On the Courcelle Conjecture

We prove a conjecture of Courcelle, which states that a graph property is definable in mso with modular counting predicates on graphs of constant treewidth if, and only if it is recognizable in the following sense: constant-width tree decompositions of graphs satisfying the property can be recognized by tree automata.
While the forward implication is a classic fact known as Courcelle's theorem, the converse direction remained open.