Numerical expressive power of logical languages with cardinality comparison

Abstract

In this paper, we investigate the numerical expressive power of various logical languages, encompassing fragments of Presburger Arithmetic (PbA), monadic second-order logic with counting with respect to finite domains (MSO$^{\phi }(\#)$) and shallow second-order graded modal logic with counting with respect to image-finite frames (SOGML$^{\textsf{s},\phi }$(#)). We show that in their respective existential fragments, the $1$-free fragment of PbA, the =-free fragment of MSO$^{\phi }(\#)$ and the graded modality-free fragment of SOGML$^{\textsf{s},\phi }$(#) possess equivalent numerical expressive power, specifically defining strongly semilinear sets. When adding universal quantifiers or adding $1$, = and graded modality to these three languages, the resulting definable sets become semilinear sets.