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It is shown that for any fixed $i>0$, the $\Sigma_{i+1}$-fragment ofPresburger arithmetic, i.e., its restriction to $i+1$ quantifier alternationsbeginning with an existential quantifier, is complete for$\mathsf{\Sigma}^{\mathsf{EXP}}_{i}$, the $i$-th level of the weak EXPhierarchy, an analogue to the polynomial-time hierarchy residing between$\mathsf{NEXP}$ and $\mathsf{EXPSPACE}$. This result completes thecomputational complexity landscape for Presburger arithmetic, a line ofresearch which dates back to the seminal work by Fischer & Rabin in 1974.Moreover, we apply some of the techniques developed in the proof of the lowerbound in order to establish bounds on sets of naturals definable in the$\Sigma_1$-fragment of Presburger arithmetic: given a $\Sigma_1$-formula$\Phi(x)$, it is shown that the set of non-negative solutions is an ultimatelyperiodic set whose period is at most doubly-exponential and that this bound istight.

Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy