Decidability of MSO Reparameterization over Countable Chains

TL;DR

This paper proves the decidability of reparameterization problems for MSO interpretations over countable linear orders, enabling simplification of complex logical structures.

Contribution

It establishes the decidability of whether an MSO interpretation over countable chains can be simplified to a fixed dimension.

Findings

Decidability of MSO reparameterization over countable chains.

Every interpretable structure with such reparameterization is equivalent to a fixed-dimensional point interpretation.

Simplification of interpretations reduces complexity in logical structure encoding.

Abstract

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.