A Class of Models with the Potential to Represent Fundamental Physics
  1. Introduction
  2. Basic Form of Models
  3. Typical Behaviors
  4. Limiting Behavior and Emergent Geometry
  5. The Updating Process for String Substitution Systems
  6. The Updating Process in Our Models
  7. Equivalence and Computation in Our Models
  8. Potential Relation to Physics
  9. Additional Material
  10. References
  11. Index
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technical introduction Typical Behaviors

3.18 Random Rules and Overall Classification of Behavior

Here are samples of random rules with various signatures (only connected results are included):

As expected from the Principle of Computational Equivalence [1:12], above a low threshold more complex rules do not generally lead to more complex behavior, although the frequencies of different kinds of behavior do change somewhat.

At a basic visual level, one can identify several general types of behavior:

  • Line-like: elements are connected primarily in sequences (lines, circles, etc.)
  • Radial: most elements are connected to just a few core elements
  • Tree-like: elements repeatedly form independent branches
  • Globular: more complex, closed structures

Inevitably, these types of behavior are neither mutually exclusive, nor precisely defined. There are certainly specific graph-theoretic and other methods that could be used to discriminate different types, but there will always be ambiguous cases (and sometimes it will even be formally undecidable what category something is in). But just like for cellular automata—or for many systems in the empirical sciences—classifications can still be useful in practice even if their definitions are not unique or precise.

As an alternative to categorical classification, one can also consider systematically arranging behaviors in a continuous feature space (e.g. [13]). The results inevitably depend on how features are extracted. Here is what happens if one takes images like the ones above, and directly applies a feature extractor trained on images of picturable nouns in human language [14]:

Here is the fairly similar result based on feature extraction of underlying adjacency matrices:

In addition to characterizing the behavior of individual rules, one can also ask to what extent behavior is clustered in rule space. Here are samples of what happens if one starts from particular 22 72 rules, then looks at a collection of “nearby” rules that differ by one element in one relation:

And what we see is that even though there are only 68 million or so 22 72 rules, changing one element (out of 14) still usually gives a rule whose overall behavior is similar.

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