On the power of ordering in linear arithmetic theories

Abstract

We study the problems of deciding whether a relation definable by a first-order formula in linear rational or linear integer arithmetic with an order relation is definable in absence of the order relation. Over the integers, this problem was shown decidable by Choffrut and Frigeri [Discret. Math. Theor. C., 12(1), pp. 21 - 38, 2010], albeit with non-elementary time complexity. Our contribution is to establish a full geometric characterisation of those sets definable without order which in turn enables us to prove coNP-completeness of this problem over the rationals and to establish an elementary upper bound over the integers. We also provide a complementary Π₂^P lower bound for the integer case that holds even in a fixed dimension. This lower bound is obtained by showing that universality for ultimately periodic sets, i.e., semilinear sets in dimension one, is Π₂^P-hard, which resolves an open problem of Huynh [Elektron. Inf.verarb. Kybern., 18(6), pp. 291 - 338, 1982].