Abstract
The Caucal hierarchy contains graphs that can be obtained from finite directed graphs by alternately applying the unfolding operation and inverse rational mappings. The goal of this work is to check whether the hierarchy is closed under interpretations in logics extending the monadic second-order logic by the unbounding quantifier, U (saying that a subformula holds for arbitrarily large finite sets). We prove that by applying interpretations described in the MSO+Ufin logic (hence also in its fragment WMSO+U) to graphs of the Caucal hierarchy we can only obtain graphs on the same level of the hierarchy. Conversely, interpretations described in the more powerful MSO+U logic can give us graphs with an undecidable MSO theory, hence outside of the Caucal hierarchy.
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