There is a symmetry set in physics called CPT Symmetry.

I have found a pair of systems that appears to have a symmetry set with similar properties. This correlation might be significant. Or it might not be.

As such I would like to get some sort of handle on symmetry with particular reference to iterated systems.

It seems to me that there are three general approaches:

1. Find a way to construct a symmetry set for any given set of rules such that the symmetry sets can be compared for different rule sets.

2. Enumerate the symmetries for different rule sets to discover the frequency of particular symmetry types.

3. (implicitly) Construct all the rule sets that are consistent with a given symmetry set.

4. Something else I haven't thought of.

CPT Symmetry

CPT stands for Charge, Parity and Time. Within (quantum) physics the behaviour of particles may be symmetric if one reverses the charge of the particles (matter becomes anti-matter), reverses the time flow of the interaction or reflects the interaction in the imaginary plane. If all three properties are inverted together - then all particle interactions are entirely symmetric. If any one or two of the properties is inverted then some interactions remain fully symmetric but some break symmetry.

So each of the individual symmetries C, P and T are partial symmetries - but the sum of all three symmetries is a total symmetry.

Note that each symmetry only has two states. Also no account is taken of geometric symmetries such as translation and rotation in space.

Issues

This has introduced the idea of partial symmetries which would seem to make the problem of clearly classifying particular symmetries intractable. With this in mind classifying symmetry sets for given rules seems to be challenging at best.

Some thoughts & illustrations

Z=Z

^{2}+ C (the equation for Mandelbrot and Julia sets) can be taken as the rule set of an iterated system. Obviously there are constraints on the geometric symmetry of this rule set (proximity to the Origin is significant). I don't know how reasonable it is to exclude geometric symmetries from consideration. It seems plausible that two distinct systems could have matching rule symmetry sets but differing geometric symmetry sets.

Ignoring adding and subtracting elements of a rule (set) - we could regard any change to a set of rules as creating some form of symmetry. Thus Z=Z

^{3}+ C might be one rule symmetry - which in this case implies an infinite number of symmetries in this aspect of the equation. Similarly Z=Z

^{2}* C might be another symmetry class. Here it isn't clear to what extent the operator might be constrained - it seems plausible this might also be an infinite set of symmetries if one allows unlimited complexity of operators. There is also the notional symmetry of Time reversal. Fortunately this can be expressed precisely as Z=(Z-C)

^{1/2}. In this case the unit of operation (equation - iteration) is definite and finite and can only occupy two states - forward and backward. This, at least, comes close to the tractable symmetry states seen with CPT symmetry.

On the face of it - any symmetry set that might be constructed for Z=Z

^{2}+ C seems unlikely to closely correspond to CPT symmetry. There doesn't seem to be any well defined limit on the number of symmetries or the way they may be constructed.

Cellular Automata (CA) might be more tractable. A one dimensional cellular automata in which the next state of a given cell depends only on itself and its two immediate neighbours is a simple class of CA. Assume initially that a single cell may be in one of two states. The subsequent state of a cell may be defined by a state table. There are 3

^{2}possible distinct entries in such a state table (cell + two neighbours = 3, each cell can have 2 states). A rule might be considered a single entry in this state table. Perhaps then the other states in the table are symmetries of that rule? In this case there are 8 symmetry states. The notional time reverse symmetry is not quite the same as one of those 8 table states. Many-to-one mappings are possible in this system.

The symmetry system for this CA seems less inclined to run away than the symmetry systems for Z=Z

^{2}+C. If one allowed each cell to have more than two states that might change. Is a tri-state system a symmetry set of the basic rules? or a completely new set of rules? In the latter case this seems much closer to CPT symmetry than Z=Z

^{2}+C - although still clearly different.

In a thread no a great distance from here I was raving about simplicity. Does the symmetry set of a system bear any relation to how simple a system may be considered? Or can the rules of a system be arbitrarily expressed in any language such that the symmetry sets depend on the expression of the rules?

If I have a set of axioms that define a system, then one might imagine that symmetries for those rules (axioms) can be generated by taking each rule in turn and looking at the ways in which that rule can be changed (with some vague constraint that the changed axiom must relate to the original rule). Thus non-euclidean space is a symmetry of Euclidean space through a symmetry of the fifth Pythagorean postulate. This might give us an approach to defining the symmetry set for a set of rules based on the number of distinct rules (this is completely ignoring the issue of how to determine the symmetry for a given rule). But... The language used to express a set of axioms might allow the rules to be expressed with fewer or more axioms. Does this mean that the symmetry sets for a set of rules can vary depending on how the rules are expressed? Is there no underlying consistency independent of language?

Conclusion

I'm all kinds of lost. It seems obvious to me that there is some structure to the symmetries possible for a given set of rules. The symmetry set of the rules of a CA is not arbitrary and is not dependent on the language used to express those rules. As such there is some means of classifying and comparing the symmetry sets for different rule systems. Something akin to Korogorov Complexity may show that my intuition is wrong.

1. Is it possible for any set of rules to specify a unique symmetry set that is unambiguous (does not depend on the language used to express those rules)?

[Semi-answer to self - I probably need to define a set of axioms to provide a solid foundation for the meaning of 'symmetry'... from there it should be possible to define a unique symmetry set for any given set of rules... I am then left with an issue of deciding whether CPT symmetry falls within my axioms - could be interesting].

Umm - well - looks like expressing my problem has made it clearer in my mind. Who would've thunk?

I'd still much appreciate any comments - At the moment I've little idea of how to choose a good set of axioms for this...