Register Automata with Extrema Constraints, and an Application to Two-Variable Logic

I will present joint work with Thomas Zeume.
In this work, we study 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.