Characterization of Properties and Relations Deened in Monadic Second Order Logic on the Nodes of Trees

A formula from monadic second order (mso) logic with one free variable can be used to deene a property of the nodes of a tree. Similarly, an mso formula with two free variables can be used to deene a binary relation between the nodes of a tree. It is proved that a node relation is mso deenable ii it can be computed by a nite-state tree-walking automaton, provided the automaton can test mso deenable properties of the nodes of the tree; if the relation is a function, the automaton is deterministic. It is also proved that a node property is mso deenable ii it can be computed by an attribute grammar of which all attributes have nitely many values. mso deenable node properties are computable in linear time, mso deenable node relations in quadratic time, and mso deenable node functions in linear time.