We show that the big-O problem for max-plus automata is decidable and PSPACE-complete. The big-O (or affine domination) problem asks whether, given two max-plus automata computing functions f and g, there exists a constant c such that f < cg+ c. This is a relaxation of the containment problem asking whether f < g, which is undecidable. Our decidability result uses Simon's forest factorisation theorem, and relies on detecting specific elements, that we call witnesses, in a finite semigroup closed under two special operations: stabilisation and flattening.

翻译：我们证明了最大加法自动机的大O问题是可判定的，且为PSPACE-完全问题。大O（或仿射支配）问题询问：给定两个分别计算函数f和g的最大加法自动机，是否存在常数c使得f < cg + c。这是包含性问题的松弛形式——该包含问题询问f < g是否成立，而它本身是不可判定的。我们的可判定性结果利用了西蒙森林分解定理，并依赖于在一个有限半群中检测特定元素（我们称之为见证者），该半群在两种特殊运算（稳定化和扁平化）下封闭。