We include this brief discussion of a P=2 system as an introduction to multi-phase time. The DM model is currently based on P=6. We have found that defining a DM with a simple, single-phase clock necessitates inelegant and clumsy processes. A 2-phase clock solves some of those problems and a 6-phase clock actually appears simplest while modeling more of physics in a natural way. If we use a 3-phase clock, we get chiral time as 1, 2, 3, 1, 2, 3… is different than 1, 3, 2, 1, 3, 2… A 3-phase clock has other problems related to being awkward with a DSOS (Digital Second Order System). A 2-phase clock does not have that inherent chiral property. Consider: 1, 2, 1, 2, 1, 2… is the same as the reverse 2, 1, 2, 1, 2, 1… Nevertheless, a RUCA can have any kind of time, any kind of spatial connectivity and still do physics exactly. The object in considering these details is nothing but simplification as opposed to necessity.
A two-phase Salt RUCA has a state that consists of an ordered pair (S2t, S2t±1). The element with an even time subscript stands for the state of the Sodium sub-array and the other element with an odd time subscript stands for the state of the Chloride sub-array. The state with an even time subscript is always the first of the pair, and the one with an odd subscript is always on the right. The state with the higher time subscript is the present and the state with the lower time subscript is the past. There are only 2 rules, R0 and R1 that are applied alternately. R1 is used when the micro-time, τ is an odd integer and R0 is used when τ is an even integer. R1 changes every local neighborhood in the past (the Sodium sub-array, which is the past when time is odd) where what the change does is dependent only on the local neighborhood in the present (the Chloride sub-array, odd subscripts). In the following example S1 refers a spatial pattern of cells in the Chloride sub-array that will, by their state, control what happens to a neighborhood of cells in the S0 neighborhood in order to transform the state of those cells into S2. Of course, the result for some neighborhoods might be that S2= S0.
R1(S0, S1)Þ( S2, S1)
R0 then changes every local neighborhood in the past (now the Chloride sub-array) where what the change does is dependent on only the local neighborhood in the present (the Sodium sub-array, even subscripts).
R0(S2, S1)Þ (S2, S3).
By limiting R to functions that have the property that Rt(Rt(u, v) )Þ(u, v) the system has a strong form of reversibility. There is no doubt that such a limitation does not make it difficult to achieve computational universality or T symmetry. The author has not yet found a simple process, P=2, that handles CPT symmetry while remaining consistent with other demands of a good DM model. It might be an easy task for someone else to do so
While R0(S2, S1)→(S2, S3), R0(S2, S3)→(S2, S1)
In a nutshell we have defined a wonderful form of reversibility, which works well in P=2 or P=6, where the continued application of the same rules in forwards or backward temporal order, will either drive the system forwards or drive the system backwards. (For “rules” think “laws of physics.”)
