Recent trends in descriptional complexity of formal languages

Formal languages can be described by several means. A basic question is how succinctly can a descriptional system represent a formal language in comparison with other descriptional systems? What is the maximal size trade-o when changing from one system to another, and can it be achieved? Here, we select some recent trends in the descriptional complexity of formal languages and discuss the problems, results, and open questions. In particular, we present the main historical development and address the basics concepts of descriptional complexity from a general abstract perspective. Then we consider the representation by two-way finite automata, multi-head finite automata, and limited automata in more detail. Finally, we discuss a few further topics in note form. The results presented are not proved but we merely draw attention to the overall picture and some of the main ideas involved.