On Presburger arithmetic extended with non-unary counting quantifiers

We consider a first-order logic for the integers with addition. This extends classical by modulo-counting, threshold-counting and exact-counting quantifiers, all applied to tuples of variables (here, residues are given as terms while moduli thresholds explicitly). Our main result shows that satisfaction this is decidable in two-fold exponential space. If only threshold- quantifiers allowed, we prove an upper bound alternating time linearly many alternations. latter almost matches Berman's exact complexity without counting quantifiers. To obtain these results, first translate into polynomial (which already proves second result). handle remaining modulo-counting tuples, reduce them doubly single elements. For provide quantifier elimination procedure similar Reddy Loveland's analyse growth coefficients, constants, appearing process. The bounds obtained way allow restrict quantification original formula bounded size which then implies mentioned above. incomparable considered Chistikov et al. 2022. They more general operations but unary move from non-unary non-trivial, since, e.g., version H\"artig results undecidable theory.