Let's consider one-dimensional cellular devices with three basic state: A, B, and C. In, each step they changes under the law (I) A <> A and B <> C, or (II) C> B> A which the next state depends only of C position. This CA is reversible. If we will replace all  B<>C  the automata will go reverse in time. It is known that any reversible device will return to the initial point.

Ours RCA (reversible cellular automatas) R3 and R4 depends on 4 or 5 sums, in corresponding cages. (The numbers of  [3] means that the sum of two numbers 0 or 1 can accept 3 values: 0, 1, 2).Boolean array gives the kind of transition - I or II.


Originally all cells are filled of A states, and only few separate cells of B and C - state. We name these initial conditions as "object".

Any object in certain RCA can behave by three different ways:

It can through after some steps to initial position. We name such object "mill".

It can move after some steps to the right or to the left without changing. We name such object as "solitons".

It can quickly start to extend by the sides. We name such object "string".

We name RCA as interesting, if in it it is not enough (or no at all) "string" and, on the contrary, we have a lot of "solitons". We will find, by means of the computer, such RCA if it exists.

Let's begin with cellular devices R3. We will make their depending on four bits and touch all of them. Total number such RCA = 2 in the power of 2 in the power of  4 = 65000 (It is visible as number 4 is critical. If it was number 5, number RCA would become 2 in the power 32). Still it is necessary to translate 3 possible conditions in 1 bit. We can do it by two ways. 0 if S> 0, and if S> 1. Eventually we have found two obvious leaders. In the first case of 3 in the power of 8 possible conditions  we have found 38 solitons and 2208 strings. In the second 26 solitons and 940 strings. We start to work with them further.
    
    

	Search 1.	Search 2.
Initial RCA	38(2208)	26(940)
Optimization R3	27(936)	20(212)
Optimization R4	28(858)	21(56)
Can reduce the number of solitons	XXX	12(0) !

Next we do improving of ours RCA. Results are resulted in the table. Wes see that "search 1" failed. And in "search 2" we come to exclusive RCA in which we have 12 solitons and NO ONE strings.(See the table of solitons).


We have chosen for everyone solitons certain Color and make the program for their collisions. We will remind that soliton + the mill cannot pass to the mill from the properties of RCA. That is nessesary to be solitons in the result..

Pressing by the left mouse button on the Color, leads to make the solitons in its beginning. Right - make the solitons going in the opposite direction in the end. In the centre it is possible to put various mills. Press DO COLLISIONS and look result.

About our command...

GroupBox - Choose solions from the list.
GroupBox - Set rand. sol. Make random group of solitons.
GroupBox - Choose various mills on the centre with displacement dX
GroupBox - Random - change youur RCA to the another
Manage - operates of our movement. It is possible to accelerate four times time and 3 times (with 256 to 768) the total number of cells.

The command of main menu View-> Play video writes down our collisions to the set of pictures in directory Video.

And finally the result....

Solitons, as a rule, transfer at a collisions in itself, by usual reflexion. The main conclusion following. At collision of three solitons that at our lengths occurs practically at once, the system passes in a new condition thet which we named "ghost". It does not break to any solitons, and expanding fills with itself to all cells. Very unusual behaviour.