It is shown that for any fixed i>0, the Σi+1-fragment of
Presburger arithmetic, i.e., its restriction to i+1 quantifier alternations
beginning with an existential quantifier, is complete for
ΣiEXP, the i-th level of the weak EXP
hierarchy, an analogue to the polynomial-time hierarchy residing between
NEXP and EXPSPACE. 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.
Moreover, we apply some of the techniques developed in the proof of the lower
bound in order to establish bounds on sets of naturals definable in the
Σ1-fragment of Presburger arithmetic: given a Σ1-formula
Φ(x), it is shown that the set of non-negative solutions is an ultimately
periodic set whose period is at most doubly-exponential and that this bound is
tight.Comment: 10 pages, 2 figure