Marrying words and trees

Closure properties of linear context-free tree languages with an application to optimality theory

Over the last decades the theory of finite automata on infinite objects has been an important source of tools for the specification and the verification of computer programs. Trees are more suitable than words to model nondeterminism and thus concurrency. In the literature, there are several examples of acceptance conditions that have been proposed for automata on infinite words and then have been fruitfully extended to infinite trees (Buchi, Rabin, and Muller conditions). The type of acceptance condition can influence both the succinctness of the corresponding class of automata and the complexity of the related decision problems. Here we consider, for automata on infinite trees, two acceptance conditions that are obtained by a relaxation of the Muller acceptance condition: the Landweber and the Muller-Superset conditions. We prove that Muller-Superset tree automata accept the same class of languages as Buchi tree automata, but using more succinct automata. Landweber tree automata, instead, define a class of languages which is not comparable with the one defined by Buchi tree automata. We prove that, for this class of automata, the emptiness problem is decidable in polynomial time, and thus we expand the class of automata with a tractable emptiness problem.

Tree automata with one memory have been introduced in 2001. They generalize both pushdown (word) automata and the tree automata with constraints of equality between brothers of Bogaert and Tison. Though it has a decidable emptiness problem, the main weakness of this model is its lack of good closure properties. We propose a generalization of the visibly pushdown automata of Alur and Madhusudan to a family of tree recognizers which carry along their (bottom-up) computation an auxiliary unbounded memory with a tree structure (instead of a symbol stack). In other words, these recognizers, called visibly Tree Automata with Memory (VTAM) define a subclass of tree automata with one memory enjoying Boolean closure properties. We show in particular that they can be determinized and the problems like emptiness, inclusion and universality are decidable for VTAM. Moreover, we propose an extension of VTAM whose transitions may be constrained by structural equality and disequality tests between memories, and show that this extension preserves the good closure and decidability properties.

Tree automata with one memory have been introduced in 2001. They generalize both pushdown (word) automata and the tree automata with constraints of equality between brothers of Bogaert and Tison. Though it has a decidable emptiness problem, the main weakness of this model is its lack of good closure properties. We propose a generalization of the visibly pushdown automata of Alur and Madhusudan to a family of tree recognizers which carry along their (bottom-up) computation an auxiliary unbounded memory with a tree structure (instead of a symbol stack). In other words, these recognizers, called Visibly Tree Automata with Memory (VTAM) define a subclass of tree automata with one memory enjoying Boolean closure properties. We show in particular that they can be determinized and the problems like emptiness, membership, inclusion and universality are decidable for VTAM. Moreover, we propose several extensions of VTAM whose transitions may be constrained by different kinds of tests between memories and also constraints a la Bogaert and Tison comparing brother subtrees in the tree in input. We show that some of these classes of constrained VTAM keep the good closure and decidability properties, and we demonstrate their expressiveness with relevant examples of tree languages.

Nested words introduced by Alur and Madhusudan are used to capture structures with both linear and hierarchical order, e.g. XML documents, without losing valuable closure properties. Furthermore, Alur and Madhusudan introduced automata and equivalent logics for both finite and infinite nested words, thus extending Büchi’s theorem to nested words. Recently, average and discounted computations of weights in quantitative systems found much interest. Here, we will introduce and investigate weighted automata models and weighted MSO logics for infinite nested words. As weight structures we consider valuation monoids which incorporate average and discounted computations of weights as well as the classical semirings. We show that under suitable assumptions, two resp. three fragments of our weighted logics can be transformed into each other. Moreover, we show that the logic fragments have the same expressive power as weighted nested word automata.Keywordsnested wordsweighted automataweighted logicsquantitative automatavaluation monoids

This article is inspired by two works from the early 90s. The first one is by Bogaert and Tison who considered a model of automata on finite ranked trees where one can check equality and disequality constraints between direct subtrees: they proved that this class of automata is closed under Boolean operations and that both the emptiness and the finiteness problem of the accepted language are decidable. The second one is by Niwinski who showed that one can compute the cardinality of any ω-regular language of infinite trees. Here, we generalise the model of automata of Tison and Bogaert to the setting of infinite binary trees. Roughly speaking we consider parity tree automata where some transitions are guarded and can be used only when the two direct sub-trees of the current node are equal/disequal. We show that the resulting class of languages encompasses the one of ω-regular languages of infinite trees while sharing most of its closure properties, in particular it is a Boolean algebra. Our main technical contribution is then to prove that it also enjoys a decidable cardinality problem. In particular, this implies the decidability of the emptiness problem.

In this article, we indicate how the category theoretical approach to tree automata, due to Betti and Kasangian, can be fruitfully combined with Walters’ categorical approach to context-free grammars to provide a simple way of establishing the well-known correspondence between context-free languages and the behaviors of non-deterministic tree automata. The connecting link between the two notions is provided by the theory of relational presheaves.

A new type of grammar for generating formal languages, called an indexed grammar, is presented. An indexed grammar is an extension of a context-free grammar, and the class of languages generated by indexed grammars has closure properties and decidability results similar to those for context-free languages. The class of languages generated by indexed grammars properly includes all context-free languages and is a proper subset of the class of context-sensitive languages. Several subclasses of indexed grammars generate interesting classes of languages.

Recently, the question has been raised whether the derivation tree languages of Minimalist grammars (MGs; [14, 16]) are closed under intersection with regular tree languages [4, 5]. Using a variation of a proof technique devised by Thatcher [17], I show that even though closure under intersection does not obtain, it holds for every MG and regular tree language that their intersection is identical to the derivation tree language of some MG modulo category labels. It immediately follows that the same closure property holds with respect to union, relative complement, and certain kinds of linear transductions. Moreover, enriching MGs with the ability to put regular constraints on the shape of their derivation trees does not increase the formalism’s weak generative capacity. This makes it straightforward to implement numerous linguistically motivated constraints on the Move operation.KeywordsMinimalist GrammarsDerivation Tree LanguagesClosure PropertiesRegular Tree LanguagesDerivational Constraints

In this survey we generalize some results on formal tree languages, tree grammars and tree automata by an algebraic treatment using semirings, fixed point theory, formal tree series and matrices. The use of these mathematical constructs makes definitions, constructions, and proofs more satisfactory from an mathematical point of view than the customary ones. The contents of this survey paper is indicated by the titles of the sections:<dl compact="compact"><dt>1.</dt><dd>Introduction</dd><dt>2.</dt><dd>Preliminaries</dd><dt>3.</dt><dd>Tree automata and systems of equations</dd><dt>4.</dt><dd>Closure properties and a Kleene Theorem for recognizable tree series</dd><dt>5.</dt><dd>Pushdown tree automata, algebraic tree systems, and a Kleene Theorem</dd><dt>6.</dt><dd>Tree series transducers</dd><dt>7.</dt><dd>Full abstract families of tree series</dd><dt>8.</dt><dd>Connections to formal power series</dd></dl>

There are many decision problems in automata theory (including membership, emptiness, emptiness of intersection, inclusion and universality problems) that for some classes of tree automata are NP-hard. The study of their parameterized complexity allows us to find new bounds of their non-polynomial time algorithmic behaviors. We present results of such a study for classical tree automata (TA), rigid tree automata (RTA), tree automata with global equality and disequality (TAGED) and t-DAG automata.KeywordsModel CheckParameterized ComplexityLinear Temporal LogicFinite AutomatonGlobal ConstraintThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We introduce context-free languages of morphisms in monoidal categories, extending recent work on the categorification of context-free languages, and regular languages of string diagrams. Context-free languages of string diagrams include classical context-free languages of words, trees, and hypergraphs, when instantiated over appropriate monoidal categories. We prove a representation theorem for context-free languages of string diagrams: every such language arises as the image under a monoidal functor of a regular language of string diagrams.

An operation of concatenation is defined for graphs. This allows strings to be viewed as expressions denoting graphs, and string languages to be interpreted as graph languages. For a class \(K\) of string languages, \({\rm Int}(K)\) is the class of all graph languages that are interpretations of languages from \(K\). For the classes REG and LIN of regular and linear context-free languages, respectively, \({\rm Int}({\rm REG}) = {\rm Int}({\rm LIN})\). \({\rm Int}({\rm REG})\) is the smallest class of graph languages containing all singletons and closed under union, concatenation and star (of graph languages). \({\rm Int}({\rm REG})\) equals the class of graph languages generated by linear HR (= Hyperedge Replacement) grammars, and \({\rm Int}(K)\) is generated by the corresponding \(K\)-controlled grammars. Two characterizations are given of the largest class \(K'\) such that \({\rm Int}(K') = {\rm Int}(K)\). For the class CF of context-free languages, \({\rm Int}({\rm CF})\) lies properly inbetween \({\rm Int}({\rm REG})\) and the class of graph languages generated by HR grammars. The concatenation operation on graphs combines nicely with the sum operation on graphs. The class of context-free (or equational) graph languages, with respect to these two operations, is the class of graph languages generated by HR grammars.

Indexed grammars are a generalization of context-free grammars and recognize a proper subset of context-sensitive languages. The class of languages recognized by indexed grammars are called indexed languages and they correspond to the languages recognized by nested stack automata. For example indexed grammars can recognize the language which is not context-free, but they cannot recognize which is context-sensitive. Indexed grammars identify a set of languages that are more expressive than context-free languages, while having decidability results that lie in between the ones of context-free and context-sensitive languages. In this work we study indexed grammars in order to formalize the relation between indexed languages and the other classes of languages in the Chomsky hierarchy. To this end, we provide a fixpoint characterization of the languages recognized by an indexed grammar and we study possible ways to abstract, in the abstract interpretation sense, these languages and their grammars into context-free and regular languages.

This work describes a formalization effort, using the Coq proof assistant, of fundamental results related to the classical theory of context-free grammars and languages. These include closure properties union, concatenation and Kleene star, grammar simplification elimination of useless symbols, inaccessible symbols, empty rules and unit rules, the existence of a Chomsky Normal Form for context-free grammars and the Pumping Lemma for context-free languages. The result is an important set of libraries covering the main results of context-free language theory, with more than 500 lemmas and theorems fully proved and checked. This is probably the most comprehensive formalization of the classical context-free language theory in the Coq proof assistant done to the present date, and includes the important result that is the formalization of the Pumping Lemma for context-free languages.