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We explore from an algebraic viewpoint the properties of the tree languagesdefinable with a first-order formula involving the ancestor predicate, usingthe description of these languages as those recognized by iterated blockproducts of forest algebras defined from finite counter monoids. Proofs ofnondefinability are infinite sequences of sets of forests, one for each levelof the hierarchy of quantification levels that defines the correspondingvariety of languages. The forests at a given level are built recursively byinserting forests from previous level at the ports of a suitable set ofmulticontexts. We show that a recursive proof exists for the syntactic algebraof every non-definable language. We also investigate certain types of uniformrecursive proofs. For this purpose, we define from a forest algebra an algebraof mappings and an extended algebra, which we also use to redefine the notionof aperiodicity in a way that generalizes the existing ones.

Proving that a Tree Language is not First-Order Definable