group action within itself. The reason is simple but profound: In both cases, the hierarchy represents a hierarchy of workspaces. Furthermore, in both cases, the workspace on any level has the same structure as the workspace on any other level. This means that a workspace moves an identical workspace about itself.

14 Recoverability

According to our theory, people call a structure an art-work, if it is maximally complex, but allows the maximal conversion of the complexity into understandability. Our theory says that this conversion is achieved by maximizing transfer and recoverability. We have begun to examine the algebraic structure of transfer. It is now necessary to bring in the factor of recoverability. By recovery, we mean the following problem:

Our first book Leyton (1992) was a 600-page analysis of this problem, and one of the main conclusions of this analysis was the following:

Asymmetry Princple. The only recoverable operations are symmetry-breaking ones. That is, a generative program is recoverable only if it is symmetry-breaking on each of the successively generated states.

Now it is clear that there are many processes in the world that a not symmetry-breaking, but are symmetry-increasing; e.g., a tank of gas settling to equilibrium under the standard entropy-increasing process. Our theory says this:

Symmetry-Increasing Processes. A symmetry-increasing process is recoverable only if it is symmetry-decreasing on successive data sets.

So, for example, you can recover the fact that the tank of gas was entropy-increasing over time, if you kept a set of records (e.g., photographs) and the records are linearly ordered, e.g., they are laid out from left to right on a table, in which case the sequence of photographs breaks the left-right symmetry of the table. In other words, the increase in spatial symmetry in the tank of gas corresponded to a decrease in spatial symmetry of the record structure.

15 Theory of Symmetry-Breaking

A basic factor emerges from the above discussion: In order to ensure recoverability, the control group must be symmetry-breaking on its fiber. Thus, the following should be observed: The transfer component of our theory leads to wreath products, and the recoverability component adds the construct that the wreath products are symmetry-breaking. Close examination reveals that this gives a far more powerful theory of symmetry-breaking than the conventional one that underlies physics and chemistry.