-WRRFYWWWWW:Y 566,6014 $(Cv)$ The table also provides us with the
rules applied to cell 0. Note, for instance, that rules 536 and 572 have the
same neighbourhood word but a different current state. Note the use of the
first rule of the table after the application of the last one was completed.

From the use of the filters which was done in Subsection 2.2, we know that it
was required that the colour of those structures could be changed: in other
words they must be programmable. Looking again at Figure 7, we can see four
Y-cells at the top of the filters. Those four Y-cells can be seen as the end
of a track arriving at the filter. We require the filter to be changed by a
B-locomotive. This time, we take cell 0 as a witness of the motion in the
filter working and we take the cell 1(1) to illustrate the change of the
filtering colour, see Figure 7.  0: same colour Y-FYRWYWWWWW:Y 604,653
Y-FYRWFWWWWW:F 616,663 F-FYRWRWWWWW:R 621,667 R-FFRWYWWWWW:Y 629,675
Y-FRRWYWWWWW:Y 637,682 0: opposite colours Y-FYRWYWWWWW:Y 604,653
Y-FYRWHWWWWW:Y 643,687 Y-FYRWRWWWWW:Y 648,691 1(1), change colour:
F-YWYWWYYYRR:F 605,654 F-YWYWWYYFRR:H 695,704 H-YWYWWYYRRR:H 700,708
H-YWYWWYYYRR:H 654,605 $(Flt)$ In the table, F and H mean opposite colours
where B and G are opposite of one another. Note that rules 700 and 708 have
the same neighbourhood word and different current states. The same observation
holds for rules 654 and 605. Note that after the change of colour, the
conservative rule for the cell 1(1) turns from rule 605,654 to rule 654,605
when the filter is B,G respectively.

###  3.5 Registers

From Sub subsection 2.2.5, we know that when the register is empty,
$\mathcal{R}$(0) is Y. From Figure 20, we know that the transformation from
$\mathcal{R}$(n) to $\mathcal{R}$(n+1) or $\mathcal{R}$(n-1) entails much
work. First, the Y-cell moves from $\mathcal{R}$(n) to $\mathcal{R}$(n+1) or
to $\mathcal{R}$(n-1) which, in the latter case, assumes that $n>0$. It means
also that the previous hat is either destroyed or modified, and that a new hat
is built from scratch or from already non empty cells.

The case when $n=0$ requires a special configuration for $\mathcal{R}$(0) in
order the decrementing operation should not be processed. Instead the
G-locomotive should go along a specific track as mentioned in Subsection
2.2.5. Now, when a locomotive goes to $\mathcal{R}$(n), when $n>1$ in order to
perform an operation, it must cross $\mathcal{R}$(0). Accordingly, there are
two possible tracks followed by a locomotive leaving $\mathcal{R}$(0),
depending on whether it could not perform the operation or it could do it as
far as it has to increment $\mathcal{R}$ or it has to operate on
$\mathcal{R}$(n) for $n>0$. The left-hand side part of Figure 20 is repeated
in Figure 26 in order to better see the situation around $\mathcal{R}$(0). The
two tracks which can be followed by a locomotive leaving $\mathcal{R}$(0)are
separate from cell 0: one track goes on to 1(3), the other goes on to 1(2). In
order to differentiate the configurations, cell 1(2) has two M-neighbours:
2(2), shared with 0 and 2(3), shared with 1(3).

###### Figure 26

The idle configuration of $\mathcal{R}$($0$). Note that the cell $1(4)$ is W
when $n>0$ and that it is Y as here, when $n=0$.

Figure 25 shows us the motions of the locomotive during an incrementing
operation and during a decrementing one, assuming that $n>1$ for those
operations.

The rules for incrementing the register are rules 535 to 684 and also rules
771 to 774. The rules of decrementing the register are rules 685 to 770. Those
rules are gathered as follows. For the incrementing operation, rules 535 up to
608 deal with the case when $n>1$. Rules 609 to 639 manage the crossing of
$\mathcal{R}$(0) and the progression of the B-locomotive to
$\mathcal{R}$($n$). Rules 640 to 684 deal with incrementing the empty
register. Rule 685 deals with the return of the B-locomotive from
$\mathcal{R}$($n$+1) including the crossing of $\mathcal{R}$(0). At last,
rules 771 to 774 deal with the incrementation of the register when $n=1$.

For the decrementing operation, rules 685 to 714 manage the decrementing of
the register when $n>1$. Rules 715 to 734 deal with the progression of a
G-locomotive from $\mathcal{R}$(0) up to $\mathcal{R}$($n$) when $n>1$. Rules
735 to 737 deal with the return of the G-locomotive from
$\mathcal{R}$($n$$-$1), also crossing $\mathcal{R}$(0). Rules 738 to 764
manage the G-locomotive when it faces an empty register. At last, rules 765 to
770 deal with the case when $n=1$ which boils down to an empty register.

Figure 27 illustrates the behaviour of the locomotive in particular cases.
First, when the register is empty, i.e. its value is 0: incrementing resets it
to 1 while decrementing fails: the return track is different in that case.
Then we have the case when the value of the register is 1. Incrementing it
sets it to 2 while decrementing it sets it to 0. At last, we consider the case
when the value of the register is 2 and the locomotive has to decrement it,
setting it to 1.

###### Figure 27

The particular cases: Top two row: $n=0$. First, decrementing fails, the
locomotive exits from another way. motion of a locomotive when the register is
empty. Second, incrementing from $0$ to $1$. Next two rows, $n=1$. First,
decrementing from $1$ to $0$. Second, incrementing from $1$ to $2$. Bottom
row: $n=2$. Decrementing from $2$ to $1$. The corresponding rules are rules
975 up to 1072 from Table 2. There are too many of those rules in order to
detail their working. Their correctness was checked by a computer program
which also created the figures illustrating that working. The same observation
holds for Figure 28 which illustrates rules 712 up to 850. That figure also
illustrate the motion of the locomotive for each operation: before performing
it and then when it completed it. That illustrates rules 851 up to 974 of
Table 2.

###### Figure 28

The case $n>1$. Top row: incrementing. Bottom row, decrementing. It is worth
having a closer look at the transformation of the hat during the operations on
a register. See Figure 29 and the instructions given at $(zIop)$ and at
$(zDop)$.

###### Figure 29

Closer look at the operations on a register when $n>1$, where $n$ is its
content. Top two rows: incrementing operation; bottom two rows: decrementing
one.

In lines under $(zIop$), we have the instructions applied to cell (3,1) and to
cell (3,10). The first cell is a good witness of the transformation touching
its neighbours and itself by an incrementing operation. Cell (3,10) belongs to
the new hat and we can see how it is changed from W to Y . We can see how the
new hat is created in 4 steps once cell (3,1) is changed from Y to M . The
cell (3,1) becomes the new marker of the end of the register after it has been
incremented.  (3,1): Y-YYWWYYYWWW:Y 718 Y-YYWWYYBWWW:M 730 M-BMWWYYRWWW:M 735
M-RYYWMYYMWW:M 750 M-WYYWMYYYWY:M 766 M-WYYYYYYYWY:Y 777 Y-WYYYYYYYYY:Y 712
(3,10): W-WWWWWWYWWW:W 15 W-WWWWWWMWWW:W 742 W-YWWWWWMWWW:W 755 W-YYWWWWMWWW:Y
780 Y-YYWWWWYWWW:Y 719 $(zIop)$ In the lines of $(zDop)$, we have the
instructions which are applied to a register with value $n$ greater than 1 to
decrement it. Cell 0 is a good witness of the transformation: its Y-neighbours
become W for 5 of them. Also, the cell (3,1) is also an interesting witness of
the transformation. For instance, rule 755 witnesses the fact that cell 0 was
changed from W to Y : indeed, cell 0 is the neighbour 1 of the cell (3,1).  0:
Y-WYYYYYYYYY:Y 712 Y-WYYYYYGYYY:Y 796 Y-WGYYYYRYYY:M 798 M-WRYYYYYYYY:M 802
M-YYWWYYYWYY:M 809 M-YYWWYYYWWR:Y 817 Y-YYWWGYYWWR:Y 821 Y-YYWWRGYWWW:Y 826
Y-YYWWYRYWWW:Y 831 Y-YYWWYYYWWW:Y 718 (3,1): Y-YYWWYYYWWW:Y 718 Y-YYWWYYGWWW:Y
800 Y-MYWWYYRWWW:W 806 W-MYWWYWYWWW:W 813 W-MWWWWWYWWW:W 754 W-YWWWWWYWWW:W 9
$(zDop)$ Table 2 displays the whole set of rules used by the automaton. We use
the same notation as what was done in the previous parts of the paper. We
remind the reader the fact that in the table, the neighbouring word is the
minimal lexicographic order among the possible equivalent forms of the rule
obtained through a circular permutation. However, on Figure 25, it is not
difficult to check the rules indicated in the text. It is needed to see that
in the text we considered rules applied to a few cells while the rules of
Table 2 were devised for all possible cells involved in the computation: cells
directly impacted by the computation as well as witnessing cells.

With Table 2, with all the tables and all the figures from Subsection 2.2, the
proof of Theorem 1 is completed. $\Box$

##  4 Conclusion

We can see that we have 1072 rules in the table, conservative, witness and
motion rules. Note that the registers alone require 361 rules and 98 of those
ones are required for the special cases when the content of the register is 0,
1 or 2, that latter case in the case of decrementing only.

We remain with the open question whether it is possible to reduce the number
of states.

Locomotives on S-track 1 W  WWWWWWWWWW  W  2 B  WWWWWWWWWW  B  3 R  WWWWWWWWWW
R  4 G  WWWWWWWWWW  G  blue locomotive $\downarrow$ motion 5 Y  YWWYWWWWWW  Y
6 Y  YWYWWWWYWW  Y  7 W  YWYYYYYBYY  W  8 Y  YYWWWWWWWW  Y  9 W  YWWWWWYWWW  W
10 Y  WBWWYYWWWW  B  11 Y  WYWWYYYWWW  Y  12 W  YWWWWWWWWW  W  13 W
WYWWWWYWWW  W  14 W  YWWWWWBWWW  W  15 W  WWWWWWYWWW  W  16 W  YWWWWWYWWY  W
17 W  YWYYWYWYYY  W  18 W  WWYYYYYWYY  W  19 Y  WYWWYYWWWW  Y  20 Y
WWWWYYWWWW  Y  21 W  YWYYYYYYYY  W  22 W  YWYBWYYWRY  W  23 W  WYWWWYYWWW  W
24 Y  WWWWRBWWWW  Y  25 B  WRWWYWYWWW  R  26 W  YWWWWRWWWW  W  27 R
BYWWWWWWWW  Y  28 W  BWWWWWRWWW  W  29 W  BWWWWYWWWW  W  30 W  RWWWWWYWWW  W
31 W  RWWWWWWWWW  W  32 W  WRWWWWBWWW  W  33 W  WWWWWWBWWW  W  34 W
WWWWWWRWWW  W  35 W  YWBYYYYRYY  W  36 B  WRWWYYWWWW  R  37 Y  WYWWYYBWWW  Y
38 Y  YBWWWWWWWW  B  39 W  YWYRWYYWYB  W  40 Y  WWWWYRWWWW  Y  41 R
WYWWBWYWWW  Y  42 Y  RYWWWWWWWW  Y  43 W  YWRYYYYYBY  W  44 R  WYWWBYWWWW  Y
45 Y  WBWWYYRWWW  B  46 B  YRWWWWWWWW  R  47 W  BWWWWWWWWW  W  48 W
YWYYWYYWYR  W  49 Y  WYWWRWYWWW  Y  50 W  YWYBYYYYRY  W  51 Y  WYWWRBWWWW  Y
52 B  WRWWYYYWWW  R  53 W  YWYYWYYWYY  W  54 W  YWYRYYYYYB  W  55 Y
WYWWYRWWWW  Y  56 R  WYWWBYYWWW  Y  57 Y  YWYWWWWBWW  Y  58 W  YWYYBYYYYR  W
59 Y  WYWWRBYWWW  Y  60 Y  YWBWWWWRWW  B  61 W  YWYYRBYYYY  W  62 Y
WYWWYRYWWW  Y  63 Y  BWWYWWWWWW  B  64 W  BWYYYRYYYY  W  65 W  YWWWWWBWWY  W
66 W  YWYYWBYWYY  W  67 B  RWWYWWWWWW  R  68 W  RWYYYYBYYY  W  69 W
BWWWWWRWWY  W  70 W  BWYYWYWYYY  W  71 W  WWYYYYBWYY  W  72 Y  BWYWWWWYWW  B
73 W  YWYYYYBYYY  W  74 W  BWYYWRYWYY  W  75 W  WYWWWYBWWW  W  76 R
YWWBWWWWWW  Y  77 W  YWYYYYRYYY  W  78 W  RWWWWWYWWY  W  79 W  RWYYWBWYYY  W
80 W  WWYYYYRWYY  W  81 B  RWYWWWWYWW  R  82 W  BWYYYYRYYY  W  83 Y
BYWWWWWWWW  B  84 W  BWWWWWYWWW  W  85 Y  YWYWWWBYWW  Y  86 W  WYWWWWBWWW  W
87 W  RWYYWYBWYY  W  88 W  WBWWWYRWWW  W  89 Y  YWWRWWWWWW  Y  90 W
YWYYWRWYYY  W  91 R  YWBWWWWYWW  Y  92 W  RWYYYBYYYY  W  93 B  RYWWWWWWWW  R
94 W  RWWWWWBWWW  W  95 Y  BWYWWWRYWW  B  96 W  WBWWWWRWWW  W  97 W
YWYYWYRWYY  W  98 W  WRWWWYYWWW  W  99 Y  YWRWWWWBWW  Y  100 W  YWYYBRYYYY  W
101 R  YBWWWWWWWW  Y  102 W  YWWWWWRWWW  W  103 B  RWYWWWYYWW  R  104 W
WRWWWWYWWW  W  105 Y  YWYWWWWRWW  Y  106 W  YWYYRYYYYB  W  107 Y  YRWWWWWWWW
Y  108 R  YWBWWWYYWW  Y  109 W  YWYBYYYYYR  W  110 Y  WYWWYBWWWW  Y  111 Y
YWRWWWYBWW  Y  112 W  YWYRYYYYBY  W  113 Y  WYWWBRWWWW  B  114 Y  YWYWWWYRWW
Y  115 W  YWYYWYWYYB  W  116 W  YWBYYYYYRY  W  117 B  WYWWRYWWWW  R  118 W
YWYBWYWYYR  W  119 W  YWRYYYYBYY  W  120 R  WBWWYYWWWW  Y  121 W  YWYRWYWYBY
W  122 W  YWYYYYYRYY  W  123 Y  WRWWYYWWWW  Y  124 W  YWBYWYWYRY  W  125 W
WWYYYBYWYY  W  126 W  YWRYWYWBYY  W  127 W  WWYYBRYWYY  W  128 W  YWYYWYWRYY
W  129 W  WWYYRYYWYB  W  130 W  WWYBYYYWYR  W  131 Y  WWWWYBWWWW  Y
$\uparrow$ motion 132 W  WWYBYYYWRY  W  133 W  WWYRYYYWYB  W

134 W  YWYYWYWBYY  W  135 W  WWYYBYYWYR  W  136 W  YWBYWYWRYY  W  137 W
WWYYRBYWYY  W  138 W  YWRYWYWYBY  W  139 W  WWYYYRYWYY  W  140 W  YWYBWYWYRY
W  141 W  YWYRWYWYYB  W  142 W  YWYYWYWYYR  W  143 Y  YWWBWWWWWW  B  144 W
YWYYWBWYYY  W  145 W  YWYYWYBWYY  W  146 W  WWBWWWYYWW  W  147 B  YWWRWWWWWW
R  148 W  BWWWWWYWWY  W  149 W  BWYYWRWYYY  W  150 W  BWYYWYRWYY  W  151 W
WWRWWWYBWW  W  152 R  BWWYWWWWWW  Y  153 W  RWWWWWBWWY  W  154 W  RWYYWYWYYY
W  155 W  RWYYWBYWYY  W  156 W  WWYWWWYRWW  W  157 Y  RWWYWWWWWW  Y  158 W
YWWWWWRWWY  W  159 W  YWYYWRYWYY  W  160 W  YWYYWYYWYB  W  161 W  YWYBWYYWYR
W  green locomotive $\downarrow$ motion 162 W  YWYYYYYGYY  W  163 Y
WGWWYYWWWW  G  164 W  YWWWWWGWWW  W  165 W  YWYGWYYWRY  W  166 Y  WWWWRGWWWW
Y  167 G  WRWWYWYWWW  R  168 R  GYWWWWWWWW  Y  169 W  GWWWWWRWWW  W  170 W
GWWWWYWWWW  W  171 W  WRWWWWGWWW  W  172 W  WWWWWWGWWW  W  173 W  YWGYYYYRYY
W  174 G  WRWWYYWWWW  R  175 Y  WYWWYYGWWW  Y  176 Y  YGWWWWWWWW  G  177 W
YWYRWYYWYG  W  178 R  WYWWGWYWWW  Y  179 W  YWRYYYYYGY  W  180 R  WYWWGYWWWW
Y  181 Y  WGWWYYRWWW  G  182 G  YRWWWWWWWW  R  183 W  GWWWWWWWWW  W  184 W
YWYGYYYYRY  W  185 Y  WYWWRGWWWW  Y  186 G  WRWWYYYWWW  R  187 W  YWYRYYYYYG
W  188 R  WYWWGYYWWW  Y  189 Y  YWYWWWWGWW  Y  190 W  YWYYGYYYYR  W  191 Y
WYWWRGYWWW  Y  192 Y  YWGWWWWRWW  G  193 W  YWYYRGYYYY  W  194 Y  GWWYWWWWWW
G  195 W  GWYYYRYYYY  W  196 W  YWWWWWGWWY  W  197 W  YWYYWGYWYY  W  198 G
RWWYWWWWWW  R  199 W  RWYYYYGYYY  W  200 W  GWWWWWRWWY  W  201 W  GWYYWYWYYY
W  202 W  WWYYYYGWYY  W  203 Y  GWYWWWWYWW  G  204 W  YWYYYYGYYY  W  205 W
GWYYWRYWYY  W  206 W  WYWWWYGWWW  W  207 R  YWWGWWWWWW  Y  208 W  RWYYWGWYYY
W  209 G  RWYWWWWYWW  R  210 W  GWYYYYRYYY  W  211 Y  GYWWWWWWWW  G  212 W
GWWWWWYWWW  W  213 Y  YWYWWWGYWW  Y  214 W  WYWWWWGWWW  W  215 W  RWYYWYGWYY
W  216 W  WGWWWYRWWW  W  217 R  YWGWWWWYWW  Y  218 W  RWYYYGYYYY  W  219 G
RYWWWWWWWW  R  220 W  RWWWWWGWWW  W  221 Y  GWYWWWRYWW  G  222 W  WGWWWWRWWW
W  223 Y  YWRWWWWGWW  Y  224 W  YWYYGRYYYY  W  225 R  YGWWWWWWWW  Y  226 G
RWYWWWYYWW  R  227 W  YWYYRYYYYG  W  228 R  YWGWWWYYWW  Y  229 W  YWYGYYYYYR
W  230 Y  WYWWYGWWWW  Y  231 Y  YWRWWWYGWW  Y  232 W  YWYRYYYYGY  W  233 Y
WYWWGRWWWW  G  234 W  YWYYWYWYYG  W  235 W  YWGYYYYYRY  W  236 G  WYWWRYWWWW
R  237 W  YWYGWYWYYR  W  238 W  YWRYYYYGYY  W  239 R  WGWWYYWWWW  Y  240 W
YWYRWYWYGY  W  241 W  YWGYWYWYRY  W  242 W  WWYYYGYWYY  W  243 W  YWRYWYWGYY
W  244 W  WWYYGRYWYY  W  245 W  WWYYRYYWYG  W  246 W  WWYGYYYWYR  W  247 Y
WWWWYGWWWW  Y  $\uparrow$ motion 248 W  WWYGYYYWRY  W  249 W  WWYRYYYWYG  W
250 W  YWYYWYWGYY  W  251 W  WWYYGYYWYR  W  252 W  YWGYWYWRYY  W  253 W
WWYYRGYWYY  W  254 W  YWRYWYWYGY  W  255 W  YWYGWYWYRY  W  256 W  YWYRWYWYYG
W  257 Y  YWWGWWWWWW  G  258 W  YWYYWGWYYY  W  259 W  YWYYWYGWYY  W  260 W
WWGWWWYYWW  W  261 G  YWWRWWWWWW  R  262 W  GWWWWWYWWY  W  263 W  GWYYWRWYYY
W  264 W  GWYYWYRWYY  W  265 W  WWRWWWYGWW  W  266 R  GWWYWWWWWW  Y  267 W
RWWWWWGWWY  W  268 W  RWYYWGYWYY  W

269 W  YWYYWYYWYG  W  270 W  YWYGWYYWYR  W  Locomotives on Z-track blue
locomotive $\downarrow$ motion 271 W  YWBYWYYYRW  W  272 Y  WRWWWWBWWW  Y  273
W  YWWWWWYYWW  W  274 W  YWYYWWYYYY  W  275 W  YWWWWYWYWW  W  276 W
WWYYYWYYYY  W  277 Y  WWYWWWWYWW  Y  278 W  YWRYWYYBYW  W  279 Y  WYWWWWRWWW
Y  280 W  YWYYWYYRYW  W  281 W  YWYYWYYYYW  W  282 W  YWYYWYBYYW  W  283 W
YWWWWBWYWW  W  284 W  BWYYWYRYYW  W  285 W  YWWWWWBYWW  W  286 W  BWYYWWYYYY
W  287 W  BWWWWRWYWW  W  288 W  WWYYYWBYYY  W  289 W  RWYYWBYYYW  W  290 W
BWWWWWRYWW  W  291 W  RWYYWWBYYY  W  292 W  RWWWWYWYWW  W  293 W  WWYYYWRYYY
W  294 W  YWYYWRYYYW  W  295 W  RWWWWWYYWW  W  296 W  YWYYWWRYYY  W  297 W
YWYYWWYBYY  W  298 W  YWBYWWYRYY  W  299 W  YWRYWWYYBY  W  300 W  YWYBWWYYRY
W  301 W  WWYYYWYBYY  W  302 W  YWYRWWYYYB  W  303 W  WWBYYWYRYY  W  304 W
YWYYWWYYYR  W  305 W  WWRYYWYYBY  W  306 W  WWYBYWYYRY  W  307 Y  WWYWWWWBWW
Y  $\uparrow$ motion 308 W  WWYBYWYYYR  W  309 W  WWYRYWYYBY  W  310 W
YWYYWWYYYB  W  311 W  WWBYYWYYRY  W  312 W  YWYBWWYYYR  W  313 W  WWRYYWYBYY
W  314 W  YWYRWWYYBY  W  315 W  WWYYYWYRYY  W  316 W  YWBYWWYYRY  W  317 W
YWRYWWYBYY  W  318 W  YWYYWWYRYY  W  319 W  YWYYWBYYYW  W  320 W  BWWWWWYYWW
W  321 W  YWYYWWBYYY  W  322 W  BWYYWRYYYW  W  323 W  RWWWWWBYWW  W  324 W
BWYYWWRYYY  W  325 W  BWWWWYWYWW  W  326 W  RWYYWYBYYW  W  327 W  YWWWWWRYWW
W  328 W  RWYYWWYYYY  W  329 W  RWWWWBWYWW  W  330 W  YWYYWYRYYW  W  331 W
YWWWWRWYWW  W  332 W  YWYYWYYBYW  W  333 W  YWBYWYYRYW  W  green locomotive
$\downarrow$ motion 334 W  YWGYWYYYRW  W  335 Y  WRWWWWGWWW  Y  336 W
YWRYWYYGYW  W  337 W  YWYYWYGYYW  W  338 W  YWWWWGWYWW  W  339 W  GWYYWYRYYW
W  340 W  YWWWWWGYWW  W  341 W  GWYYWWYYYY  W  342 W  GWWWWRWYWW  W  343 W
WWYYYWGYYY  W  344 W  RWYYWGYYYW  W  345 W  GWWWWWRYWW  W  346 W  RWYYWWGYYY
W  347 W  YWYYWWYGYY  W  348 W  YWGYWWYRYY  W  349 W  YWRYWWYYGY  W  350 W
YWYGWWYYRY  W  351 W  WWYYYWYGYY  W  352 W  YWYRWWYYYG  W  353 W  WWGYYWYRYY
W  354 W  WWRYYWYYGY  W  355 W  WWYGYWYYRY  W  356 Y  WWYWWWWGWW  Y
$\uparrow$ motion 357 W  WWYGYWYYYR  W  358 W  WWYRYWYYGY  W  359 W
YWYYWWYYYG  W  360 W  WWGYYWYYRY  W  361 W  YWYGWWYYYR  W  362 W  WWRYYWYGYY
W  363 W  YWYRWWYYGY  W  364 W  YWGYWWYYRY  W  365 W  YWRYWWYGYY  W  366 W
YWYYWGYYYW  W  367 W  GWWWWWYYWW  W  368 W  YWYYWWGYYY  W  369 W  GWYYWRYYYW
W  370 W  RWWWWWGYWW  W  371 W  GWYYWWRYYY  W  372 W  GWWWWYWYWW  W  373 W
RWYYWYGYYW  W  374 W  RWWWWGWYWW  W  375 W  YWYYWYYGYW  W  376 W  YWGYWYYRYW
W  passive fixed switch blue locomotive $\bullet$ $\rightarrow$ 377 W
WYWWWWYYWW  W  378 W  YWYBWYWWRY  W  379 W  WWWWWYYWWW  W  380 Y  YWYYWWWYRY
Y  381 Y  YYWWRYWWWW  Y  382 Y  YRWWYWYWWW  Y  383 R  YYWRWWYWRR  R  384 W
RWWWRRYWWW  W  385 R  RRWWRWWWWW  R  386 W  WRRWWWYRWW  W  387 R  RWWWWWRWWW
R  388 W  RWWWWWRWWW  W  389 W  RWWWWRWWWW  W  390 R  WRWWWWRWWW  R  391 Y
YWWWWWWWWW  Y  392 W  YWYRWYWWYB  W  393 W  YWYYWYWWYR  W

394 W  YWYYWYWWYY  W  395 Y  YWYYWWWBRY  Y  396 Y  YBWWRYWWWW  B  397 Y
YWBYWWWRRY  B  398 B  YRWWRYWWWW  R  399 Y  YRWWYWBWWW  Y  400 R  YBWRWWYWRR
R  401 W  RWWWRRBWWW  W  402 W  YWYYWBWWYY  W  403 B  YWRYWWWYRY  R  404 R
BYWWRYWWWW  Y  405 Y  BRWWYWRWWW  Y  406 R  YRWRWWBWRR  R  407 W  RWWWRRRWWW
W  408 W  WBYYYYYYYY  W  409 W  WYWWWWBYWW  W  410 W  BWYYWRWWYY  W  411 W
WWWWWYBWWW  W  412 R  BWYYWWWYRY  Y  413 Y  RYWWRYWWWW  Y  414 Y  RRWWYWYWWW
Y  415 R  YYWRWWRWRR  R  416 W  WBWWWWRYWW  W  417 W  RWYYWYWWYY  W  418 W
WWWWWYRWWW  W  419 Y  RWYYWWWYRY  Y  420 W  WRWWWWYBWW  W  421 W  WYWWWWYRWW
W  422 Y  BWWWWWWWWW  Y  $\leftarrow$ $\bullet$ 423 Y  YWYYWWWYRB  Y  424 Y
YRWWBWYWWW  B  425 Y  YWYBWWWYRR  B  426 Y  YYWWRBWWWW  Y  427 B  YRWWRWYWWW
R  428 R  BYWRWWYWRR  R  429 W  WRRWWWBRWW  W  430 B  YWYRWWWYRY  R  431 W
BWYYYYYYYY  W  432 Y  BYWWRRWWWW  Y  433 R  BRWWYWYWWW  Y  434 R  RYWRWWBWRR
R  435 W  WRRWWWRRWW  W  green locomotive $\bullet$ $\rightarrow$ 436 W
YWYGWYWWRY  W  437 W  YWYRWYWWYG  W  438 Y  YWYYWWWGRY  Y  439 Y  YGWWRYWWWW
G  440 Y  YWGYWWWRRY  G  441 G  YRWWRYWWWW  R  442 Y  YRWWYWGWWW  Y  443 R
YGWRWWYWRR  R  444 W  RWWWRRGWWW  W  445 W  YWYYWGWWYY  W  446 G  YWRYWWWYRY
R  447 R  GYWWRYWWWW  Y  448 Y  GRWWYWRWWW  Y  449 R  YRWRWWGWRR  R  450 W
WGYYYYYYYY  W  451 W  WYWWWWGYWW  W  452 W  GWYYWRWWYY  W  453 W  WWWWWYGWWW
W  454 R  GWYYWWWYRY  Y  455 W  WGWWWWRYWW  W  456 W  WRWWWWYGWW  W  457 Y
GWWWWWWWWW  Y  $\leftarrow$ $\bullet$ 458 Y  YWYYWWWYRG  Y  459 Y  YRWWGWYWWW
G  460 Y  YWYGWWWYRR  G  461 Y  YYWWRGWWWW  Y  462 G  YRWWRWYWWW  R  463 R
GYWRWWYWRR  R  464 W  WRRWWWGRWW  W  465 G  YWYRWWWYRY  R  466 W  GWYYYYYYYY
W  467 Y  GYWWRRWWWW  Y  468 R  GRWWYWYWWW  Y  469 R  RYWRWWGWRR  R  fork blue
locomotive 470 W  WYWWWWYBWW  W  471 W  WBWWWWYRWW  W  472 W  WRWWWWBYWW  W
473 W  WYWWWWRYWW  W  474 Y  YYWRRYWWWW  Y  475 Y  YRRWYWYWWW  Y  476 R
YWWWWRWWWW  R  477 R  YYWWWRYRWW  R  478 R  YWWWWWRWWW  R  479 W  RRWWWWYWWW
W  480 W  BWYYWYWWYY  W  481 Y  BWYYWWWYRY  B  482 W  RWYYWBWWYY  W  483 B
RWYYWWWYRY  R  484 Y  BYWRRYWWWW  B  485 Y  BRRWYWYWWW  B  486 R  YYWWWRBRWW
R  487 W  YWYYWRWWYY  W  488 R  YWBBWWWYRY  Y  489 B  RYWRRBWWWW  R  490 B
RRRWYWBWWW  R  491 R  BWWWWRWWWW  R  492 R  BBWWWRRRWW  R  493 R  BWWWWWRWWW
R  494 W  RRWWWWBWWW  W  495 Y  YWRRWWWBRB  Y  496 R  YBWRRRWWWW  Y  497 R
YRRWBWRWWW  Y  498 W  RWWWWRBWWW  W  499 R  RRWWWRYRWW  R  500 W  RWWWWBRWWW
W  501 W  RRWWWWRWWW  W  502 Y  YWYYWWWRRR  Y  503 Y  YRWRRYWWWW  Y  504 Y
YRRWRWYWWW  Y  505 W  YWWWWRRWWW  W  506 W  YWYYWYWWYB  W  507 W  YWYBWYWWYR
W  green locomotive 508 W  WYWWWWYGWW  W  509 W  WGWWWWYRWW  W  510 W
WRWWWWGYWW  W  511 W  GWYYWYWWYY  W  512 Y  GWYYWWWYRY  G  513 W  RWYYWGWWYY
W  514 G  RWYYWWWYRY  R  515 Y  GYWRRYWWWW  G  516 Y  GRRWYWYWWW  G  517 R
YYWWWRGRWW  R  518 R  YWGGWWWYRY  Y  519 G  RYWRRGWWWW  R  520 G  RRRWYWGWWW
R  521 R  GWWWWRWWWW  R  522 R  GGWWWRRRWW  R  523 R  GWWWWWRWWW  R  524 W
RRWWWWGWWW  W

525 Y  YWRRWWWGRG  Y  526 R  YGWRRRWWWW  Y  527 R  YRRWGWRWWW  Y  528 W
RWWWWRGWWW  W  529 W  RWWWWGRWWW  W  530 W  YWYYWYWWYG  W  531 W  YWYGWYWWYR
W  converters from blue to green 532 Y  WYRGYWWWWW  Y  533 Y  YWYWWRWWWW  Y
534 R  YWRRWGYWWW  R  535 W  RYWWWRYWWW  W  536 G  YWGGWYRWWW  G  537 W
GRWWWGYRWW  W  538 G  GWWWWGWWWW  G  539 G  GWWWWWGWWW  G  540 W  GWWWWWGWWW
W  541 W  GGWWWWGWWW  W  542 W  YGWWWYYGWW  W  543 W  WGWWWWGWWW  W  544 Y
RWWWWWWWWW  Y  545 Y  YWBWWWGWWW  B  546 W  YGWWWBYGWW  W  547 Y  WYRGBWWWWW
G  548 G  YWGGWBRWWW  G  549 B  YWRWWWGWWW  R  550 W  BGWWWRYGWW  W  551 G
WYRGRWWWWW  R  552 Y  GWYWWRWWWW  G  553 R  GWRRWGYWWW  R  554 W  RYWWWRGWWW
W  555 G  GWGGWRRWWW  G  556 W  GRWWWGGRWW  W  557 R  GWYWWWGWWW  Y  558 W
RGWWWYGGWW  W  559 R  WGRGYWWWWW  Y  560 G  RWYWWRWWWW  R  561 R  RWRRWGGWWW
R  562 W  RGWWWRRWWW  W  563 G  RWGGWYRWWW  G  564 W  GRWWWGRRWW  W  565 W
YGWWWYRGWW  W  566 Y  WRRGYWWWWW  Y  567 R  YWGWWRWWWW  Y  568 R  YWRRWGRWWW
R  569 W  RRWWWRYWWW  W  from green to blue 570 Y  WYRBYWWWWW  Y  571 R
YWRRWBYWWW  R  572 B  YWGGWYRWWW  B  573 W  BRWWWGYRWW  W  574 G  BWWWWGWWWW
G  575 G  BWWWWWGWWW  G  576 W  GWWWWWBWWW  W  577 W  GGWWWWBWWW  W  578 W
YBWWWYYGWW  W  579 W  WGWWWWBWWW  W  580 Y  YWGWWWBWWW  G  581 W  YBWWWGYGWW
W  582 Y  WYRBGWWWWW  B  583 B  YWGGWGRWWW  B  584 G  YWRWWWBWWW  R  585 W
GBWWWRYGWW  W  586 B  WYRBRWWWWW  R  587 Y  BWYWWRWWWW  B  588 R  BWRRWBYWWW
R  589 W  RYWWWRBWWW  W  590 B  BWGGWRRWWW  B  591 W  BRWWWGBRWW  W  592 R
BWYWWWBWWW  Y  593 W  RBWWWYBGWW  W  594 R  WBRBYWWWWW  Y  595 B  RWYWWRWWWW
R  596 R  RWRRWBBWWW  R  597 W  RBWWWRRWWW  W  598 B  RWGGWYRWWW  B  599 W
BRWWWGRRWW  W  600 W  YBWWWYRGWW  W  601 Y  WRRBYWWWWW  Y  602 R  YWBWWRWWWW
Y  603 R  YWRRWBRWWW  R  filters blue filter blue locomotive 604 Y  BYRWYWWWWW
Y  605 B  YWYWWYYYRR  B  606 W  BYWYYYYWWY  W  607 W  BRWWRWYWWW  W  608 R
WYWWWWBRWW  R  609 Y  YWYWWRBWWW  Y  610 W  YBRWWYYWWW  W  611 R  WWWWWWBWWW
R  612 R  YWRRWWYWWW  R  613 W  WRWWWWYRWW  W  614 Y  YWYWWBWWWW  Y  615 Y
YWBWWBWWWW  B  616 Y  BYRWBWWWWW  B  617 B  YWYWWYBYRR  B  618 W  BBWYYYYWWY
W  619 W  YWWWWBRWWW  W  620 B  YWRWWBWWWW  R  621 B  BYRWRWWWWW  R  622 B
BWYWWYRYRR  B  623 W  BRWYYYBWWY  W  624 Y  BWYWWRBWWW  B  625 W  YBRWWYBWWW
W  626 R  BWRRWWYWWW  R  627 W  BWWWWRRWWW  W  628 R  BWYWWBWWWW  Y  629 R
BBRWYWWWWW  Y  630 B  RWYWWBYYRR  B  631 W  BYWYYYRWWY  W  632 B  RWYWWRBWWW
R  633 W  BBRWWYRWWW  W  634 R  RWRRWWBWWW  R  635 W  WRWWWWRRWW  W  636 Y
RWYWWBWWWW  Y  637 Y  BRRWYWWWWW  Y  638 B  YWYWWRYYRR  B  639 R  YWBWWRBWWW
Y  640 W  RBRWWBYWWW  W  641 R  YWRRWWRWWW  R  green locomotive 642 Y
YWGWWBWWWW  G  643 Y  BYRWGWWWWW  Y  644 B  YWYWWYGYRR  B  645 W  BGWYYYYWWY
W  646 W  YWWWWGRWWW  W  647 G  YWRWWBWWWW  R  648 Y  BYRWRWWWWW  Y  649 B
YWYWWYRYRR  B  650 W  BRWYYYYWWY  W  651 R  YWYWWBWWWW  Y  652 W  RWWWWYYWWW
W

green filter green locomotive 653 Y  GYRWYWWWWW  Y  654 G  YWYWWYYYRR  G  655
W  GYWYYYYWWY  W  656 W  GRWWRWYWWW  W  657 R  WYWWWWGRWW  R  658 Y
YWYWWRGWWW  Y  659 W  YGRWWYYWWW  W  660 R  WWWWWWGWWW  R  661 Y  YWYWWGWWWW
Y  662 Y  YWGWWGWWWW  G  663 Y  GYRWGWWWWW  G  664 G  YWYWWYGYRR  G  665 W
GGWYYYYWWY  W  666 G  YWRWWGWWWW  R  667 G  GYRWRWWWWW  R  668 G  GWYWWYRYRR
G  669 W  GRWYYYGWWY  W  670 Y  GWYWWRGWWW  G  671 W  YGRWWYGWWW  W  672 R
GWRRWWYWWW  R  673 W  GWWWWRRWWW  W  674 R  GWYWWGWWWW  Y  675 R  GGRWYWWWWW
Y  676 G  RWYWWGYYRR  G  677 W  GYWYYYRWWY  W  678 G  RWYWWRGWWW  R  679 W
GGRWWYRWWW  W  680 R  RWRRWWGWWW  R  681 Y  RWYWWGWWWW  Y  682 Y  GRRWYWWWWW
Y  683 G  YWYWWRYYRR  G  684 R  YWGWWRGWWW  Y  685 W  RGRWWGYWWW  W  blue
locomotive 686 Y  YWBWWGWWWW  B  687 Y  GYRWBWWWWW  Y  688 G  YWYWWYBYRR  G
689 W  GBWYYYYWWY  W  690 B  YWRWWGWWWW  R  691 Y  GYRWRWWWWW  Y  692 G
YWYWWYRYRR  G  693 W  GRWYYYYWWY  W  694 R  YWYWWGWWWW  Y  changing colour
from blue to green 695 B  YWYWWYYBRR  G  696 W  BYWYBYYWWR  W  697 Y
BBWRRWWWWW  Y  698 B  YWRWWWBYWW  R  699 W  YWWWWRBWWW  W  700 G  YWYWWYYRRR
G  701 W  GYWYRYYWWY  W  702 Y  GRWRRWWWWW  Y  703 R  YWYWWWGYWW  Y  from
green to blue 704 G  YWYWWYYBRR  B  705 W  GYWYBYYWWR  W  706 Y  GBWRRWWWWW  Y
707 B  YWRWWWGYWW  R  708 B  YWYWWYYRRR  B  709 W  BYWYRYYWWY  W  710 Y
BRWRRWWWWW  Y  711 R  YWYWWWBYWW  Y  registers general case incrementing 712 Y
WYYYYYYYYY  Y  713 W  WYYWYYYYYB  W  714 W  WYYWBYYYYR  W  715 W  WYWWRBYWWW
W  716 Y  YYWWYYWWWW  Y  717 R  BWWWWWWWWW  Y  718 Y  YYWWYYYWWW  Y  719 Y
YYWWWWYWWW  Y  720 Y  YYWWYWYWWW  Y  721 W  WYYWRBYYYY  W  722 W  WYWWYRYWWW
W  723 Y  WYYYYYBYYY  Y  724 W  WYYWYRYYYY  W  725 W  WYWWYYYWWW  W  726 Y
YBWWYYWWWW  B  727 Y  WBYYYYRYYY  B  728 W  WYYWYYYYYY  W  729 B  YRWWYYWWWW
R  730 Y  YYWWYYBWWW  M  731 Y  YBWWWWYWWW  M  732 B  WRMYYYYMYY  R  733 R
BYWWMMWWWW  Y  734 W  RWWWWMWWWW  W  735 M  BMWWYYRWWW  M  736 M  MRWWWWBWWW
Y  737 W  MWWWWWMWWW  Y  738 W  MWWWWYWWWW  W  739 W  MWWWWWRWWW  W  740 W
MWWWWWWWWW  W  741 W  WMWWWWMWWW  M  742 W  WWWWWWMWWW  W  743 Y  BYWWYYMWWW
Y  744 Y  YMWWWWBWWW  M  745 W  YWWWWWMWWW  W  746 Y  BYWWYWYWWW  B  747 Y
YYWWWWBWWW  R  748 R  WYMYBYYYMR  W  749 Y  RYWWYMWWWW  Y  750 M  RYYWMYYMWW
M  751 Y  MYWWMYRWWW  Y  752 Y  MMWWWWYWWW  Y  753 W  MWWWWMMWWW  M  754 W
MWWWWWYWWW  W  755 W  YWWWWMWWWW  W  756 M  YYWWWWMWWW  Y  757 Y  RMWWRBMWWW
Y  758 M  YMWWWWRWWW  M  759 W  BWWWWWMWWW  W  760 W  WMWWWWBWWW  W  761 B
RRWWYWYWWW  R  762 R  BYWWWWRWWW  Y  763 W  WYMYRBYYMY  W  764 W  WBYWYYRYYY
W  765 Y  WYWWYMWWWW  Y  766 M  WYYWMYYYWY  M  767 Y  MYWWYYWWWW  Y  768 Y
MYWWWWYWWW  Y  769 W  MWWWYMYWWW  Y  770 W  MWWWWMYWWW  W  771 Y  YYWWWWMWWW
Y  772 Y  WMWWYRMWWW  Y  773 M  YMYWWWWWWW  Y  774 Y  MWWWWWMWWW  Y  775 W
WYMYYRYYYY  W  776 W  WRYWYYYBYY  W  777 M  WYYYYYYYWY  Y  778 Y  MYWWWYYWWW
Y  779 Y  MWWWYYYWWW  Y

780 W  YYWWWWMWWW  Y  781 Y  WYWWYYMWWW  Y  782 Y  YMYWWWWYWW  Y  783 W
WYBWYYYRYY  W  784 W  WYYWYYBYYY  W  785 W  WYRWYYYYBY  W  786 W  WYYWYYYYRY
W  787 W  WYWWYYBWWW  W  788 W  WYRWYYYYYY  W  789 R  WYWWYWYWWW  Y  790 W
WYWWYYRWWW  W  decrementing 791 W  WYYWYYYYYG  W  792 W  WYYWGYYYYR  W  793 W
WYWWRGYWWW  W  794 R  GWWWWWWWWW  Y  795 W  WYYWRGYYYY  W  796 Y  WYYYYYGYYY
Y  797 Y  YGWWYYWWWW  G  798 Y  WGYYYYRYYY  M  799 G  YRWWYYWWWW  R  800 Y
YYWWYYGWWW  Y  801 Y  YGWWWWYWWW  Y  802 M  WRYYYYYYYY  M  803 W  MYYWYRYYYY
Y  804 R  MYWWYYWWWW  Y  805 Y  RWYWWWMWWW  Y  806 Y  MYWWYYRWWW  W  807 Y
YRWWWWMWWW  W  808 Y  MYWWYYYWWW  W  809 M  YYWWYYYWYY  M  810 Y  MYYWYYYYYY
Y  811 Y  YYWWYYMWWW  Y  812 Y  YYYWWWMWWW  Y  813 W  MWWWYWYWWW  W  814 W
MYWWYYWWWW  W  815 Y  WWWWWWMWWW  W  816 Y  YWWWWWMWWW  R  817 M  YYWWYYYWWR
Y  818 W  MWWWRYWWWW  W  819 Y  MRWWYYWWWW  G  820 R  YWWWWWMWWW  R  821 Y
YYWWGYYWWR  Y  822 Y  YYYWYYGYYY  Y  823 Y  YGWWYYYWWW  G  824 W  YWWWRGWWWW
W  825 R  GWWWWWYWWW  W  826 Y  YYWWRGYWWW  Y  827 Y  YGYWYYRYYY  Y  828 G
YRWWYYYWWW  R  829 Y  YYWWYWGWWW  Y  830 R  YWWWGYWWWW  Y  831 Y  YYWWYRYWWW
Y  832 Y  YRYWYYYYYY  Y  833 R  YYWWYYYWWW  R  834 Y  YYWWYWRWWW  Y  835 Y
YRWWWWYWWW  G  836 Y  YWWWRYWWWW  Y  837 Y  YRYWYYYGYY  Y  838 R  YYWWGYYWWW
Y  839 Y  YGWWYWRWWW  G  840 G  YRWWWWYWWW  R  841 Y  YYGWYYYRYY  Y  842 G
YRWWYWYWWW  R  843 W  YYYWYYGYYY  W  844 Y  WGWWYYYWWW  G  845 Y  YYRWYYYYGY
Y  846 R  YYWWGWYWWW  Y  847 Y  YYYWYYYYRY  Y  848 Y  YYWWRWYWWW  Y  849 W
WYYWYYGYYY  W  850 W  WGYWYYRYYY  W  $\downarrow$ in the register blue
locomotive 851 Y  WYYWYWMMWW  Y  852 W  YWWWMYYWWW  W  853 Y  YMYWMYWWWW  Y
854 M  YWWWWYYWWW  M  855 Y  YWWYWWMWWW  Y  856 W  YWWWWMYWWW  W  857 W
YMWWWWYWWW  W  858 Y  YMWYWWYWWW  Y  859 M  YYWWWWYWWW  M  860 W  YWWWWYMWWW
W  861 Y  YWWYWWWWYW  Y  862 W  YWWWYYWWWW  W  863 Y  WYYWBWMMWW  B  864 W
YWWWMYBWWW  W  865 B  WYYWRWMMWW  R  866 W  BWWWMYRWWW  W  867 Y  BMYWMYWWWW
Y  868 M  YWWWWYBWWW  M  869 Y  BMWYWWYWWW  B  870 M  YYWWWWBWWW  M  871 R
WYBWYWMMWW  Y  872 W  RWWWMYYWWW  W  873 Y  RMYWMBWWWW  Y  874 M  YWWWWYRWWW
M  875 B  RMWYWWYWWW  R  876 M  BYWWWWRWWW  M  877 W  BWWWWYMWWW  W  878 Y
WYRWYWMMWW  Y  879 Y  YMYWMRWWWW  Y  880 R  YMWBWWYWWW  Y  881 M  RYWWWWYWWW
M  882 W  RWWWWBMWWW  W  883 W  WMWWWWRWWW  W  884 Y  BWWYWWWWYW  B  885 Y
YMWRWWYWWW  Y  886 W  YWWWWRMWWW  W  887 B  RWWYWWWWYW  R  888 W  BWWWYYWWWW
W  889 R  YWWBWWWWYW  Y  890 W  RWWWYBWWWW  W  891 Y  BWWWWWRWWW  Y  892 Y
YWWRWWWWYW  Y  893 W  YWWWYRWWWW  W  894 R  YYWWWWWWWW  Y  895 Y  RWWWWWYWWW
Y  green locomotive 896 Y  WYYWGWMMWW  G  897 W  YWWWMYGWWW  W  898 G
WYYWRWMMWW  R  899 W  GWWWMYRWWW  W  900 Y  GMYWMYWWWW  Y  901 M  YWWWWYGWWW
M  902 Y  GMWYWWYWWW  G  903 M  YYWWWWGWWW  M  904 R  WYGWYWMMWW  Y  905 Y
RMYWMGWWWW  Y  906 G  RMWYWWYWWW  R  907 M  GYWWWWRWWW  M  908 W  GWWWWYMWWW
W  909 W  WMWWWWGWWW  W  910 R  YMWGWWYWWW  Y

911 W  RWWWWGMWWW  W  912 Y  GWWYWWWWYW  G  913 G  RWWYWWWWYW  R  914 W
GWWWYYWWWW  W  915 R  YWWGWWWWYW  Y  916 W  RWWWYGWWWW  W  917 Y  GWWWWWRWWW
Y  $\uparrow$ in the register blue locomotive 918 W  WWWYYYWWYY  W  919 W
WWWWYWYWWW  W  920 W  WYYWWWYYWW  W  921 Y  WYYWWWYWWW  Y  922 Y  YMYYWWYWYW
Y  923 M  YYWYYYYWYY  M  924 Y  YYWYWWYWWW  Y  925 W  MWWWYYYWWW  W  926 W
WYWWYYYYWW  W  927 W  YWYWWYYYYY  W  928 W  YWYWWBYYWW  W  929 Y  WWWWWWWWWW
Y  930 W  WWWYYYWWBY  W  931 W  BWYWWRYYWW  W  932 W  WWWBYYWWRY  W  933 W
RWYWWYBYWW  W  934 W  WWWRYYWWYB  W  935 W  YWYWWYRYWW  W  936 W  WWWYBYWWYR
W  937 W  WYWWYYBYWW  W  938 W  WWWYRBWWYY  W  939 W  YWYWWYYYWW  W  940 W
WBWWYYRYWW  W  941 W  WWWYYRWWYY  W  942 W  WRWWYYYBWW  W  943 W  WYYWWWBYWW
W  944 W  WYWWYYYRWW  W  945 W  WBYWWWRYWW  W  946 Y  WYYWWWBWWW  Y  947 Y
YBWWWYWWWW  B  948 W  WRYWWWYBWW  W  949 Y  WBYWWWRWWW  Y  950 B  YRWWWYWWWW
R  951 W  WYYWWWYRWW  W  952 Y  WYWWRYWWWW  Y  953 Y  WRYWWWYWWW  Y  954 R
YYWWWYWWWW  Y  955 W  YRWWWWYWWW  W  956 W  WYRWWWYYYW  W  green locomotive
957 W  YWYWWGYYWW  W  958 W  WWWYYYWWGY  W  959 W  GWYWWRYYWW  W  960 W
WWWGYYWWRY  W  961 W  RWYWWYGYWW  W  962 W  WWWRYYWWYG  W  963 W  WWWYGYWWYR
W  964 W  WYWWYYGYWW  W  965 W  WWWYRGWWYY  W  966 W  WGWWYYRYWW  W  967 W
WRWWYYYGWW  W  968 W  WYYWWWGYWW  W  969 W  WGYWWWRYWW  W  970 Y  WYYWWWGWWW
Y  971 Y  YGWWWYWWWW  G  972 W  WRYWWWYGWW  W  973 Y  WGYWWWRWWW  Y  974 G
YRWWWYWWWW  R  particular cases $n=0$ incrementing 975 Y  WYYYYWMMYW  Y  976 Y
YMWWYYYWWW  Y  977 Y  WYYYBWMMYW  B  978 B  WYYYRWMMYW  R  979 Y  BMWWYYYWWW
B  980 R  WYBBYWMMMW  Y  981 B  RMWWMBYWWW  Y  982 M  RYWWWWRWWW  M  983 W
RWWWWWMWWW  W  984 W  BWWWWMWWWW  M  985 B  RMYYWYBYYY  R  986 M  BBWWYYRWWW
M  987 W  MWWWWWBWWW  Y  988 Y  WYYRYWMMMW  Y  989 Y  YMWMMRYWWY  Y  990 W
YWWWWMMWWW  W  991 M  RYWWMYYWWW  M  992 Y  WYYWYWMMMW  Y  993 Y  YMWYMWYWWY
Y  994 Y  WYYWYWMMYW  Y  995 Y  YMWYYWYWWY  Y  996 W  YWYWWYYBWW  W  997 W
YWYWWYYRWW  W  no decrementing 998 Y  WYYYGWMMYW  M  999 M  WYYYRWMMYW  M
1000 W  MWWWMYRWWW  W  1001 Y  MMYWMYWWWW  G  1002 M  YWWWWYMWWW  M  1003 Y
MMWWYYYWWW  Y  1004 Y  MYYYWRYYYY  Y  1005 R  MWYWWWYWWW  Y  1006 W
RYYYWYMYYY  W  1007 M  WGYYYWMYYW  M  1008 W  MWWWMGYWWW  W  1009 G
MMYWYYWWWW  R  1010 M  GWWWWYMWWW  M  1011 Y  GWWYWWMWWW  G  1012 W
GWWWWYYWWW  W  1013 W  YMWWWWGWWW  W  1014 Y  MYWWYYGWWW  Y  1015 Y
YGWWWWMWWW  M  1016 Y  MYYYWYYYYY  Y  1017 Y  MWYWWWYWWW  Y  1018 W
YYYYWYMYYY  W  1019 M  WRYYYWMMYW  Y  1020 W  MWWWMRYWWW  W  1021 R
MMGWMYWWWW  Y  1022 M  RWWWWGMWWW  M  1023 G  RWWYWWMWWW  R  1024 W
RWWWWMGWWW  W  1025 W  GMWWWWRWWW  W  1026 Y  MMWWYYRWWW  Y  1027 M
YRWWWWMWWW  M  1028 Y  YMRWMYWWWW  Y  1029 M  YWWWWRYWWW  M  1030 R
YWWGWWMWWW  Y  1031 W  YWWWWMRWWW  W  1032 Y  YWWRWWMWWW  Y  1033 R
YWYWWWWWWW  Y  $n=1$ incrementing 1034 Y  WYYWBWMMYW  B  1035 B  WYYWRWMMYW  R
1036 Y  BMWYYWYWWY  B  1037 R  WYBWYWMMYW  Y

1038 B  RMWYYWYWWY  R  1039 Y  WYRWYWMMBW  Y  1040 R  YMWMBWYWWM  Y  1041 W
RWWWWMMWWW  W  1042 Y  WYYWYWMMRW  Y  1043 Y  YMWYRWYWWM  Y  1044 Y
YMWYWWYWWM  Y  1045 Y  YMWYWWYWWY  Y  decrementing 1046 Y  WYYWGWMMYW  G  1047
G  WYYWRWMMYW  R  1048 Y  GMWYYWYWWY  G  1049 R  WYGWYWMMYW  Y  1050 G
RMWYYWYWWY  R  1051 Y  WYRWYWMMMW  Y  1052 R  YMWYMWYWWY  Y  1053 W
RWWWWYMWWW  W  1054 Y  WYYYYWMMMW  Y  1055 Y  YMWWMYYWWW  Y  $n=2$
decrementing 1056 M  YYWWWWYYWW  M  1057 Y  MYMWYWYWWW  Y  1058 M  YWWWWWYWWW
M  1059 Y  WYWWYWMWWW  Y  1060 Y  MWYWGYWWYW  G  1061 M  YYWWWWGYWW  M  1062 Y
MGMWYWYWWW  Y  1063 M  YWWWWWGWWW  M  1064 G  MWYWRYWWYW  R  1065 M
GYWWWWRYWW  M  1066 Y  MRMWYWGWWW  Y  1067 M  YWWWWWRWWW  M  1068 R
MWGWYYWWYW  Y  1069 M  RYWWWWYYWW  M  1070 Y  MYMWYWRWWW  Y  1071 Y
MWRWYYWWMW  Y  1072 Y  MWYYYYWWMW  Y

###### Table 2

Table of the rules used by the cellular automaton.

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