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.

翻译：解释是数理逻辑中的基本工具，允许通过逻辑定义将结构编码到其他结构中。我们研究MSO多维点解释，其中被解释结构的元素由宿主结构的元素元组表示，并探讨通过降低表示维度来简化此类解释的问题。为形式化简化，我们使用“重参数化”概念。主要结果表明：在可数标号线性序类上，给定MSO公式是否允许d维重参数化是可判定的。由此，每个其定义域允许此类重参数化的解释都等价于一个d维点解释。