Developments in Higher-Dimensional Automata Theory

We develop the language theory of higher-dimensional automata (HDAs). We show a pumping lemma which allows us to expose a class of non-regular ipomset languages. We also give an example of a regular language with unbounded ambiguity. Then we pass to decision and closure properties of regular languages. We show that inclusion of regular languages is decidable (hence is emptiness), and that intersections of regular languages are again regular. On the other hand, no universal finite HDA exists, so complements of regular languages are not regular. We introduce a width-bounded complement and show that width-bounded complements of regular languages are again regular.