Undecidability of MSO+"ultimately periodic"
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We prove that MSO on ω-words becomes undecidable if allowing to quantify over sets of positions that are ultimately periodic, i.e, sets X such that for some positive integer p, ultimately either both or none of positions x and x+p belong to X. We obtain it as a corollary of the undecidability of MSO on ω-words extended with the second-order predicate U_1[X] which says that the distance between consecutive positions in a set X ⊆N is unbounded. This is achieved by showing that adding U_1 to MSO gives a logic with the same expressive power as MSO+U, a logic on ω-words with undecidable satisfiability.
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