Weighted Tree Automata Research Articles

Weighted Tree Automata with Constraints

The HOM problem, which asks whether the image of a regular tree language under a given tree homomorphism is again regular, is known to be decidable [Godoy & Giménez: The HOM problem is decidable. JACM 60(4), 2013]. However, the problem remains open for regular weighted tree languages. It is demonstrated that the main notion used in the unweighted setting, the tree automaton with equality and inequality constraints , can straightforwardly be generalized to the weighted setting and can represent the image of any regular weighted tree language under any nondeleting and nonerasing tree homomorphism. Several closure properties as well as decision problems are also investigated for the weighted tree languages generated by weighted tree automata with constraints.

A Comparison of Sets of Recognizable Weighted Tree Languages Over Specific Sets of Bounded Lattices

We consider weighted tree automata (wta) over bounded lattices, locally finite bounded lattices, distributive bounded lattices, and finite chains and we consider the weighted tree languages defined by their run semantics and by their initial algebra semantics. Due to these eight combinations, there are eight sets of weighted tree languages. Moreover, we consider the four sets of weighted tree languages which are recognized by crisp-deterministic wta over the above mentioned sets of weight algebras. We give all inclusion relations among all these twelve sets of weighted tree languages. More precisely, we prove that eleven of the twelve sets are equal, and this set is a proper subset of the set of weighted tree languages recognized by wta over bounded lattices with initial algebra semantics.

Equivalence, Unambiguity, and Sequentiality of Finitely Ambiguous Max-Plus Tree Automata

We show that the equivalence, unambiguity, and sequentiality problems are decidable for finitely ambiguous max-plus tree automata. A max-plus tree automaton is a weighted tree automaton over the max-plus semiring. A max-plus tree automaton is called finitely ambiguous if the number of accepting runs on every tree is bounded by a global constant and it is called unambiguous if there exists at most one accepting run on every tree. For the equivalence problem, we show that for two finitely ambiguous max-plus tree automata, it is decidable whether both assign the same weight to every tree. For the unambiguity and sequentiality problems, we show that for every finitely ambiguous max-plus tree automaton, both the existence of an equivalent unambiguous automaton and the existence of an equivalent deterministic automaton are decidable.

Ambiguity Hierarchies for Weighted Tree Automata

Weighted tree automata (WTA) extend classical weighted automata (WA) to the non-linear structure of trees. The expressive power of WA with varying degrees of ambiguity has been extensively studied. Unambiguous, finitely ambiguous, and polynomially ambiguous WA over the tropical (as well as the arctic) semiring strictly increase in expressive power. The recently developed pumping results of Mazowiecki and Riveros (STACS 2018) are lifted to trees in order to achieve the same strict hierarchy for WTA over the tropical (as well as the arctic) semiring.

A Theorem on Supports of Weighted Tree Automata Over Strong Bimonoids

We investigate weighted tree automata over strong bimonoids. Intuitively, strong bimonoids are semirings which may lack the distributivity laws. For instance, each (not necessarily distributive) bounded lattice is a strong bimonoid. In general, for a weighted tree automaton over a strong bimonoid, its run semantics is different from its initial algebra semantics. We prove that for positive strong bimonoids the supports of these semantics are equal, and we also deal with a kind of reverse statement.

Improved N-Best Extraction with an Evaluation on Language Data

Abstract We show that a previously proposed algorithm for the N-best trees problem can be made more efficient by changing how it arranges and explores the search space. Given an integer N and a weighted tree automaton (wta) M over the tropical semiring, the algorithm computes N trees of minimal weight with respect to M. Compared with the original algorithm, the modifications increase the laziness of the evaluation strategy, which makes the new algorithm asymptotically more efficient than its predecessor. The algorithm is implemented in the software Betty, and compared to the state-of-the-art algorithm for extracting the N best runs, implemented in the software toolkit Tiburon. The data sets used in the experiments are wtas resulting from real-world natural language processing tasks, as well as artificially created wtas with varying degrees of nondeterminism. We find that Betty outperforms Tiburon on all tested data sets with respect to running time, while Tiburon seems to be the more memory-efficient choice.

Finite Sequentiality of Finitely Ambiguous Max-Plus Tree Automata

We show that the finite sequentiality problem is decidable for finitely ambiguous max-plus tree automata. A max-plus tree automaton is a weighted tree automaton over the max-plus semiring. A max-plus tree automaton is called finitely ambiguous if the number of accepting runs on every tree is bounded by a global constant. The finite sequentiality problem asks whether for a given max-plus tree automaton, there exist finitely many deterministic max-plus tree automata whose pointwise maximum is equivalent to the given automaton.

Kleene and Büchi theorems for weighted forest languages over M-monoids

We consider forests as tuples of trees and introduce weighted forest automata (wfa) over M-monoids. A wfa acts on each individual tree in a forest like a weighted tree automaton over the same M-monoid. In order to combine the values of trees in forests, a wfa uses an associative multiplication operation that distributes over the addition of the M-monoid. We continue by introducing two semantics for wfa, an “initial algebra”-like semantic and a run-semantic, and prove that these semantics are equal for distributive M-monoids. We prove that our automaton model accepts finite products of recognizable weighted tree languages over M-monoids. Next, we introduce rational weighted forest expressions and forest M-expressions over M-monoids and show that the classes of languages generated by these formalisms coincide with recognizable weighted forest languages under certain conditions on the underlying M-monoid.

From generic partition refinement to weighted tree automata minimization

Partition refinement is a method for minimizing automata and transition systems of various types. Recently,we have developed a partition refinement algorithm that is generic in the transition type of the given systemand matches the run time of the best known algorithms for many concrete types of systems, e.g. deterministicautomata as well as ordinary, weighted, and probabilistic (labelled) transition systems. Genericity is achieved bymodelling transition types as functors on sets, and systems as coalgebras. In the present work, we refine the runtime analysis of our algorithm to cover additional instances, notably weighted automata and, more generally,weighted tree automata. For weights in a cancellative monoid we match, and for non-cancellative monoids suchas (the additive monoid of) the tropical semiring even substantially improve, the asymptotic run time of the bestknown algorithms. We have implemented our algorithm in a generic tool that is easily instantiated to concretesystem types by implementing a simple refinement interface. Moreover, the algorithm and the tool are modular,and partition refiners for new types of systems are obtained easily by composing pre-implemented basic functors.Experiments show that even for complex system types, the tool is able to handle systems with millions oftransitions.

Approximate minimization of weighted tree automata

This paper studies the following approximate minimization problem: given a minimal weighted tree automaton A with n states recognizing a weighted tree language f, can we construct a smaller automaton Aˆ with nˆ<n states recognizing a language fˆ that is a good approximation of f? The corresponding problem for weighted automata on strings was recently studied by Balle et al. [16,15], where the authors introduced a new canonical form for weighted automata called singular value automata inspired by spectral methods, and showed that truncating such canonical form yields a solution for the problem satisfying a certain approximation criteria. In this paper we take a similar approach and show that in the tree case one can obtain an analogous canonical form that we call singular value tree automata, and use it to study the approximate minimization problem for weighted tree automata. We first establish the existence of this canonical form for weighted tree automata and then provide bounds on the quality of the resulting approximation method based on truncation. We also study the problem of computing the canonical form given a minimal weighted tree automaton and show that in the tree case this task is considerably more complicated than in the string case. In particular, computing the canonical form reduces to solving a system of polynomial equations. By further reducing this problem to the computation of generalized partition functions for weighted tree automata, we propose and analyze two methods for computing the canonical form based on iterative methods: fixed point iteration and Newton's method. Our analysis of Newton's method unveils a connection between iterative methods and sequences of sets of trees satisfying a certain technical condition that might be of independent interest.

A Kleene theorem for weighted tree automata over tree valuation monoids

We investigate weighted tree automata over Cauchy tree valuation monoids, a new type of weight structure which includes all commutative semirings and, in addition, average and discounted computations of weights for trees. We define rational tree series over these weight structures, and we prove Kleene's classical theorem for this setting: a tree series over a Cauchy tree valuation monoid is recognizable by a weighted tree automaton if and only if it is rational. The proof works via direct automata-theoretic constructions. Along the way, our results yield a new characterization of weighted tree automata over commutative semirings by rational expressions.

Efficient enumeration of weighted tree languages over the tropical semiring

We generalise a search algorithm by Mohri and Riley from strings to trees. The original algorithm takes as input a nondeterministic weighted automaton M over the tropical semiring and an integer N, and outputs N strings of minimal weight with respect to M. In our setting, M is a weighted tree automaton, again over the tropical semiring, and the output is a set of N trees with minimal weight in this language. We prove that the algorithm is correct, and that its time complexity is a low polynomial in N and the relevant size parameters of M.

Finding the N best vertices in an infinite weighted hypergraph

We propose an algorithm for computing the N best vertices in a weighted acyclic hypergraph over a nice semiring. A semiring is nice if it is finitely-generated, idempotent, and has 1 as its minimal element. We then apply the algorithm to the problem of computing the N best trees with respect to a weighted tree automaton, and complement theoretical correctness and complexity arguments with experimental data. The algorithm has several practical applications in natural language processing, for example, to derive the N most likely parse trees with respect to a probabilistic context-free grammar.

Characterizations of recognizable weighted tree languages by logic and bimorphisms

We give a general definition of weighted tree automata (wta) and define three instances which differ in the underlying weight algebras: semirings, multi-operator monoids, and tree-valuation monoids. Also, we define a general concept of weighted expressions based on monadic second-order logics. In the same way as for wta, we define three instances corresponding to the above-mentioned weight algebras. We prove that wta over semirings are equivalent to weighted expressions over semirings, and prove the same equivalence over tree-valuation monoids. For wta over semirings and for wta over tree-valuation monoids we prove characterizations in terms of bimorphisms.

A link between multioperator and tree valuation automata and logics

Weighted tree languages over semirings lack the expressive power to model computations like taking the average or the discounting of weights in a straightforward manner. This limitation was overcome by weighted tree automata and logics using (a) tree valuation monoids and (b) multioperator monoids. We compare the expressive power of these two solutions and show that a weighted tree language recognizable (resp. definable) over a tree valuation monoid is also recognizable (resp. definable) using a multioperator monoid. For this, we provide direct, semantic-preserving transformations between the automata models and between the respective logics.

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.

Minimization of Deterministic Fuzzy Tree Automata

Until now, some methods for minimizing deterministic fuzzy finite tree automata (DFFTA) and weighted tree automata have been established by researchers. Those methods are language preserving, but the behavior of original automata and minimized one may be different. This paper, considers both language preserving and behavior preserving in minimization process. We drive Myhill-Nerode kind theorems corresponding to each proposed method and introduce PTIME algorithms for behaviorally and linguistically minimization. The proposed minimization algorithms are based on two main steps. The first step includes finding dependency between equivalency of states, according to the set of transition rules of DFFTA, and making merging dependency graph (MDG). The second step is refinement of MDG and making minimization equivalency set (MES). Additionally, behavior preserving minimization of DFFTA requires a pre-processing for modifying fuzzy membership grade of rules and final states, which is called normalization.

THE CATEGORY OF SIMULATIONS FOR WEIGHTED TREE AUTOMATA

Simulations of weighted tree automata (wta) are considered. It is shown how such simulations can be decomposed into simpler functional and dual functional simulations also called forward and backward simulations. In addition, it is shown in several cases (fields, commutative rings, NOETHERIAN semirings, semiring of natural numbers) that all equivalent wta M and N can be joined by a finite chain of simulations. More precisely, in all mentioned cases there is a single wta that simulates both M and N. Those results immediately yield decidability of equivalence provided that the semiring is finitely (and effectively) presented.

A Büchi-Like Theorem for Weighted Tree Automata over Multioperator Monoids

A multioperator monoid $\mathcal{A}$ is a commutative monoid with additional operations on its carrier set. A weighted tree automaton over $\mathcal{A}$ is a finite state tree automaton of which each transition is equipped with an operation of $\mathcal{A}$. We define M-expressions over $\mathcal{A}$ in the spirit of formulas of weighted monadic second-order logics and, as our main result, we prove that if $\mathcal{A}$ is absorptive, then the class of tree series recognizable by weighted tree automata over $\mathcal{A}$ coincides with the class of tree series definable by M-expressions over $\mathcal{A}$. This result implies the known fact that for the series over semirings recognizability by weighted tree automata is equivalent with definability in syntactically restricted weighted monadic second-order logic. We prove this implication by providing two purely syntactical transformations, from M-expressions into formulas of syntactically restricted weighted monadic second-order logic, and vice versa.

Automata-Based Axiom Pinpointing

Axiom pinpointing has been introduced in description logics (DL) to help the user understand the reasons why consequences hold by computing minimal subsets of the knowledge base that have the consequence in question (MinA). Most of the pinpointing algorithms described in the DL literature are obtained as extensions of tableau-based reasoning algorithms for computing consequences from DL knowledge bases. In this paper, we show that automata-based algorithms for reasoning in DLs and other logics can also be extended to pinpointing algorithms. The idea is that the tree automaton constructed by the automata-based approach can be transformed into a weighted tree automaton whose so-called behaviour yields a pinpointing formula, i.e., a monotone Boolean formula whose minimal valuations correspond to the MinAs. We also develop an approach for computing the behaviour of a given weighted tree automaton. We use the DL $\mathcal{SI}$ as well as Linear Temporal Logic (LTL) to illustrate our new pinpointing approach.