Walking automata in the free inverse monoid

In this paper, we study languages of birooted trees or, following Scheiblich-Munn’s theorem, subsets of free inverse monoids. Extending the classical notion of rational languages with a projection operator that maps every set of birooted trees to the subset of its idempotent elements it is first shown that the hierarchy induced by the nesting depth of that projection operator simply correspond the hierarchy induced by the number of (invisible) pebbles used in tree walking automata extended to birooted trees (with complete run semantics). Then, analyzing further the behavior of these walking automata by allowing partial accepting runs runs that are no longer required to traverse the complete input structure it is also shown that finite boolean combinations of languages recognizable by finite state walking automata (with partial run semantics) are equivalent to languages recognizable by means of (some computable notion of) premorphisms from free inverse monoids into finite partially ordered monoids. The various classes of definable languages that are considered in this paper are compared with the class of languages definable in Monadic Second Order (MSO) logic : a typical yardstick of expressive power.