Chapter 2 Languages , Decidability , and Complexity

Control problems for discrete-event systems or hybrid systems typically involve manipulation of languages that describe system behaviors. This chapter introduces basic automata and grammar models for generating and analyzing languages of the Chomsky hierarchy, as well as their associated decision problems, which are necessary for the understanding of other parts of this book. Notions of decidability of a problem (that is, is there an algorithm solving the given problem?) and of computational complexity (that is, how many computation steps are necessary to solve the given problem?) are introduced. The basic complexity classes are recalled. This chapter is not intended to replace a course on these topics but merely to provide basic notions that are used further in this book, and to provide references to the literature. In the following, we introduce the basic terminology, notation, definitions, and results concerning the Chomsky hierarchy of formal languages to present the necessary prerequisites for understanding the topic of this chapter, that is, the devices recognizing and generating languages. As this is only an introductory material, not all details and proofs are presented here. Usually, only the basic ideas or sketches of proofs are presented. Further details can be found in the literature, see, e.g., [4, 5, 10, 12, 13, 15].