Complexity of Sets of Two-Dimensional Patterns

We study the two-dimensional pattern matching implemented using the two-dimensional on-line tessellation automaton, which is a restricted type of the cellular automaton able to simulate the Baker-Bird algorithm, proposed as the first algorithm for the two-dimensional pattern matching. We further explore capabilities of this automaton to carry out the matching task against an arbitrary set of equally sized patterns. To measure amount of resources needed to accomplish it, we introduce the pattern complexity of a picture language. We show that this complexity spreads in a wide range. It is demonstrated by giving examples, deriving general techniques and proving some lower bounds.