Abstract
Lower bounds are established on the computational complexity of the decision problem and on the inherent lengths of proofs for two classical decidable theories of logic: the first-order theory of the real numbers under addition, and Presburger arithmetic — the first-order theory of addition on the natural numbers. There is a fixed constant c > 0 such that for every (nondeterministic) decision procedure for determining the truth of sentences of real addition and for all sufficiently large n, there is a sentence of length n for which the decision procedure runs for more than 2cn steps. In the case of Presburger arithmetic, the corresponding bound is \({2^{{2^{cn}}}}\). These bounds apply also to the minimal lengths of proofs for any complete axiomatization in which the axioms are easily recognized.
Reprinted from SIAM-AMS Proceedings, Volume VII, 1974, pp. 27–41, by permission of the American Mathematical Society.
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© 1998 Springer-Verlag/Wien
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Fischer, M.J., Rabin, M.O. (1998). Super-Exponential Complexity of Presburger Arithmetic. In: Caviness, B.F., Johnson, J.R. (eds) Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-9459-1_5
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DOI: https://doi.org/10.1007/978-3-7091-9459-1_5
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