Narrow Results

We consider the state complexity of several combined operations. Those results show that the state complexity of a combined operation is in general very different from the composition of the state complexities of the participating individual operations. We also consider general estimation methods for the state complexity of combined operations. In particular, estimation through nondeterministic state complexity is studied. It is shown that the method is very promising for a large class of combined operations.

In this paper, we study the state complexities of two particular combinations of operations: catenation combined with union and catenation combined with intersection. We show that the state complexity of the former combined operation is considerably less than the mathematical composition of the state complexities of catenation and union, while the state complexity of the latter one is equal to the mathematical composition of the state complexities of catenation and intersection.

We discuss a number of essential questions concerning the state complexity research. The questions include why many basic problems were not studied earlier, whether there is a general algorithm for state complexity of combined operations, and whether there is a new and effective approach in this area of research. The concept of state complexity approximation is also discussed. We show that state complexity approximation can be used to obtain good results when the exact state complexities are difficult to find and when the exact state complexities are too complex to comprehend. We also list a number of questions for future research in this area.