On Presburger arithmetic extended with non-unary counting quantifiers
Peter Habermehl, Dietrich Kuske
TL;DR
The paper investigates FO over ℤ with addition extended by modulo-counting, threshold-counting, and exact-counting quantifiers on tuples and proves decidability in space 2^{2^{O(n)}} with corresponding alternating-time bounds for unary variants. The authors establish a two-step reduction: first eliminating threshold and exact counting to obtain FO[∃^(t,p) x̄], and then reducing non-unary modulo-counting to unary modulo-counting in doubly exponential time, ultimately enabling a quantifier-elimination approach for unary modulo-counting. A quantified-elimination procedure for FO[∃^(q,p) x] is developed, with explicit bounds on coefficient/constant/modulus growth, and extended to the full non-unary setting via careful reductions. The construction yields a principled pathway to decide the full logic by translating to FO and Presburger-like procedures, clarifying the expressive boundary between unary and non-unary counting quantifiers in arithmetic theories. The results advance understanding of counting quantifiers in Presburger-like settings, providing elementary (2-EXPSPACE/2-EXPTIME) bounds and linking to prior work on Chistikov et al. and Berman’s classical results.
Abstract
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 (here, residues are given as terms while moduli and thresholds are given explicitly). 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 restrict 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 by Chistikov et al. in 2022. 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 Härtig quantifier results in an undecidable theory.