  Cellular automata are fully-discrete, spatially-extended dynamical systems
that evolve by simultaneously applying a local update function. Despite their
simplicity, the induced global dynamic produces a stunning array of
richly-structured, complex behaviors. These behaviors present a challenge to
traditional closed-form analytic methods. In certain cases, specifically when
the local update is additive, powerful techniques may be brought to bear,
including characteristic polynomials, the ergodic theorem with Fourier
analysis, and endomorphisms of compact Abelian groups. For general dynamics,
though, where such analytics generically do not apply, behavior-driven analysis
shows great promise in directly monitoring the emergence of structure and
complexity in cellular automata. Here we detail a surprising connection between
generalized symmetries in the spacetime fields of configuration orbits as
revealed by the behavior-driven local causal states, invariant sets of spatial
configurations, and additive subdynamics which allow for closed-form analytic
methods.
