Following a recently considered generalization of linear equations to unordered data vectors, we perform a further generalization to ordered data vectors. These generalized equations naturally appear in the analysis of vector addition systems (or Petri nets) extended with ordered data. We show that nonnegative-integer solvability of linear equations is computationally equivalent (up to an exponential blowup) to the reachability problem for (plain) vector addition systems. This high complexity

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... surprising, and contrasts with NP-completeness for unordered data vectors. This also contrasts with our second result, namely polynomial time complexity of the solvability problem when the nonnegative-integer restriction on solutions is relaxed.