Tree Automata with Memory, Visibility and Structural Constraints

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.

We define an extension of tree automata by adding some tests in the rules. The goal is to handle non linearity. We obtain a family which has good closure and decidability properties and we give some applications.KeywordsTree Automatanon linearity

Rigid tree automata and applications

Traditionally, data that has both linear and hierarchical structure, such as annotated linguistic data, is modeled using ordered trees and queried using tree automata. In this paper, we argue that nested words and automata over nested words offer a better way to capture and process the dual structure. Nested words generalize both words and ordered trees, and allow both word and tree operations. We study various classes of automata over nested words, and show that while they enjoy expressiveness and succinctness benefits over word and tree automata, their analysis complexity and closure properties are analogous to the corresponding word and tree special cases. In particular, we show that finite-state nested word automata can be exponentially more succinct than tree automata, and pushdown nested word automata include the two incomparable classes of context-free word languages and context-free tree languages.

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.

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>

International audience

Connections between Tree Automata and Term Rewriting are now well known. Whereas tree automata can be viewed as a subclass of ground rewrite systems, tree automata are successfully used as decision tools in rewriting theory. Furthermore, applications, including rewriting theory, have influenced the definition of new classes of tree automata. In this talk, we will first present a short and not exhaustive reminder of some fruitful applications of tree automata in rewriting theory. Then, we will focus on extensions of tree automata, specially tree automata with local or/and global (dis-)equality constraints: we will emphasize new results, compare different extensions, and sketch some applications.

Closure properties and decision problems of dag automata

We study two-way tree automata modulo equational theories. We deal with the theories of Abelian groups (ACUM), idempotent commutative monoids (ACUI), and the theory of exclusive-or (ACUX), as well as some variants including the theory of commutative monoids (ACU). We show that the one-way automata for all these theories are closed under union and intersection, and emptiness is decidable. For two-way automata the situation is more complex. In all these theories except ACUI, we show that two-way automata can be effectively reduced to one-way automata, provided some care is taken in the definition of the so-called push clauses. (The ACUI case is open.) In particular, the two-way automata modulo these theories are closed under union and intersection, and emptiness is decidable. We also note that alternating variants have undecidable emptiness problem for most theories, contrarily to the non-equational case where alternation is essentially harmless.KeywordsEquational TheoryPredicate SymbolFunctional TermTree AutomatonFunctional SupportThese 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 the class of Rigid Tree Automata (RTA), an extension of standard bottom-up automata on ranked trees with distinguished states called rigid. Rigid states define a restriction on the computation of RTA on trees: RTA can test for equality in subtrees reaching the same rigid state. RTA are able to perform local and global tests of equality between subtrees, non-linear tree pattern matching, and restricted disequality tests as well. Properties like determinism, pumping lemma, boolean closure, and several decision problems are studied in detail. In particular, the emptiness problem is shown decidable in linear time for RTA whereas membership of a given tree to the language of a given RTA is NP-complete. Our main result is the decidability of whether a given tree belongs to the rewrite closure of a RTA language under a restricted family of term rewriting systems, whereas this closure is not a RTA language. This result, one of the first on rewrite closure of languages of tree automata with constraints, is enabling the extension of model checking procedures based on finite tree automata techniques. Finally, a comparison of RTA with several classes of tree automata with local and global equality tests, and with dag automata is also provided.

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

Closure of Tree Automata Languages under Innermost Rewriting

Tree automata with equality constraints modulo equational theories