Regular Languages: To Finite Automata and Beyond Succinct Descriptions and Optimal Simulations

It is well known that regular (or Type 3) languages are equivalent to finite automata. Nevertheless, many other characterizations of this class of lan- guages in terms of computational devices and generative models are present in the literature. For example, by suitably restricting more general models such as context-free grammars, pushdown automata, and Turing machines, that characterize wider classes of languages, it is possible to obtain formal models that generate or recognize regular languages only. These restricted formalisms provide alternative representations of Type 3 languages that may be significantly more concise than other models that share the same express- ing power. The goal of this work is to provide an overview of old and recent re- sults on these formal systems from a descriptional complexity perspective, that is the study of the relationships between the sizes of such devices. We also present some results related to the investigation of the famous ques- tion posed by Sakoda and Sipser in 1978, concerning the size blowups from nondeterministic finite automata to two-way deterministic finite automata.