Inexpressibility Results for Regular Languages in Nonregular Settings

My ostensible purpose in this talk is to describe some new results (found in collaboration with Amitabha Roy) on expressibility of regular languages in certain generalizations of first-order logic. [10]. This provides me with a good excuse for describing some the work on the algebraic theory of regular languages in what one might call “nonregular settings”. The syntactic monoid and syntactic morphism of a regular language provide a highly effective tool for proving that a given regular language is not expressible or recognizable in certain compuational models, as long as the model is guaranteed to produce only regular languages. This includes finite automata, of course. but also formulas of propositional temporal logic, and first-order logic, provided one is careful to restrict the expressive power of such logics. (For example, by only allowing the order relation in first-order formulas.) Things become much harder, and quite a bit more interesting, when we drop this kind of restriction on the model. The questions that arise are important (particularly in computational complexity), and most of them are unsolved. They all point to a rich theory that extends the reach of algebraic methods beyond the domain of finite automata