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Generic Complexity of Presburger Arithmetic

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Abstract

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).

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Authors and Affiliations

  1. Omsk Branch of the Institute of Mathematics of SB RAS, Pevtsova, 13, Omsk, 644099, Russia

    Alexander N. Rybalov

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  1. Alexander N. Rybalov
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Correspondence to Alexander N. Rybalov.

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Supported by the Russian Foundation for Basic Research, grants 07-01-00392 and 08-01-00067.

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Rybalov, A.N. Generic Complexity of Presburger Arithmetic. Theory Comput Syst 46, 2–8 (2010). https://doi.org/10.1007/s00224-008-9120-3

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  • DOI: https://doi.org/10.1007/s00224-008-9120-3

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