Tree Automata Research Articles

Plant Theory

Abstract Over a hundred years after the publication of the scientist Jagadish Chandra Bose’s 1922 Abyakta, a book of essays in Bangla where he writes about his experiments on plant life, the response of plants to external stimuli, particularly electrical energy, a nervous system in plants, and his work towards revealing what he calls ‘torulipi’, a plant script, so as to be able to prove that plants were indeed living beings, scientists are now proving many of the Indian scientist’s findings. Simultaneous with this has been a public enthusiasm for the work of writers and artists that feature speculation over, and the subsequent creation of, a plant script. This review of works published in 2023 considers two key publications that illuminate the prescience of Bose’s thought for the contemporary investment in the plant humanities: Katie Holten’s edited anthology The Language of Trees and the spring 2023 special issue of Orion on ‘The Language of Nature’. This review therefore focuses on literature about the plant script while also recording and contextualizing the zeitgeist that compels and indulges such an imagining.

In their own words: A mixed-methods exploration of public perceptions of affordable housing and their connections to support

One of the main barriers faced by proposed affordable housing developments is local public opposition. Consequently, this exploratory study seeks: 1) to determine the U.S. public's current top-of-mind perceptions of affordable housing, and 2) to understand which of these perceptions are related to self-reported support for affordable housing in one's own neighborhood. We employ a mixed-methods approach, administering a nation-wide online survey (N = 534) of close- and open-ended questions. While no majority understood definition emerged, through our discursive analysis and natural language process (NLP) topic modeling we uncover common and persisting perceptions of current affordable housing buildings as federally supported apartments that are run-down and in undesirable neighborhoods. Narratives about residents are limited, though when prompted participants have a clear perceived racial profile of residents. Using a conditional inference regression tree (CI-Tree) and NLP language feature associations, we also find that perceptions focused on government involvement, subsidies, and unsafe neighborhoods are significant predictors of lower support for proposed developments. Alternatively, mentioning the financial aspects of affordable housing is significantly associated with higher support. These findings demonstrate the potential of machine learning methods in uncovering pathways to support affordable housing and can be leveraged for effective framing of proposed developments.

Rational weighted tree languages with storage

We define the class of rational weighted tree languages with storage over complete, not necessarily commutative, semirings and we repeat its characterization by weighted regular tree grammars with storage. Moreover, we show an alternative proof of the fact that the class of rational weighted tree languages with storage is closed under the rational operations, i.e., top-concatenation, scalar multiplication, sum, tree concatenation, and Kleene-star, where the latter two closure results require that the storage has a reset instruction.

The impact of state merging on predictive accuracy in probabilistic tree automata: Dietze's conjecture revisited

Dietze's conjecture concerns the problem of equipping a tree automaton M with weights to make it probabilistic, in such a way that the resulting automaton N predicts a given corpus C as accurately as possible. The conjecture states that the accuracy cannot increase if the states in M are merged with respect to an equivalence relation ∼ so that the result is a smaller automaton M∼. Put differently, merging states can never improve predictions. This is under the assumption that both M and M∼ are bottom-up deterministic and accept every tree in C. We prove that the conjecture holds, using a construction that turns any probabilistic version N∼ of M∼ into a probabilistic version N of M, such that N assigns at least as great a weight to each tree in C as N∼ does.

Parsing Unranked Tree Languages, Folded Once

A regular unranked tree folding consists of a regular unranked tree language and a folding operation that merges (i.e., folds) selected nodes of a tree to form a graph; the combination is a formal device for representing graph languages. If, in the process of folding, the order among edges is discarded so that the result is an unordered graph, then two applications of a fold operation are enough to make the associated parsing problem NP-complete. However, if the order is kept, then the problem is solvable in non-uniform polynomial time. In this paper, we address the remaining case, where only one fold operation is applied, but the order among the edges is discarded. We show that, under these conditions, the problem is solvable in non-uniform polynomial time.

Combined Approximations for Uniform Operational Consistent Query Answering

Operational consistent query answering (CQA) is a recent framework for CQA based on revised definitions of repairs, which are built by applying a sequence of operations (e.g., fact deletions) starting from an inconsistent database until we reach a database that is consistent w.r.t. the given set of constraints. It has been recently shown that there is an efficient approximation for computing the percentage of repairs that entail a given query when we focus on primary keys, conjunctive queries, and assuming the query is fixed (i.e., in data complexity). However, it has been left open whether such an approximation exists when the query is part of the input (i.e., in combined complexity). We show that this is the case when we focus on self-join-free conjunctive queries of bounded generelized hypertreewidth. We also show that it is unlikely that efficient approximation schemes exist once we give up one of the adopted syntactic restrictions, i.e., self-join-freeness or bounding the generelized hypertreewidth. Towards the desired approximation, we introduce a counting complexity class, called SpanTL, show that each problem in it admits an efficient approximation scheme by using a recent approximability result about tree automata, and then place the problem of interest in SpanTL.

Forward linearised tree pattern matching using tree pattern border array

We define a tree pattern border array as a property of linearised trees analogous to border arrays from the string domain. We consider tree patterns with node wildcards and subtree wildcards. Then we use the tree pattern border array in a new forward tree pattern matching algorithm for ordered ranked trees. The algorithm finds all occurrences of a single linearised tree pattern in a linearised input tree. As with the classical Morris–Pratt algorithm for searching in strings, the tree pattern border array is used to determine shift lengths in the searching phase of the tree pattern matching algorithm. We compare this new algorithm with the best performing previously existing algorithms based on backward linearised tree pattern matching algorithms, (non-)linearised tree pattern matching algorithms using finite tree automata or stringpath matchers. We show that our new algorithm outperforms other tree pattern matching algorithms in single tree pattern matching.

Regular planar monoidal languages

We introduce regular languages of morphisms in free monoidal categories, with their associated grammars and automata. These subsume the classical theory of regular languages of words and trees, but also open up a much wider class of languages of planar string diagrams. We give a pumping lemma for monoidal languages, generalizing the one for words and trees. We use the algebra of monoidal and cartesian restriction categories to investigate the properties of regular monoidal languages, and provide sufficient conditions for their recognizability by deterministic monoidal automata.

A Nivat Theorem for Weighted Alternating Automata over Commutative Semirings

This paper connects the classes of weighted alternating finite automata (WAFA), weighted finite tree automata (WFTA), and polynomial automata (PA). First, we investigate the use of trees in the run semantics for weighted alternating automata and prove that the behavior of a weighted alternating automaton can be characterized as the composition of the behavior of a weighted finite tree automaton and a specific tree homomorphism, if weights are taken from a commutative semiring. Based on this, we give a Nivat-like characterization for weighted alternating automata. Moreover, we show that the class of series recognized by weighted alternating automata is closed under inverses of homomorphisms, but not under homomorphisms. Additionally, 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. We prove: A weighted language is recognized by a weighted alternating automaton iff its reversal in recognized by a polynomial automaton. Using the corresponding result for polynomial automata, we are able to prove that the ZERONESS problem for weighted alternating automata with weights taken from the rational numbers decidable.

Complexity of the emptiness problem for graph-walking automata and for tilings with star subgraphs

This paper proves the decidability of the emptiness problem for two models which recognize finite graphs: graph-walking automata, and tilings of graphs by star subgraphs (star automata). Furthermore, it is proved that the non-emptiness problem for graph-walking automata (that is, whether a given automaton accepts at least one graph) is NEXP-complete. For star automata, which generalize nondeterministic tree automata to the case of graphs, it is proved that their non-emptiness problem is NP-complete.

Tree-Based Generation of Restricted Graph Languages

Order-preserving DAG grammars (OPDGs) is a formalism for representing languages of structurally restricted graphs. As demonstrated in [17], they are sufficiently expressive to model abstract meaning representations in natural language processing, a graph-based form of semantic representation in which nodes encode objects and edges relations. At the same time, they can be parsed in [Formula: see text], where [Formula: see text] and [Formula: see text] are the sizes of the grammar and the input graph, respectively. In this work, we provide an initial algebra semantic for OPDGs, which allows us to view them as regular tree grammars under an equivalence theory. This makes it possible to transfer results from the field of formal tree languages to the domain of OPDGs, both in the unweighted and the weighted case. In particular, we show that deterministic OPDGs can be minimised efficiently, and that they are learnable under the “minimal adequeate teacher” paradigm, that is, by querying an oracle for equivalence between languages, and membership of individual graphs. To conclude, we demonstrate that the languages generated by OPDGs are definable in monadic second-order logic.

Aperiodicity, Star-freeness, and First-order Logic Definability of Operator Precedence Languages

A classic result in formal language theory is the equivalence among non-counting, or aperiodic, regular languages, and languages defined through star-free regular expressions, or first-order logic. Past attempts to extend this result beyond the realm of regular languages have met with difficulties: for instance it is known that star-free tree languages may violate the non-counting property and there are aperiodic tree languages that cannot be defined through first-order logic. We extend such classic equivalence results to a significant family of deterministic context-free languages, the operator-precedence languages (OPL), which strictly includes the widely investigated visibly pushdown, alias input-driven, family and other structured context-free languages. The OP model originated in the '60s for defining programming languages and is still used by high performance compilers; its rich algebraic properties have been investigated initially in connection with grammar learning and recently completed with further closure properties and with monadic second order logic definition. We introduce an extension of regular expressions, the OP-expressions (OPE) which define the OPLs and, under the star-free hypothesis, define first-order definable and non-counting OPLs. Then, we prove, through a fairly articulated grammar transformation, that aperiodic OPLs are first-order definable. Thus, the classic equivalence of star-freeness, aperiodicity, and first-order definability is established for the large and powerful class of OPLs. We argue that the same approach can be exploited to obtain analogous results for visibly pushdown languages too.

Transduction from trees to graphs through folding

We introduce a fold operation that realises a tree-to-graph transduction by merging selected nodes in the input tree to form a possibly cyclic output graph. The work is motivated by the increasing use of graph-based representations in semantic parsing. We show that a suitable class of graphs languages can be generated by applying the fold operation to regular unranked tree languages. We investigate two versions of the fold operation, one that preserves a depth-first ordering between the edges, and one that does not. Finally, we demonstrate that the time complexity for the associated non-uniform membership problem is solvable in polynomial time for the order-preserving version, and NP-complete for the order-cancelling one.

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.

Checking in polynomial time whether or not a regular tree language is deterministic top-down

It is well known that for a given bottom-up tree automaton it can be decided whether or not an equivalent deterministic top-down tree automaton exists. Recently it was claimed that such a decision can be carried out in polynomial time (Leupold and Maneth, FCT'2021); but their procedure and corresponding property is wrong. Here we address this mistake and present a correct property which allows to determine in polynomial time whether or not a given tree language can be recognized by a deterministic top-down tree automaton. Furthermore, our new property is stated for arbitrary deterministic bottom-up tree automata, and not only for minimal such automata (as before).

Hybrid tree automata and the yield theorem for constituent tree automata

We introduce an automaton model for recognizing sets of hybrid trees, the hybrid tree automaton (HTA). Special cases of hybrid trees are constituent trees and dependency trees, as they occur in natural language processing. This includes the cases of discontinuous constituent trees and non-projective dependency trees. In general, a hybrid tree is a tree over a ranked alphabet in which a symbol can additionally be equipped with a natural number, called index; in a hybrid tree, each index occurs at most once. The yield of a hybrid tree is a sequence of strings over those symbols which occur in an indexed form; the corresponding indices determine the order within these strings; the borders between two consecutive strings are determined by the gaps in the sequence of indices. As a special case of HTA, we define constituent tree automata (CTA) which recognize sets of constituent trees. We introduce the notion of CTA-inductively recognizable and we show that the set of yields of a CTA-inductively recognizable set of constituent trees is an LCFRS language, and vice versa.

Homomorphisms and inverse homomorphisms on graph-walking automata

Graph-walking automata analyze an input graph by moving between its nodes, following the edges. This paper investigates the effect of node-replacement graph homomorphisms and inverse homomorphisms on recognizability by these automata. For deterministic graph-walking automata, it is shown that the family of graph languages they recognize is closed under inverse homomorphisms: for an n-state automaton, the inverse homomorphic images of the graphs it accepts can be recognized by an automaton with at most kn+1 states, where k is the number of labels of edge end-points in the pre-image graphs. At the same time, it is proved that in the worst case these inverse homomorphic images require a deterministic graph-walking automaton with at least kn states. The upper bound kn+1 also holds for nondeterministic graph-walking automata. The second result is that already for tree-walking automata, both deterministic and nondeterministic, the families they recognize are not closed under injective homomorphisms. Here the proof is based on a new homomorphic characterization of regular tree languages: every regular tree language is representable as h−1(g(all trees)), for some injective node-replacement homomorphisms g and h.

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.

Tree Automata and Pigeonhole Classes of Matroids: II

Let $\psi$ be a sentence in the counting monadic second-order logic of matroids and let F be a finite field. Hlineny's Theorem says that we can test whether F-representable matroids satisfy $\psi$ using an algorithm that is fixed-parameter tractable with respect to branch-width. In a previous paper we proved there is a similar fixed-parameter tractable algorithm that can test the members of any efficiently pigeonhole class. In this sequel we apply results from the first paper and thereby extend Hlineny's Theorem to the classes of fundamental transversal matroids, lattice path matroids, bicircular matroids, and H-gain-graphic matroids, when H is a finite group. As a consequence, we can obtain a new proof of Courcelle's Theorem.

The Language of Trees: A Rewilding of Literature and Landscape. By Katie Holten

<i>The Language of Trees: A Rewilding of Literature and Landscape</i>. By Katie Holten