On the descriptional and algorithmic complexity of regular languages

Gedruckt auf alterungsbeständigem Papier nach ISO 9706 (säure-, holz-und chlorfrei). Digital ist besser Tocotronic vi Preface This thesis is a research monograph. As such, it targets at an audience of experts, primarily in the fields of foundations of computer science and discrete mathematics. Nevertheless, already several persons showed interest in understanding at least the main theme of this thesis, but have only little background in mathematics and computer science. This preface can not serve a bluffer's guide to the main body of work, but we try at best to explain at least the central abstractions at an informal level. The present work deals with mathematical models of computational processes. Several such models exist, each with its own advantages and characteristics. We will concentrate on the simplest of these models, namely on finite automata. An example drawn from everyday life naturally modeled as finite automaton is a vending machine. The observable behavior of a such a vending machine is described as sequence of atomic events. For simplicity, let us assume the possible events are: a coin worth one or two units of money is inserted, in symbols ‚ or ƒ, a cup of coffee ∪ is requested and brewed, and alternatively the cancel button © can be hit. Thereafter some coins might be returned, actions for which we introduce the symbols x or y. Now the behavior of a correctly operating vending machine is described as a set of sequences made up from these symbols. We expect that most users of the machine will be content with the sequence ‚ ƒ x ∪ ‚ ‚ © y ƒ ∪ , except possibly for the price of two money units for a cup of coffee. In contrast, the sequence ƒ ∪ ‚ ‚ is certainly not expected to be observed on a machine that operates correctly. We thus model the observable behavior of a vending machine as a set of sequences over some finite alphabet—in our example, this set will contain infinitely many correct sequences, but it will not include all possible sequences. The internal behavior can be modeled as a control unit, which is at each point in time in one of finitely many possible internal states. At this point, we mention that our vending machine always immediately returns overpaid amounts. In this way, we can retain a finite number of states. Otherwise it would be possible to insert arbitrarily large …