Tree Automata Research Articles

Complexity results on register context-free grammars and related formalisms

Register context-free grammars (RCFG), register pushdown automata (RPDA) and register tree automata (RTA) are extensions of their classical counterparts to handle data values in a restricted way. These extended models are paid attention as models of query languages for structured documents such as XML with data values. This paper investigates the computational complexity of the basic decision problems for the models. We show that the membership and emptiness problems for RCFG are EXPTIME-complete and also show the membership problem becomes PSPACE-complete and NP-complete for ε-rule free RCFG and growing RCFG, respectively while the emptiness problem remains EXPTIME-complete for these subclasses. The complexities of these problems for RPDA and RTA as well as their language expressive powers are also investigated.

On the Expressive Power of the Normal Form for Branching-Time Temporal Logics

With the emerging applications that involve complex distributed systems branching-time specifications are specifically important as they reflect dynamic and non-deterministic nature of such applications. We describe the expressive power of a simple yet powerful branching-time specification framework -- branching-time normal form (BNF), which has been developed as part of clausal resolution for branching-time temporal logics. We show the encoding of Buchi Tree Automata in the language of the normal form, thus representing, syntactically, tree automata in a high-level way. Thus we can treat BNF as a normal form for the latter. These results enable us (1) to translate given problem specifications into the normal form and apply as a verification method a deductive reasoning technique -- the clausal temporal resolution; (2) to apply one of the core components of the resolution method -- the loop searching to extract, syntactically, hidden invariants in a wide range of complex temporal specifications.

Finite-image property of weighted tree automata over past-finite monotonic strong bimonoids

We consider weighted tree automata over strong bimonoids (for short: wta). A wta A has the finite-image property if its recognized weighted tree language 〚A〛 has finite image; moreover, A has the preimage property if the preimage under 〚A〛 of each element of the underlying strong bimonoid is a recognizable tree language. For each wta A over a past-finite monotonic strong bimonoid we prove the following results. In terms of A's structural properties, we characterize whether it has the finite-image property. We characterize those past-finite monotonic strong bimonoids such that for each wta A it is decidable whether A has the finite-image property. In particular, the finite-image property is decidable for wta over past-finite monotonic semirings. Moreover, we prove that A has the preimage property. All our results also hold for weighted string automata.

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.

Gram Stain Prediction with Machine Learning Techniques Using Biochemical Parameters in ICU Patients with Urinary Tract Infections

Purpose: The aim of this study was to develop a useful algorithm based on complete blood count (CBC), urinalysis, and biochemical parameters that could be an alternative to gram staining in the prediction of UTI and the determination of initial antibiotic treatment in ICU patients. Methods: All the specimens included in the study were obtained from ICU patients and were subjected to gram staining in the laboratory. Simultaneously, CBC, urinalysis, and biochemical tests were performed for each specimen. A classification based on biochemical parameters was performed for the estimation of gram-negative and gram-positive bacteria, as an alternative to gram staining. Results: Classification was achieved using multiple classification systems including Artificial Neural Networks (ANN), Support Vector Machine (SVM), the K-Nearest Neighbors (KNN), and Decision Tree Language (DTL) and the best classification performance was achieved by ANN, with an accuracy of 84.6%, sensitivity of 88.5%, and specificity of 73.5%. Conclusion: The high specificity and accuracy of the algorithm indicated that this algorithm can be effectively used in the selection of empirical antibiotic treatment for ICU patients with UTI and can provide more advanced and technological opportunities by combining laboratory parameters with machine learning techniques.

Tree Automata and Pigeonhole Classes of Matroids: I

Hliněný’s Theorem shows that any sentence in the monadic second-order logic of matroids can be tested in polynomial time, when the input is limited to a class of {mathbb {F}}-representable matroids with bounded branch-width (where {mathbb {F}} is a finite field). If each matroid in a class can be decomposed by a subcubic tree in such a way that only a bounded amount of information flows across displayed separations, then the class has bounded decomposition-width. We introduce the pigeonhole property for classes of matroids: if every subclass with bounded branch-width also has bounded decomposition-width, then the class is pigeonhole. An efficiently pigeonhole class has a stronger property, involving an efficiently-computable equivalence relation on subsets of the ground set. We show that Hliněný’s Theorem extends to any efficiently pigeonhole class. In a sequel paper, we use these ideas to extend Hliněný’s Theorem to the classes of fundamental transversal matroids, lattice path matroids, bicircular matroids, and H-gain-graphic matroids, where H is any finite group. We also give a characterisation of the families of hypergraphs that can be described via tree automata: a family is defined by a tree automaton if and only if it has bounded decomposition-width. Furthermore, we show that if a class of matroids has the pigeonhole property, and can be defined in monadic second-order logic, then any subclass with bounded branch-width has a decidable monadic second-order theory.

Regular matching problems for infinite trees

We study the matching problem of regular tree languages, that is, "$\exists \sigma:\sigma(L)\subseteq R$?" where $L,R$ are regular tree languages over the union of finite ranked alphabets $\Sigma$ and $\mathcal{X}$ where $\mathcal{X}$ is an alphabet of variables and $\sigma$ is a substitution such that $\sigma(x)$ is a set of trees in $T(\Sigma\cup H)\setminus H$ for all $x\in \mathcal{X}$. Here, $H$ denotes a set of "holes" which are used to define a "sorted" concatenation of trees. Conway studied this problem in the special case for languages of finite words in his classical textbook "Regular algebra and finite machines" published in 1971. He showed that if $L$ and $R$ are regular, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma(L)\subseteq R$?" is decidable. Moreover, there are only finitely many maximal solutions, the maximal solutions are regular substitutions, and they are effectively computable. We extend Conway's results when $L,R$ are regular languages of finite and infinite trees, and language substitution is applied inside-out, in the sense of Engelfriet and Schmidt (1977/78). More precisely, we show that if $L\subseteq T(\Sigma\cup\mathcal{X})$ and $R\subseteq T(\Sigma)$ are regular tree languages over finite or infinite trees, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma_{\mathrm{io}}(L)\subseteq R$?" is decidable. Here, the subscript "$\mathrm{io}$" in $\sigma_{\mathrm{io}}(L)$ refers to "inside-out". Moreover, there are only finitely many maximal solutions $\sigma$, the maximal solutions are regular substitutions and effectively computable. The corresponding question for the outside-in extension $\sigma_{\mathrm{oi}}$ remains open, even in the restricted setting of finite trees.

Characterization of Tree Automata Based on Quantum Logic

Characterization of Tree Automata Based on Quantum Logic

A Congruence-Based Perspective on Finite Tree Automata

We provide new insights on the determinization and minimization of tree automata using congruences on trees. From this perspective, we study a Brzozowski’s style minimization algorithm for tree automata. First, we prove correct this method relying on the following fact: when the automata-based and the language-based congruences coincide, determinizing the automaton yields the minimal one. Such automata-based congruences, in the case of word automata, are defined using pre and post operators. Now we extend these operators to tree automata, a task that is particularly challenging due to the reduced expressive power of deterministic top-down (or equivalently co-deterministic bottom-up) automata. We leverage further our framework to offer an extension of the original result by Brzozowski for word automata.

Synthesizing fuzzy tree automata

Fuzzy tree automata are mathematical devices for modeling and analyzing vaguely defined tree structures. The behavior of a fuzzy tree automaton generates a fuzzy tree language by mapping a set of regular trees on a ranked alphabet to fuzzy membership values. It calculates the membership grade of trees using a set of rules that process their structural characteristics. This paper deals with constructing fuzzy tree automata models that their behavior satisfies a set of given logical propositions (called properties) on the structure of trees. Our goal is uncertainty modeling by synthesizing fuzzy tree automata whose behavior is described by fuzzy linguistic variables. In this regard, we first provide several patterns and heuristic tricks and techniques for constructing fuzzy tree automata that satisfy simple properties. Then, we develop a method for modeling complex propositional formulas based on the conversion of a logical formula into a computation tree, as well as a step-by-step combination of models.

On substructures of semigroups of inductive terms

<abstract><p>An inductive composition is an operation generalizing from a superposition $ S^n $ on the set of all $ n $-ary terms of type $ \tau $. A binary operation called <italic>inductive product</italic> is obtainable from such composition. It is a generalization of a tree language product but on the set of all $ n $-ary terms of type $ \tau $. Unlike the original one, this inductive product is not associative on the mentioned set. Nonetheless, it turns to be associative on some restricted set. A semigroup arising in this way is the main focus of this paper. We consider its special subsemigroups and semigroup factorizations.</p></abstract>

Photo Art Project ‘Female Multi-Component Associative Image ‘Fern Blossom’. Part 1

The author’s idea. What the phenomenon of ‘Fern blossom’ is, whereas according to scientific data, the fern does not bloom and does not form inflorescences. Fern flower in the East Slavic mythology has the character of a magical plant that gives a person magical power. With the help of the fern flower, the owner of it could understand the language of animals and trees, see hidden precious treasures under the ground, heal people from various diseases, predict the future and more. It is believed that the fern blossom can be found only on Ivan Kupala night. This holiday is traditional in Ukraine and is named after the Christian Saint John the Baptist, but originates in the distant past from the pagan faith.

A coalgebraic take on regular and $\omega$-regular behaviours

We present a general coalgebraic setting in which we define finite and infinite behaviour with B\"uchi acceptance condition for systems whose type is a monad. The first part of the paper is devoted to presenting a construction of a monad suitable for modelling (in)finite behaviour. The second part of the paper focuses on presenting the concepts of a (coalgebraic) automaton and its ($\omega$-) behaviour. We end the paper with coalgebraic Kleene-type theorems for ($\omega$-) regular input. The framework is instantiated on non-deterministic (B\"uchi) automata, tree automata and probabilistic automata.

Lattice-valued tree pushdown automata: Pumping lemma and closure properties

In this paper, we introduce the concept of lattice-valued tree pushdown automata that extends the concepts of lattice-valued pushdown automata, lattice-valued tree automata and fuzzy tree pushdown automata. Also, we consider fuzzy tree languages recognized by lattice-valued (deterministic) tree pushdown automata (called lattice-valued context-free tree languages). Then, we prove the pumping lemma that can be used in proving that certain sets of tree languages are not recognizable. Also, we study closure properties of lattice-valued context-free tree languages with respect to some operations. We show that lattice-valued context-free tree languages recognized by lattice-valued deterministic tree pushdown automata are closed under union, concatenation, star, linear and alphabetic tree homomorphism and inverse tree homomorphism. By applying the pumping lemma on an example, we show that lattice-valued context-free tree languages are not closed under intersection. However, we prove that the intersection of lattice-valued context-free tree language and lattice-valued regular tree language is recognizable by a lattice-valued tree pushdown automaton. Also, we show that lattice-valued non-deterministic tree pushdown automata are only closed under union. However, if L satisfies distributivity, then non-deterministic tree pushdown automata are closed under all of these operations.

The tree-generative capacity of combinatory categorial grammars

The generative capacity of combinatory categorial grammars (CCGs) as generators of tree languages is investigated. It is demonstrated that the tree languages generated by CCGs can also be generated by simple monadic context-free tree grammars. However, the important subclass of pure combinatory categorial grammars cannot even generate all regular tree languages. Additionally, the tree languages generated by combinatory categorial grammars with limited rule degrees are characterized: If only application rules are allowed, then these grammars can generate only a proper subset of the regular tree languages, whereas they can generate exactly the regular tree languages once first-degree composition rules are permitted.

Linear weighted tree automata with storage and inverse linear tree homomorphisms

We introduce linear weighted tree automata with storage and show that this model generalizes linear pushdown tree automata. We prove that the class of weighted tree languages recognizable by our linear automaton model is closed under inverse linear tree homomorphisms.

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.

Study of the subtyping machine of nominal subtyping with variance

This is a study of the computing power of the subtyping machine behind Kennedy and Pierce's nominal subtyping with variance. We depict the lattice of fragments of Kennedy and Pierce's type system and characterize their computing power in terms of regular, context-free, deterministic, and non-deterministic tree languages. Based on the theory, we present Treetop---a generator of C# implementations of subtyping machines. The software artifact constitutes the first feasible (yet POC) fluent API generator to support context-free API protocols in a decidable type system fragment.

A Nivat Theorem for Weighted Alternating Automata over Commutative Semirings

In this paper, we give a Nivat-like characterization for weighted alternating automata over commutative semirings (WAFA). To this purpose we prove that weighted alternating can be characterized as the concatenation of weighted finite tree automata (WFTA) and a specific class of tree homomorphism. We show that the class of series recognized by weighted alternating automata is closed under inverses of homomorphisms, but not under homomorphisms. We give a logical characterization of weighted alternating automata, which uses weighted MSO logic for trees. Finally we investigate the strong connection between weighted alternating automata and polynomial automata. Using the corresponding result for polynomial automata, we are able to prove that the ZERONESS problem for weighted alternating automata with the rational numbers as weights is decidable.

Ambiguity Hierarchy of Regular Infinite Tree Languages

An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is k-ambiguous (for k > 0) if for every input it has at most k accepting computations. An automaton is boundedly ambiguous if it is k-ambiguous for some $k \in \mathbb{N}$. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. The degree of ambiguity of a regular language is defined in a natural way. A language is k-ambiguous (respectively, boundedly, finitely, countably ambiguous) if it is accepted by a k-ambiguous (respectively, boundedly, finitely, countably ambiguous) automaton. Over finite words every regular language is accepted by a deterministic automaton. Over finite trees every regular language is accepted by an unambiguous automaton. Over $\omega$-words every regular language is accepted by an unambiguous B\"uchi automaton and by a deterministic parity automaton. Over infinite trees Carayol et al. showed that there are ambiguous languages. We show that over infinite trees there is a hierarchy of degrees of ambiguity: For every k > 1 there are k-ambiguous languages that are not k - 1 ambiguous; and there are finitely (respectively countably, uncountably) ambiguous languages that are not boundedly (respectively finitely, countably) ambiguous.