Short Presburger arithmetic is hard
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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_P^m-complete or Π_P^m-complete, when the first quantifier is ∃ or ∀, respectively. Counting versions and restricted systems are also analyzed. Further application are given to hardness of two natural problems in Integer Optimizations.
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