Elementary Theory of Ordinals with Addition and Left Translation by ω

Abstract

After Buchi it has become very natural to interprete formulae of certain logical theories as finite automata, i.e., as recognizing devices. This recognition aspect though, was neglected by the inventor of the concept and the study of the families of linear structures that could be accepted in the language theory sense of the term, was carried out by other authors. The most popular field of application of Buchi type automata is nowadays connected with model checking by considering a process as a possibly infinite sequence of events. For over a decade, the original model has been enriched by adding a time parameter in order to model reactive systems and their properties. Originally Buchi was interested in the monadic second order theory with the successor over ω but he later considered the theory of countable ordinals for which he was led to propose new notions of finite automata. Again these constructs can be viewed as recognizing devices ofwords over a finite alphabet whose length are countable ordinals. They were studied by other authors, mainly Choueka and Wojciechowski to who we owe two Theorems “a la Kleene” asserting the equivalence between expressions using suitable rational operators and subsets (languages) of transfinite words, [6] and [13].