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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^ (\#) ) and shallow second-order graded modal logic with counting with respect to image-finite frames (SOGML^s, (#) ). We show that in their respective existential fragments, the 1-free fragment of PbA, the =-free fragment of MSO^ (\#) and the graded modality-free fragment of SOGML^s, (#) 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.
Fu et al. (2024) studied this question.