Generic Complexity of Presburger Arithmetic

Fischer and Rabin proved in (Proceedings of the SIAM-AMS Symposium in Applied Mathematics, vol. 7, pp. 27–41, 1974) that the decision problem for Presburger Arithmetic has at least double exponential worst-case complexity (for deterministic and for nondeterministic Turing machines). In Kapovich et al. (J. Algebra 264(2):665–694, 2003) a theory of generic-case complexity was developed, where algorithmic problems are studied on “most” inputs instead of set of all inputs. A question rises about existing of more efficient (say, polynomial) generic algorithm deciding Presburger Arithmetic on a set of closed formulas of asymptotic density 1. We prove in this paper that there is not an exponential generic decision algorithm working correctly on an input set of asymptotic density exponentially converging to 1 (so-called strongly generic sets).