Hyper-Minimization for Deterministic Weighted Tree Automata

Hyper-minimization is a state reduction technique that allows a finite change in the semantics. The theory for hyper-minimization of deterministic weighted tree automata is provided. The presence of weights slightly complicates the situation in comparison to the unweighted case. In addition, the first hyper-minimization algorithm for deterministic weighted tree automata, weighted over commutative semifields, is provided together with some implementation remarks that enable an efficient implementation. In fact, the same run-time O(m log n) as in the unweighted case is obtained, where m is the size of the deterministic weighted tree automaton and n is its number of states.Comment: In Proceedings AFL 2014, arXiv:1405.527

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

Survey : Weighted extended top-down tree transducers part I. : basics and expressive power

Weighted extended top-down tree transducers (transducteurs généralisés descendants [Arnold, Dauchet: Bi-transductions de forêts. ICALP'76. Edinburgh University Press, 1976]) received renewed interest in the field of Natural Language Processing, where they are used in syntax-based machine translation. This survey presents the foundations for a theoretical analysis of weighted extended top-down tree transducers. In particular, it discusses essentially complete semirings, which are a novel concept that can be used to lift incomparability results from the unweighted case to the weighted case even in the presence of infinite sums. In addition, several equivalent ways to define weighted extended top-down tree transducers are presented and the individual benefits of each presentation is shown on a small result

Random Generation of Nondeterministic Finite-State Tree Automata

Algorithms for (nondeterministic) finite-state tree automata (FTAs) are often tested on random FTAs, in which all internal transitions are equiprobable. The run-time results obtained in this manner are usually overly optimistic as most such generated random FTAs are trivial in the sense that the number of states of an equivalent minimal deterministic FTA is extremely small. It is demonstrated that nontrivial random FTAs are obtained only for a narrow band of transition probabilities. Moreover, an analytic analysis yields a formula to approximate the transition probability that yields the most complex random FTAs, which should be used in experiments.Comment: In Proceedings TTATT 2013, arXiv:1311.5058. Andreas Maletti and Daniel Quernheim were financially supported by the German Research Foundation (DFG) grant MA/4959/1-

Multiple Context-Free Tree Grammars: Lexicalization and Characterization

Multiple (simple) context-free tree grammars are investigated, where "simple" means "linear and nondeleting". Every multiple context-free tree grammar that is finitely ambiguous can be lexicalized; i.e., it can be transformed into an equivalent one (generating the same tree language) in which each rule of the grammar contains a lexical symbol. Due to this transformation, the rank of the nonterminals increases at most by 1, and the multiplicity (or fan-out) of the grammar increases at most by the maximal rank of the lexical symbols; in particular, the multiplicity does not increase when all lexical symbols have rank 0. Multiple context-free tree grammars have the same tree generating power as multi-component tree adjoining grammars (provided the latter can use a root-marker). Moreover, every multi-component tree adjoining grammar that is finitely ambiguous can be lexicalized. Multiple context-free tree grammars have the same string generating power as multiple context-free (string) grammars and polynomial time parsing algorithms. A tree language can be generated by a multiple context-free tree grammar if and only if it is the image of a regular tree language under a deterministic finite-copying macro tree transducer. Multiple context-free tree grammars can be used as a synchronous translation device.Comment: 78 pages, 13 figure

Pumping Lemmata for Recognizable Weighted Languages over Artinian Semirings

Pumping lemmata are the main tool to prove that a certain language does not belong to a class of languages like the recognizable languages or the context-free languages. Essentially two pumping lemmata exist for the recognizable weighted languages: the classical one for the Boolean semiring (i.e., the unweighted case), which can be generalized to zero-sum free semirings, and the one for fields. A joint generalization of these two pumping lemmata is provided that applies to all Artinian semirings, over which all finitely generated semimodules have a finite bound on the length of chains of strictly increasing subsemimodules. Since Artinian rings are exactly those that satisfy the Descending Chain Condition, the Artinian semirings include all fields and naturally also all finite semirings (like the Boolean semiring). The new pumping lemma thus covers most previously known pumping lemmata for recognizable weighted languages.Comment: In Proceedings AFL 2023, arXiv:2309.0112

Synchronous forest substitution grammars

The expressive power of synchronous forest (tree-sequence) substitution grammars (SFSGs) is studied in relation to multi bottom-up tree transducers (MBOTs). It is proved that SFSGs have exactly the same expressive power as compositions of an inverse MBOT with an MBOT. This result is used to derive complexity results for SFSGs and the fact that compositions of an MBOT with an inverse MBOT can compute tree translations that cannot be computed by any SFSG, although the class of tree translations computable by MBOTs is closed under composition