On Presburger arithmetic extended with non-unary counting quantifiers

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