The monadic theory of finite representations of infinite words

In this paper, we consider existential monadic second-order logic (monadic Σ1) on finite unary graphs, that is finite graphs with functional edge relation. These can be seen as finite encodings of ultimately periodic words. We show that, on these graphs, the bisimulation-invariant fragment of monadic Σ1 equals the bisimulation-invariant fragment of monadic second-order logic itself, and that MSO-definable bisimulation closed classes of graphs coincide with classes of graphs definable by means of (an extension of) finite state ω-word automata. This result can be seen as a translation, onto finite representations of infinite words, of Büchi's automata-theoretic characterization of S1S. In terms of descriptive complexity, this result contrasts with the situation on arbitrary unary structures where bisimulation invariant monadic Σ1 properties only define languages that are closed with respect to the prefix topology.