Andreas Maletti

The Power of Extended Top-Down Tree Transducers

Extended top-down tree transducers (transducteurs generalises descendants; see [A. Arnold and M. Dauchet, Bi-transductions de forets, in Proceedings of the 3rd International Colloquium on Automata, Languages and Programming, Edinburgh University Press, Edinburgh, 1976, pp. 74-86]) received renewed interest in the field of natural language processing. Here those transducers are extensively and systematically studied. Their main properties are identified and their relation to classical top-down tree transducers is exactly characterized. The obtained properties completely explain the Hasse diagram of the induced classes of tree transformations. In addition, it is shown that most interesting classes of transformations computed by extended top-down tree transducers are not closed under composition.

Compositions of extended top-down tree transducers

Unfortunately, the class of transformations computed by linear extended top-down tree transducers with regular look-ahead is not closed under composition. It is shown that the class of transformations computed by certain linear bimorphisms coincides with the previously mentioned class. Moreover, it is demonstrated that every linear epsilon-free extended top-down tree transducer with regular look-ahead can be implemented by a linear multi bottom-up tree transducer. The class of transformations computed by the latter device is shown to be closed under composition, and to be included in the composition of the class of transformations computed by top-down tree transducers with itself. More precisely, it constitutes the composition closure of the class of transformations computed by finite-copying top-down tree transducers.

Extended multi bottom–up tree transducers: Composition and decomposition

Extended multi bottom–up tree transducers are defined and investigated. They are an extension of multi bottom–up tree transducers by arbitrary, not just shallow, left-hand sides of rules; this includes rules that do not consume input. It is shown that such transducers, even linear ones, can compute all transformations that are computed by linear extended top–down tree transducers, which are a theoretical model for syntax-based machine translation. Moreover, the classical composition results for bottom–up tree transducers are generalized to extended multi bottom–up tree transducers. Finally, characterizations in terms of extended top–down tree transducers and tree bimorphisms are presented.

Backward and forward bisimulation minimisation of tree automata

We improve an existing bisimulation minimisation algorithm for tree automata by introducing backward and forward bisimulations and developing minimisation algorithms for them. Minimisation via forward bisimulation is also effective for deterministic automata and faster than the previous algorithm. Minimisation via backward bisimulation generalises the previous algorithm and is thus more effective but just as fast. We demonstrate implementations of these algorithms on a typical task in natural language processing.

Backward and forward bisimulation minimization of tree automata

We improve on an existing [P.A. Abdulla, J. Hogberg, L. Kaati, Bisimulation minimization of tree automata, International Journal of Foundations of Computer Science 18(4) (2007) 699-713] bisimulation minimization algorithm for finite-state tree automata by introducing backward and forward bisimulation and developing minimization algorithms for them. Minimization via forward bisimulation is also effective on deterministic tree automata, faster than the previous algorithm, and yields the minimal equivalent deterministic tree automaton. Minimization via backward bisimulation generalizes the previous algorithm and can yield smaller automata but is just as fast. We demonstrate implementations of these algorithms on a typical task in natural language processing.

RELATING TREE SERIES TRANSDUCERS AND WEIGHTED TREE AUTOMATA

Bottom-up tree series transducers (tst) over the semiring are implemented with the help of bottom-up weighted tree automata (wta) over an extension of . Therefore bottom-up -weighted tree automata (-wta) with a distributive Ω-algebra are introduced. A -wta is essentially a wta but uses as transition weight an operation symbol of the Ω-algebra instead of a semiring element. The given tst is implemented with the help of a -wta, essentially showing that -wta are a joint generalization of tst (using IO-substitution) and wta. Then a semiring and a wta are constructed such that the wta computes a formal representation of the semantics of the -wta. The applicability of the obtained presentation result is demonstrated by deriving a pumping lemma for deterministic finite -wta from a known pumping lemma for deterministic finite wta. Finally, it is observed that the known decidability results for emptiness cannot be applied to obtain decidability of emptiness for finite -wta. Thus with help of a weaker version of the derived pumping lemma, decidability of the emptiness problem for finite -wta is shown under mild conditions on .

Weighted Extended Tree Transducers

Weighted extended tree transducers (wxtts) over countably complete semirings are systematically explored. It is proved that the extension in the left-hand sides of a wxtt can be simulated by the inverse of a linear and nondeleting tree homomorphism. In addition, a characterization of the class of weighted tree transformations computable by bottom-up wxtts in terms of bimorphisms is provided. Backward and forward application to recognizable weighted tree languages are standard operations for wxtts. It is shown that the backward application of a linear wxtt preserves recognizability and that the domain of an arbitrary bottom-up wxtt is recognizable. Examples demonstrate that neither backward nor forward application of arbitrary wxtts preserves recognizability. Finally, a HASSE diagram relates most of the important subclasses of weighted tree transformations computable by wxtts.

Minimizing deterministic weighted tree automata

An apparatus for positioning a workpiece comprises four measuring devices which collectively render possible the fabrication of a continuous bore at a workpiece. One part of the bore is produced from one side or face of the workpiece and the other part of the bore from the opposite side or face of the workpiece and both parts of the bore are in alignment with one another. The first measuring device serves to enable rotating a rotatable table through any desired angle, the second measuring device for enabling displacing a slide through any desired amount in the X-direction. According to the invention the third measuring device serves for the exact positioning of the rotatable table at 0 DEG , 90 DEG , 180 DEG and 270 DEG . Finally, the fourth measuring device ensures that the rotatable table and a work spindle axis precisely intersect.

Parsing Algorithms based on Tree Automata

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.

Compositions 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 bottom-up 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.