Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy

ABSTRACT. It is shown that for any fixed $i>0$, the $\Sigma_{i+1}$-fragment of Presburger arithmetic, \emph{i.e.}, its restriction to $i+1$ quantifier alternations beginning with an existential quantifier, is complete for $\Sigma^{\text{EXP}}_{i}$, the $i$-th level of the weak EXP hierarchy. This result completes the computational complexity landscape for Presburger arithmetic, a line of research which dates back to the seminal work by Fischer \& Rabin in 1974. The second part of the paper applies some techniques developed in the first part 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 solutions is an ultimately periodic set whose period can be doubly-exponentially bounded from below and above.