We define weighted tree-walking automata. We show that the class of tree series recognizable by weighted tree-walking automata over a commutative semiring K is a subclass of the class of regular tree series over K. If K is not a ring, then the inclusion is strict.

We provide a hierarchy of tree languages recognised by nondeterministic parity tree automata with priorities in {0, 1, 2}, whose length exceeds the first fixed point of the ε operation (that itself enumerates the fixed points of x 7→ ω). We conjecture that, up to Wadge equivalence, it exhibits all regular tree languages of index [0, 2].

A tree controlled grammar is specified as a pair (G,G) where G is a context-free grammar and G is a regular grammar. Its language consists of all terminal words with a derivation in G such that all levels of the corresponding derivation tree – except the last level – belong to L(G). We define the nonterminal complexity Var(H) of H = (G,G) as the sum of the numbers of nonterminals of G and G. In...

We present two enhancements to Jiang’s tree alignment algorithm, motivated by experience with its use for RNA structure alignment. One enhancement is the introduction of an affine gap model, which can be accommodated with a runtime increase by a constant factor. The second enhancement is a speed-up of the alignment algorithm when certain nodes in the trees are pre-aligned by a socalled anchorin...

We propose a new algebraic framework to discuss and classify recognizable tree languages, and to characterize interesting classes of such languages. Our algebraic tool, called preclones, encompasses the classical notion of syntactic Σ-algebra or minimal tree automaton, but adds new expressivity to it. The main result in this paper is a variety theorem à la Eilenberg, but we also discuss importa...

Every rational number p q defines a rational base numeration system in which every integer has a unique finite representation, up to leading zeroes. This work is a contribution to the study of the set of the representations of integers. This prefix-closed subset of the free monoid is naturally represented as a highly non-regular tree. Its nodes are the integers, its edges bear labels taken in {...

We consider varieties of tree languages which are not restricted to a fixed ranked alphabet, varieties of finite algebras that contain algebras of all finite types, and a matching notion of varieties of congruences. A variety theorem that yields isomorphisms between the lattices formed by these three types of varieties is proved. To achieve this, some basic universal algebra is suitably general...

We introduce probabilistic regular tree expressions and give a Kleene-like theorem for probabilistic tree automata (PTA). Furthermore, we define probabilistic MSO logic. This logic is more expressive than PTA. We define bottom-up PTA, which are strictly more expressive than PTA. Using bottom-up PTA, we prove a Büchi-like theorem for probabilistic MSO logic. We obtain a Nivat-style theorem as an...

When rewriting a regular language with a left-linear term rewriting system, if the set of reachable terms is regular, we show that equational tree automata completion can compute it. This was known to be true for some known TRS classes preserving regularity, but was still an open question in the general case. The proof is not constructive: it assumes that the set of reachable terms is regular, ...

W e present a decision procedure, based on tree automata techniques, for satisfiability of systems of set constraints including negated subset relationships. This result extends all previous works on set constraints solving (Heintze and Jaffar [HJ$Oa] ; Aiken and Wimmers [AW$2] ; Gilleron, Tison and Tommasi [GTT93a] ; Bachmair, Ganzinger and Waldmann [BG W$3]) and solves a problem which was lef...