A new universal cellular automaton on the ternary heptagrid

As indicated in the abstract, this paper is a significant improvement of the first result about a universal cellular automaton on the ternary heptagrid which was obtained by the same author and Y. Song, see [13]. This time we have a weakly universal cellular automaton on the ternary heptagrid which is the smallest universal cellular automaton obtained in the hyperbolic plane, up to date. As noticed in the quoted paper, the translation of the present result to the pentagrid is not straightforward and would require at least one more state with the same pattern of simulation. We remind that for the pentagrid, the best result was obtained also by the authors of [13] in [12,14], also see [10]. The latter result was a significant improvement of the first result established in the pentagrid, see [1]. For the pentagrid, papers [12,14,10] reduce the number of states from 22 down to 9. In [13], we proved that there is a weakly universal cellular automaton on the heptagrid with six states.

In this paper, we reduce the number of states to four ones as indicated in the following: Theorem 1 (Margenstern) -There is a cellular automaton on the ternary heptagrid which is weakly universal and which has four states. Moreover, the rules of the cellular automaton are rotation invariant. The cellular automaton has an infinite initial configuration which is ultimately periodic along two different rays of mid-points r 1 and r 2 of the ternary heptagrid and finite in the complement of the parts attached to r 1 and r 2 .

Our present cellular automaton also simulates a railway circuit, as this used in papers [1,12,14,10,13]. In order to make this paper self-contained, Section 2 reminds the principles of this simulation. In Section 3, we remind the reader about hyperbolic geometry and cellular automata on the ternary heptagrid. Still in Section 3, we give the general features of the implementation of a railway circuit in the ternary heptagrid and in Section 4, we precisely define the implementation in the heptagrid. In Section 5, we give the format of the rules and the transition table of the automaton whose action is described in Section 4. We also indicate how a computer program contributed to the construction of the table.

As initially devised in [15] and then mentioned in [3,1,12,13,10], the circuit uses tracks represented by lines and quarters of circles and switches. There are three kinds of switches: the fixed, the memory and the flip-flop switches. They are represented by the schemes given in Fig. 1. 00 00 11 11 00 00 11 11 Figure 1 The three kinds of switches. From left to right: fixed, flip-flop and memory switches.

Note that a switch is an oriented structure: on one side, it has a single track u and, on the the other side, it has two tracks a and b. This defines two ways of crossing a switch. Call the way from u to a or b active. Call the other way, from a or b to u passive. The names comes from the fact that in a passive way, the switch plays no role on the trajectory of the locomotive. On the contrary, in an active crossing, the switch indicates which track between a and b will be followed by the locomotive after running on u: the new track is called the selected track.

With the help of these three kind of switches, we define an elementary circuit as in [15], which exactly contains one bit of information. The circuit is illustrated by Fig. 2, above. It can be remarked that the working of the circuit strongly depends on how the locomotive enters it. If the locomotive enters the circuit through E, it leaves the circuit through O 1 or O 2 , depending on the selected track of the memory switch which stands near E. If the locomotive enters through U , the application of the given definitions shows that the selected track at the switches near E and U are both changed: the switch at U is a flipflop which is changed by the very active passage of the locomotive and the switch at E is a memory one which is changed because it is passively crossed by the locomotive and through the non-selected track. The just described actions of the locomotive correspond to a read and a write operation on the bit contained by the circuit which consists of the configurations of the switches at E and at U . It is assumed that the write operation is triggered when we know that we have to change the bit which we wish to rewrite. From this element, it is easy to devise circuits which represent different parts of a register machine. As an example, Fig. 3 illustrates an implementation of a unit of a register. Figure 3 Here, we have two consecutive units of a register. A register contains infinitely many copies of units. Note the tracks i, d, r, j1 and j2. For incrementing, the locomotive arrives at a unit through i and it leaves the unit through r. For decrementing, it arrives though d and it leaves also through r if decrementing the register was possible, otherwise, it leaves through j1 or j2.

As indicated by its name, the fixed switch is left uncha

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