Decidability of MSO Reparameterization over Countable Chains
For logicians studying MSO interpretations over linear orders, this provides a decidability result for simplifying interpretations by reducing dimension.
The paper shows that over countable labelled linear orders, it is decidable whether a given MSO formula admits a d-dimensional reparameterization, implying equivalence to a d-dimensional point interpretation.
Interpretations are a fundamental tool in mathematical logic, allowing structures to be encoded within other structures via logical definitions. We study $\MSO$ \emph{multidimensional point interpretations}, where elements of an interpreted structure are represented by tuples of elements of the host structure, and address the problem of simplifying such interpretations by reducing their representation dimension. To formalize simplification, we use the notion of \emph{reparameterization}. Our main result shows that, over the class of countable labelled linear orders, it is decidable whether a given $\MSO$ formula admits a $d$-dimensional reparameterization. As a consequence, every interpretation whose domain admits such a reparameterization is equivalent to a $d$-dimensional point interpretation.