2 research outputs found
Proving that a Tree Language is not First-Order Definable
We explore from an algebraic viewpoint the properties of the tree languages
definable with a first-order formula involving the ancestor predicate, using
the description of these languages as those recognized by iterated block
products of forest algebras defined from finite counter monoids. Proofs of
nondefinability are infinite sequences of sets of forests, one for each level
of the hierarchy of quantification levels that defines the corresponding
variety of languages. The forests at a given level are built recursively by
inserting forests from previous level at the ports of a suitable set of
multicontexts. We show that a recursive proof exists for the syntactic algebra
of every non-definable language. We also investigate certain types of uniform
recursive proofs. For this purpose, we define from a forest algebra an algebra
of mappings and an extended algebra, which we also use to redefine the notion
of aperiodicity in a way that generalizes the existing ones