A combinatorial theorem for trees applications to monadic logic and infinite structures

Abstract

Following the idea developed by I. Simon in his theorem of Ramseyan factorisation forests, we develop a result of 'deterministic factorisations'. This extra determinism property makes it usable on trees (finite or infinite). We apply our result for proving that, over trees, every monadic interpretation is equivalent to the composition of a first-order interpretation (with access to the ancestor relation) and a monadic marking. Using this remark, we give new characterisations for prefix-recognisable structures and for the Caucal hierarchy. Furthermore, we believe that this approach has other potential applications. © Springer-Verlag Berlin Heidelberg 2007.