A Combinatorial Approach to the Theory of omega-Automata

A combinatorial lemma is proved and used here to derive new results on ω -automata and to give simpler proofs of known ones. In particular, we reprove McNaughton's fundamental theorem (characterizing the ω -regular sequence-sets), without having to construct a sophisticated ω -automaton. The theorem is obtained by coding the behaviour of automata in a second-order language and a simple application of the lemma. In close analogy (now referring to a first-order language) a theory of counter-free ω-automata is developed; it is shown that these automata are appropriate for characterizing the ω -star-free sequence-sets. Finally, the lemma is applied in mathematical logic: Here new normal form theorems and also decidability results are proved for the first-order and the monadic second-order theory of certain structures over the ordering of natural numbers.