Tree Grammars Research Articles

Automating Unrealizability Logic: Hoare-Style Proof Synthesis for Infinite Sets of Programs

Automated verification of all members of a (potentially infinite) set of programs has the potential to be useful in program synthesis, as well as in verification of dynamically loaded code, concurrent code, and language properties. Existing techniques for verification of sets of programs are limited in scope and unable to create or use interpretable or reusable information about sets of programs. The consequence is that one cannot learn anything from one verification problem that can be used in another. Unrealizability logic (UL), proposed by Kim et al. as the first Hoare-style proof system to prove properties over sets of programs (defined by a regular tree grammar), presents a theoretical framework that can express and use reusable insight. In particular, UL features nonterminal summaries ---inductive facts that characterize recursive nonterminals (analogous to procedure summaries in Hoare logic). In this work, we design the first UL proof synthesis algorithm, implemented as Wuldo. Specifically, we decouple the problem of deciding how to apply UL rules from the problem of synthesizing/checking nonterminal summaries by computing proof structure in a fully syntax-directed fashion. We show that Wuldo, when provided nonterminal summaries, can express and prove verification problems beyond the reach of existing tools, including establishing how infinitely many programs behave on infinitely many inputs. In some cases, Wuldo can even synthesize the necessary nonterminal summaries. Moreover, Wuldo can reuse previously proven nonterminal summaries across verification queries, making verification 1.96 times as fast as when summaries are instead proven from scratch.

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.

A Survey on Syntactic Pattern Recognition Methods in Bioinformatics

Formal tools and models of syntactic pattern recognition which are used in bioinformatics are introduced and characterized in the paper. They include, among others: stochastic (string) grammars and automata, hidden Markov models, programmed grammars, attributed grammars, stochastic tree grammars, Tree Adjoining Grammars (TAGs), algebraic dynamic programming, NLC- and NCE-type graph grammars, and algebraic graph transformation systems. The survey of applications of these formal tools and models in bioinformatics is presented.

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.

Combinatory categorial grammars as generators of weighted forests

Combinatory Categorial Grammar (CCG) is an extension of categorial grammar that is well-established in computational linguistics. It is mildly context-sensitive, so it is efficiently parsable and reaches an expressiveness that is suitable for describing natural languages. Weighted CCG (wCCG) are introduced as a natural extension of CCG with weights taken from an arbitrary commutative semiring. Their expressive power is compared to other weighted formalisms with special emphasis on the weighted forests generated by wCCG since the ability to express the underlying syntactic structure of an input sentence is a vital feature of CCG in the area of natural language processing. Building on recent results for the expressivity in the unweighted setting, the corresponding results are derived for the weighted setting for any commutative semiring. More precisely, the weighted forests generatable by wCCG are also generatable by weighted simple monadic context-free tree grammar (wsCFTG). If the rule system is restricted to application rules and composition rules of first degree, then the generatable weighted forests are exactly the regular weighted forests. Finally, when only application rules are allowed, then a proper subset of the regular weighted forests is generatable.

Tree generating context-free grammars and regular tree grammars are equivalent

We show that it is decidable whether the language generated by a given context-free grammar over a tree alphabet is a tree language. Furthermore, if the answer to this question is “yes”, then we can even effectively construct a regular tree grammar which generates that tree language.

The Component Hierarchy of Chain-free Cooperating Distributed Regular Tree Grammars Revisited

We denote by [Formula: see text] the class of tree languages generated by chain-free distributed regular tree grammars of [Formula: see text] components cooperating with terminal strategy. Dányi and Fülöp [2] showed that the hierarchy [Formula: see text], [Formula: see text] contains an infinite proper hierarchy. We improve their result showing that for each [Formula: see text], [Formula: see text].

Recursive tree grammar autoencoders

Machine learning on trees has been mostly focused on trees as input. Much less research has investigated trees as output, which has many applications, such as molecule optimization for drug discovery, or hint generation for intelligent tutoring systems. In this work, we propose a novel autoencoder approach, called recursive tree grammar autoencoder (RTG-AE), which encodes trees via a bottom-up parser and decodes trees via a tree grammar, both learned via recursive neural networks that minimize the variational autoencoder loss. The resulting encoder and decoder can then be utilized in subsequent tasks, such as optimization and time series prediction. RTG-AEs are the first model to combine three features: recursive processing, grammatical knowledge, and deep learning. Our key message is that this unique combination of all three features outperforms models which combine any two of the three. Experimentally, we show that RTG-AE improves the autoencoding error, training time, and optimization score on synthetic as well as real datasets compared to four baselines. We further prove that RTG-AEs parse and generate trees in linear time and are expressive enough to handle all regular tree grammars.

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.

Weighted parsing for grammar-based language models over multioperator monoids

We develop a general framework for weighted parsing which is built on top of grammar-based language models and employs multioperator monoids (M-monoids) as weight algebras. It generalizes previous work in that area, e.g., semiring parsing and weighted deductive parsing, and also covers applications outside the classical scope of parsing, e.g., algebraic dynamic programming. More specifically, we introduce weighted RTG-based language models (where RTG stands for regular tree grammar) and define the M-monoid parsing problem. We show an algorithm which is supposed to solve this problem and prove in detail that, for a large class of weighted RTG-based language models, the algorithm terminates and, indeed, solves this problem. We compare our algorithm with semiring parsing and weighted deductive parsing regarding applicability and complexity.

Error-Tolerant Anytime Approach to Plan Recognition Using a Particle Filter

Classical plan recognition approaches require restrictive assumptions and are generally off-line. However, many real-world plan recognition applications must deal with real-time constraints, noisy information, temporal relations in actions, agent preferences, and so on. Many existing approaches have tried to relax assumptions, but none can deal with the above-cited needs. This paper proposes an extension of previous works on plan recognition based on plan tree grammar. Our anytime topdown approach uses a particle filter. This approach manages to give a quick reliable solution to the plan recognition problem while dealing with noisy observations and without reducing the expressiveness of plan libraries. Empirical results on simulated problems show the efficiency of our approach.

Principal abstract families of weighted tree languages

We introduce semiring-weighted regular tree grammars with storage where the semiring is complete. We show that the class of weighted tree languages generated by them is a principal abstract family of weighted tree languages provided that the semiring is commutative, the rank of symbols occurring in trees is bounded by a global parameter, and that the storage is finitely encoded and contains a reset instruction. Moreover, we prove the same statement for the iterated pushdown (instead of finitely encoded storage containing a reset instruction).

Congruence-based proofs of the recognizability theorems for free many-sorted algebras

Abstract We generalize several recognizability theorems for free single-sorted algebras to free many-sorted algebras and provide, in a uniform way and without using either regular tree grammars or tree automata, purely algebraic proofs of them based on congruences.

Idempotent Anti-unification

In this article, we address two problems related to idempotent anti-unification. First, we show that there exists an anti-unification problem with a single idempotent symbol that has an infinite minimal complete set of generalizations. It means that anti-unification with a single idempotent symbol has infinitary or nullary generalization type, similar to anti-unification with two idempotent symbols, shown earlier by Loïc Pottier. Next, we develop an algorithm that takes an arbitrary idempotent anti-unification problem and computes a representation of its solution set in the form of a regular tree grammar. The algorithm does not depend on the number of idempotent function symbols in the input terms. The language generated by the grammar is the minimal complete set of generalizations of the given anti-unification problem, which implies that idempotent anti-unification is infinitary.

Universal Tree Source Coding Using Grammar-Based Compression

The problem of universal source coding for binary trees is considered. Zhang, Yang, and Kieffer derived upper bounds on the average-case redundancy of codes based on directed acyclic graph (DAG) compression for binary tree sources with certain properties. In this paper, a natural class of binary tree sources is presented such that the demanded properties are fulfilled. Moreover, for both subclasses considered in the paper of Zhang, Yang, and Kieffer, their result is improved by deriving bounds on the maximal pointwise redundancy (or worst-case redundancy) instead of the average-case redundancy. Finally, using context-free tree grammars instead of DAGs, upper bounds on the maximal pointwise redundancy for certain binary tree sources are derived. This yields universal codes for new classes of binary tree sources.

Linear context-free tree languages and inverse homomorphisms

We prove that the class of linear context-free tree languages is not closed under inverse linear tree homomorphisms. In fact, we prove a stronger result: we encode Dyck words into a linear context-free tree language and prove that its preimage under a certain linear tree homomorphism cannot be generated by any context-free tree grammar. A positive result can still be obtained: the linear monadic context-free tree languages are closed under inverse linear tree homomorphisms.

An algebraic language for RNA pseudoknots comparison

BackgroundRNA secondary structure comparison is a fundamental task for several studies, among which are RNA structure prediction and evolution. The comparison can currently be done efficiently only for pseudoknot-free structures due to their inherent tree representation.ResultsIn this work, we introduce an algebraic language to represent RNA secondary structures with arbitrary pseudoknots. Each structure is associated with a unique algebraic RNA tree that is derived from a tree grammar having concatenation, nesting and crossing as operators. From an algebraic RNA tree, an abstraction is defined in which the primary structure is neglected. The resulting structural RNA tree allows us to define a new measure of similarity calculated exploiting classical tree alignment.ConclusionsThe tree grammar with its operators permit to uniquely represent any RNA secondary structure as a tree. Structural RNA trees allow us to perform comparison of RNA secondary structures with arbitrary pseudoknots without taking into account the primary structure.

Colored operads, series on colored operads, and combinatorial generating systems

We introduce bud generating systems, which are used for combinatorial generation. They specify sets of various kinds of combinatorial objects, called languages. They can emulate context-free grammars, regular tree grammars, and synchronous grammars, allowing us to work with all these generating systems in a unified way. The theory of bud generating systems uses colored operads. Indeed, an object is generated by a bud generating system if it satisfies a certain equation in a colored operad. To compute the generating series of the languages of bud generating systems, we introduce formal power series on colored operads and several operations on these. Series on colored operads are crucial to express the languages specified by bud generating systems and allow us to enumerate combinatorial objects with respect to some statistics. Some examples of bud generating systems are constructed; in particular to specify some sorts of balanced trees and to obtain recursive formulas enumerating these.

On Transforming Narrowing Trees into Regular Tree Grammars Generating Ranges of Substitutions

The grammar representation of a narrowing tree for a syntactically deterministic conditional term rewriting system and a pair of terms is a regular tree grammar that generates expressions for substitutions obtained by all possible innermost-narrowing derivations that start with the pair and end with particular non-narrowable terms. In this paper, under a certain syntactic condition, we show a transformation of the grammar representation of a narrowing tree into another regular tree grammar that overapproximately generates the ranges of ground substitutions generated by the grammar representation. In our previous work, such a transformation is restricted to the ranges w.r.t. a given single variable, and thus, the usefulness is limited. We extend the previous transformation by representing the range of a ground substitution as a tuple of terms, which is obtained by the coding for finite trees. We show a precise definition of the transformation and prove that the language of the transformed regular tree grammar is an overapproximation of the ranges of ground substitutions generated by the grammar representation. We leave an experiment to evaluate the usefulness of the transformation as future work.

Inclusion between the frontier language of a non-deterministic recursive program scheme and the Dyck language is undecidable

In 1970's, Nivat studied recursive program schemes (a.k.a order-1 higher-order recursion schemes in modern terminology), first-order tree grammars for generating possibly infinite trees. We consider the inclusion problem between the frontier language of a non-deterministic recursive program scheme (equivalently, an order-2 word language or indexed language) and the Dyck language, and prove that it is undecidable by a reduction from the undecidability of Hilbert's 10th problem. Essentially the same result has recently been proved by Uezato and Minamide, but our proof is arguably more direct, demonstrating the expressive power of higher-order grammars.