Better-Quasi-Orders: Extensions and Abstractions

We generalise the notion of �-scattered to partial orders and prove that some large classes of �-scattered partial orders are better-quasi-ordered under embeddability. This generalises theorems of Laver, Corominas and Thomass�e regarding �-scattered linear orders, �-scattered trees, countable pseudo-trees and N-free partial orders. In particular, a class of countable partial orders is better-quasi-ordered whenever the class of indecomposable subsets of its members satis�es a natural strengthening of better-quasi-order. We prove that some natural classes of structured �-scattered pseudo-trees are betterquasi- ordered, strengthening similar results of K�r���z, Corominas and Laver. We then use this theorem to prove that some large classes of graphs are better-quasi-ordered under the induced subgraph relation, thus generalising results of Damaschke and Thomass�e. We investigate abstract better-quasi-orders by modifying the normal de�nition of better-quasi-order to use an alternative Ramsey space rather than exclusively the Ellentuck space as is usual. We classify the possible notions of well-quasi-order that can arise by generalising in this way, before proving that the corresponding notion of better-quasi-order is closed under taking iterated power sets, as happens in the usual case. We consider Shelah's notion of better-quasi-orders for uncountable cardinals, and prove that the corresponding modi�cation of his de�nition using fronts instead of barriers is equivalent. This gives rise to a natural version of Simpson's de�nition of better-quasiorder for uncountable cardinals, even in the absence of any Ramsey-theoretic results. We give a classi�cation of the fronts on [�]!, providing a description of how far away a front is from being a barrier

Logical definability and query languages over ranked and unranked trees

We study relations on trees defined by first-order constraints over a vocabulary that includes the tree extension relation T ≺ T ′ , holding if and only if every branch of T extends to a branch of T ′, unary node-tests, and a binary relation checking if the domains of two trees are equal. We consider both ranked and unranked trees. These are trees with and without a restriction on the number of children of nodes. We adopt the model-theoretic approach to tree relations and study relations definable over the structure consisting of the set of all trees and the above predicates. We relate definability of sets and relations of trees to computability by tree automata. We show that some natural restrictions correspond to familiar logics in the more classical setting, where every tree is a structure over a fixed vocabulary, and to logics studied in the context of XML pattern languages. We then look at relational calculi over collections of trees, and obtain quantifier-restriction results that give us bounds on the expressive power and complexity. As unrestricted relational calculi can express problems complete for each level of the polynomial hierarchy, we look at their restrictions, corresponding to the restricted logics over the family of all unranked trees, and find several calculi with low (NC 1) data complexity, while still expressing properties important for database an