Complexity of Two-Dimensional Patterns

In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a {\it language} or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy, with which we can measure the complexity of these sets of sequences, and thus the complexity of the systems which produce them. In this paper, we look at the first few levels of a hierarchy of complexity for two-or-more-dimensional patterns. We show that several definitions of ``regular language'' or ``local rule'' that are equivalent in $d=1$ lead to distinct classes in $d \geq 2$. We explore the closure properties and computational complexity of these classes, including certain undecidability and {\bf NP}-completeness results. We apply these classes to cellular automata, in particular to their sets of fixed and periodic points, finite-time images, and limit sets. We show that it is undecidable whether a CA in $d \geq 2$ has a periodic point of a given period, and that certain ``local lattice languages'' are not finite-time images or limit sets of any CA. We also show that the entropy of a $d$-dimensional CA's finite-time image cannot decrease faster than $t^{-d}$ unless it maps every initial condition to a single homogeneous state.