We investigate the determinization of nondeterministic bottom-up/top-down finite state weighted tree automata over some semiring A and compare the resulting four classes of formal tree series with each other. In fact, we generalize well known theorems on classes of tree languages (cf. [GS84] Chapter II, Theorems 2.6 and 2.10, Example 2.11), viz. if A is a commutative and locally finite semifield, then (i) nondeterministic bottom-up, (ii) deterministic bottom-up, and (iii) nondeterministic top-down finite state weighted tree automata are equally powerful. Moreover, if the input alphabet is not trivial, then deterministic top-down finite state weighted tree automata are strictly less powerful than the aforementioned classes.

Speech recognition has been a very active area of research over the past twenty years. Despite an evident progress, it is generally agreed by the practitioners of the field that performance of the current speech recognition systems is rather suboptimal and new ap-proaches are needed. The motivation behind the undertaken research is an observation that the notion of representation of objects and concepts that once was considered to be central in the early days of pattern recognition, has been largely marginalised by the ad-vent of statistical approaches. As a consequence of a predominantly statistical approach to speech recognition problem, due to the numeric, feature vector-based, nature of rep-resentation, the classes inductively discovered from real data using decision-theoretic techniques have little meaning outside the statistical framework. This is because deci-sion surfaces or probability distributions are difficult to analyse linguistically. Because of the later limitation it is doubtful that the gap between speech recognition and lin-guistic research can be bridged by the numeric representations. This thesis investigates an alternative, structural, approach to spoken language representation and categorisa-

We investigate several algorithms related to the parsing problem for weighted automata, under the assumption that the input is a string rather than a tree. This assumption is motivated by several natural language processing applications. We provide algorithms for the computation of parse-forests, best tree probability, inside probability (called partition function), and prefix probability. Our algorithms are obtained by extending to weighted tree automata the Bar-Hillel technique, as defined for context-free grammars.

The relationship between classes of tree-to-tree-series and o-tree-to-tree-series transformations, which are computed by restricted deterministic bottom-up weighted tree transducers, is investigated. Essentially, these transducers are deterministic bottom-up tree series transducers, except that the former are defined over monoids whereas the latter are defined over semirings and only use the multiplicative monoid thereof. In particular, the common restrictions of nondeletion, linearity, totality, and homomorphism can equivalently be defined for deterministic bottom-up weighted tree transducers. Using well-known results of classical tree transducer theory and also new results on deterministic weighted tree transducers, classes of tree-to-tree-series and o-tree-to-tree-series transformations computed by restricted deterministic bottomup weighted tree transducers are ordered by set inclusion. More precisely, for every commutative monoid and all sensible combinations of the above mentioned restrictions, the inclusion relation of the classes of tree-to-tree-series and o-tree-to-treeseries transformations is completely conveyed by means of Hasse diagrams.

Tree series transformations computed by polynomial topdown and bottom-up tree series transducers are considered. The hierarchy of tree series transformations obtained in [Fülöp, Gazdag, Vogler: Hierarchies of Tree Series Transformations. Theoret. Comput. Sci. 314(3), p. 387–429, 2004] for commutative izz-semirings (izz abbreviates idempotent, zero-sum and zero-divisor free) is generalized to arbitrary positive (i. e., zero-sum and zero-divisor free) commutative semirings. The latter class of semirings includes prominent examples such as the natural numbers semiring and the least common multiple semiring, which are not members of the former class.

Two examples of automata-theoretic models for the validation of xml documents against user-de ned schema are the stepwise unranked tree automaton (suta) and the parallel unranked tree automaton (puta). By adding a weight, taken from some semiring, to every transition we generalise these two qualitative automata models to quantitative models, thereby obtaining weighted stepwise unranked tree automata (wsuta) and weighted parallel unranked tree automata (wputa); the qualitative automata models are reobtained by choosing the Boolean semiring. We deal with the minimisation problem of wsuta and wputa by using (forward and backward) bisimulations and we prove the following results: (1) for every wsuta an equivalent forward (resp. backward) bisimulation minimal wsuta can be computed in time O(mn) where n is the number of states and m is the number of transitions of the given wsuta; (2) the same result is proved for wputa instead of wsuta; (3) if the semiring is additive cancellative or the Boolean semiring, then the bound can be improved to O(m log n) for both wsuta and wputa; (4) for every deterministic puta we can compute a minimal equivalent deterministic puta in time O(m log n); (5) the automata models wsuta, wputa, and weighted unranked tree automaton have the same computational power.

Abstract. Recently, an algorithm- DEES- was proposed for learning rational stochastic tree languages. Given a sample of trees independently and identically drawn according to a distribution de ned by a rational stochastic language, DEES outputs a linear representation of a rational series which converges to the target. DEES can then be used to identify in the limit with probability one rational stochastic tree languages. However, when DEES deals with nite samples, it often outputs a rational tree series which does not de ne a stochastic language. Moreover, the linear representation can not be directly used as a generative model. In this paper, we show that any representation of a rational stochastic tree language can be transformed in a reduced normalised representation that can be used to generate trees from the underlying distribution. We also study some properties of consistency for rational stochastic tree languages and discuss their implication for the inference. We nally consider the applicability of DEES to trees built over an unranked alphabet. 1

Abstract: Kleene's theorem on the equivalence of recognizability and rationality for formal tree series over distributive multioperator monoids is proved. As a consequence of this, Kleene's theorem for weighted tree automata over arbitrary, i.e., not necessarily commutative, semirings is derived. 1

Abstract: We define a weighted monadic second order logic for XML-documents (weighted uMSO-logic), viewed as unranked trees. Such a logic allows us to pose quantitative queries to XML-databases. Also we define the concept of unranked weighted tree automata and prove that, for every commutative semiring K, they have the same expressive power as (1) restricted weighted uMSO-logic, (2) almost existential weighted uMSO-logic whenever the additive monoid of K is locally finite, (3) (unrestricted) weighted uMSO-logic whenever K is locally finite. Moreover, if K is a computable field, then the equivalence of any two sentences of restricted weighted uMSO-logic is decidable and there is an effective procedure to construct the automaton from the sentence. 1

A tree series over a semiring with partially ordered carrier set can be considered as a fuzzy set. We investigate conditions under which it can also be understood as a fuzzi ed recognizable tree language. In this sense, su cient conditions are presented which, when imposed, ensure that every cut set, i.e., the pre-image of a prime lter of the carrier set, is a recognizable tree language. Moreover, such conditions are also presented for cut sets of recognizable tree series. 1