Pure and O-Substitution

The basic properties of distributivity and deletion of pure and o-substitution are investigated. The obtained results are applied to show preservation of recognizability in a number of surprising cases. It is proved that linear and recognizable tree series are closed under o-substitution provided that the underlying semiring is commutative, continuous, and additively idempotent. It is known that, in general, pure substitution does not preserve recognizability (not even for linear target tree series), but it is shown that recognizable linear probability distributions (represented as tree series) are closed under pure substitution

Composition of Tree Series Transformations

Tree series transformations computed by bottom-up and top-down tree series transducers are called bottom-up and top-down tree series transformations, respectively. (Functional) compositions of such transformations are investigated. It turns out that the class of bottomup tree series transformations over a commutative and complete semiring is closed under left-composition with linear bottom-up tree series transformations and right-composition with boolean deterministic bottom-up tree series transformations. Moreover, it is shown that the class of top-down tree series transformations over a commutative and complete semiring is closed under right-composition with linear, nondeleting top-down tree series transformations. Finally, the composition of a boolean, deterministic, total top-down tree series transformation with a linear top-down tree series transformation is shown to be a top-down tree series transformation

Hierarchies of tree series transformations

AbstractWe study bottom-up and top-down tree series transducers over a semiring A and denote the tree series transformation classes computed by them by BOTt−ts(A) and TOPt−ts(A), respectively. We present the inclusion diagram of the classes p-BOTt−tsn(A), p-TOPt−tsn(A), p-BOTt−tsn+1(A), and p-TOPt−tsn+1(A) and prove its correctness, where A is a commutative izz-semiring (izz=idempotent, zero-divisor free, and zero-sum free) and the prefix p stands for polynomial. This inclusion diagram implies the properness of the following four hierarchies: p-TOPt−ts(A)⊆p-TOPt−ts2(A)⊆p-TOPt−ts3(A)⊆⋯,p-BOTt−ts(A)⊆p-BOTt−ts2(A)⊆p-BOTt−ts3(A)⊆⋯,p-TOPt−ts(A)⊆p-BOTt−ts2(A)⊆p-TOPt−ts3(A)⊆p-BOTt−ts4(A)⊆⋯,p-BOTt−ts(A)⊆p-TOPt−ts2(A)⊆p-BOTt−ts3(A)⊆p-TOPt−ts4(A)⊆⋯,where the first hierarchy generalizes the famous top-down tree transformation hierarchy of Engelfriet (Math. Systems Theory 15 (1982) 95–125). As the second main result we prove that the first two hierarchies are proper even for arbitrary (i.e., not necessarily commutative) izz-semirings

A pumping lemma and decidability problems for recognizable tree series

In the present paper we show that given a tree series S, which is accepted by (a) a deterministic bottom-up finite state weighted tree automaton (for short: bu-w-fta) or (b) a non-deterministic bu-w-fta over a locally finite semiring, there exists for every input tree t E supp(S) a decomposition t = C'[C[s]] into contexts C, C' and an input tree s as well as there exist semiring elements a, a', b, b', c such that the equation (S,C'[Cn[s]]) = a'OanOcObnOb' holds for every non-negative integer n. In order to prove this pumping lemma we extend the power-set construction of classical theories and show that for every non-deterministic bu-w-fta over a locally finite semiring there exists an equivalent deterministic one. By applying the pumping lemma we prove the decidability of a tree series S being constant on its support, S being constant, S being boolean, the support of S being the empty set, and the support of S being a finite set provided that S is accepted by (a) a deterministic bu-w-fta over a commutative semiring or (b) a non-deterministic bu-w-fta over a locally finite commutative semiring

Inclusion Diagrams for Classes of Deterministic Bottom-up Tree-to-Tree-Series Transformations

In this paper we investigate the relationship between classes of tree-to-tree-series (for short: t-ts) and o-tree-to-tree-series (for short: o-t-ts) transformations computed by restricted deterministic bottom-up weighted tree transducers (for short: deterministic bu-w-tt). Essentially, deterministic bu-w-tt are deterministic bottom-up tree series transducers [EFV02, FV03, ful, FGV04], but the former are de ned over monoids whereas the latter are de ned over semirings and only use the multiplicative monoid thereof. In particular, the common restrictions of non-deletion, linearity, totality, and homomorphism [Eng75] can equivalently be de ned for deterministic bu-w-tt. Using well-known results of classical tree transducer theory (cf., e.g., [Eng75, Fül91]) and also new results on deterministic bu-w-tt, we order classes of t-ts and o-t-ts transformations computed by restricted deterministic bu-w-tt by set inclusion. More precisely, for every commutative monoid we completely specify the inclusion relation of the classes of t-ts and o-t-ts transformations for all sensible combinations of restrictions by means of inclusion diagrams

Tree series transformations that respect copying

SIGLEAvailable from TIB Hannover: RR 7739(02-01) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Multioperator Weighted Monadic Datalog

In this thesis we will introduce multioperator weighted monadic datalog (mwmd), a formal model for specifying tree series, tree transformations, and tree languages. This model combines aspects of multioperator weighted tree automata (wmta), weighted monadic datalog (wmd), and monadic datalog tree transducers (mdtt). In order to develop a rich theory we will define multiple versions of semantics for mwmd and compare their expressiveness. We will study normal forms and decidability results of mwmd and show (by employing particular semantic domains) that the theory of mwmd subsumes the theory of both wmd and mdtt. We conclude this thesis by showing that mwmd even contain wmta as a syntactic subclass and present results concerning this subclass

Weighted Tree Automata -- May it be a little more?

This is a book on weighted tree automata. We present the basic definitions and some of the important results in a coherent form with full proofs. The concept of weighted tree automata is part of Automata Theory and it touches the area of Universal Algebra. It originated from two sources: weighted string automata and finite-state tree automata