Short Presburger Arithmetic Is Hard

We study the computational complexity of short sentences in Presburger arithmetic (SHORT-PA). Here by “short” we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integer coefficients involved in the linear inequalities. We prove that satisfiability of SHORT-PA sentences with m+2 alternating quantifiers is Σ
_m
^P
-complete or Π
_m
^P
-complete, when the first quantifier is ∃ or ∀, respectively. Counting versions and restricted systems are also analyzed.