On Expressive Power of Regular Expressions over Infinite Orders
Two fundamental results of classical automata theory are the Kleene theorem and the Büchi-Elgot-Trakhtenbrot theorem. Kleene’s theorem states that a language of finite words is definable by a regular expression iff it is accepted by a finite state automaton. Büchi-Elgot-Trakhtenbrot’s theorem states that a language of finite words is accepted by a finite-state automaton iff it is definable in the weak monadic second-order logic. Hence, the weak monadic logic and regular expressions are expressively equivalent over finite words. We generalize this to words over arbitrary linear orders.
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