We introduce a model of register automata over infinite trees with extrema constraints. Such an automaton can store elements of a linearly ordered domain in its registers, and can compare those values to the suprema and infima of register values in subtrees. We show that the emptiness problem for these automata is decidable. As an application, we prove decidability of the countable satisfiability problem for two-variable logic in the presence of a tree order, a linear order, and arbitrary atoms that are MSO definable from the tree order. As a consequence, the satisfiability problem for two-variable logic with arbitrary predicates, two of them interpreted by linear orders, is decidable.

中文翻译：

---

用极值约束注册自动机，并将其应用于二变量逻辑

我们引入了具有极值约束的无限树上的寄存器自动机模型。这样的自动机可以将线性排序域的元素存储在其寄存器中，并且可以将这些值与子树中寄存器值的上限值和下限值进行比较。我们证明这些自动机的空性问题是可以判定的。作为应用，我们证明了在存在树顺序，线性顺序和可由树顺序定义的MSO的任意原子的情况下，二变量逻辑的可数可满足性问题的可判定性。结果，可以确定具有任意谓词的两个变量逻辑的可满足性问题，其中两个谓词由线性顺序解释。