Rigid tree automata and applications

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.

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.

We define a new class of tree automata with constraints and prove decidability of the emptiness problem for this class in exponential time. As a consequence, we obtain several EXPTIME-completeness results for problems on images of regular tree languages under tree homomorphisms, like set inclusion, regularity (HOM problem), and finiteness of set difference. Our result also has implications in term rewriting, since the set of reducible terms of a term rewrite system can be described as the image of a tree homomorphism. In particular, we prove that inclusion of sets of normal forms of term rewrite systems can be decided in exponential time. Analogous consequences arise in the context of XML typechecking, since types are defined by tree automata and some type transformations are homomorphic.

We present a decision procedure, based on tree automata techniques, for satisfiability of systems of set constraints including negated subset relationships. This result extends all previous works on set constraints solving and solves a problem which was left open by L. Bachmair et al. (1993). We prove in a constructive way that a non empty set of solutions always contains a regular solution, that is a tuple of regular tree languages. Moreover, we think that the new class of tree automata described here could be interesting in its own. >

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.

Closure properties and decision problems of dag automata

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, 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.

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.

The emptiness problem for tree automata with at least one global disequality constraint is NP-hard

The notion of comparative similarity ‘X is more similar or closer to Y than to Z’ has been investigated in both foundational and applied areas of knowledge representation and reasoning, e.g., in concept formation, similarity-based reasoning and areas of bioinformatics such as protein sequence alignment. In this paper we analyse the computational behaviour of the ‘propositional’ logic with the binary operator ‘closer to a set τ 1 than to a set τ 2’ and nominals interpreted over various classes of distance (or similarity) spaces. In particular, using a reduction to the emptiness problem for certain tree automata, we show that the satisfiability problem for this logic is ExpTime-complete for the classes of all finite symmetric and all finite (possibly non-symmetric) distance spaces. For finite subspaces of the real line (and higher dimensional Euclidean spaces) we prove the undecidability of satisfiability by a reduction of the solvability problem for Diophantine equations. As our ‘closer’ operator has the same expressive power as the standard operator > of conditional logic, these results may have interesting implications for conditional logic as well.KeywordsModal LogicDescription LogicComparative SimilarityDiophantine EquationDistance SpaceThese 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.

Automata-theoretic decision of timed games

We consider graph decision problems on partial 3-trees that can be solved by a finite-state, leaf-to-root tree automaton processing a width-3 tree decomposition of the given graph. The class of yes-instances of such a problem is said to be recognizable by the tree automaton. In this paper we show that any such class of recognizable partial 3-trees is definable by a sentence in CMS logic. Here, CMS logic is the extension of Monadic Second-order logic with predicates to count the cardinality of sets modulo fixed integers. We also generalize this result to show that recognizability (by a tree automaton that processes width-k tree decompositions) implies CMS-definability for k -connected partial k -trees. The converse—definability implies recognizability—is known to hold for any class of partial k -trees, and even for any graph class with an appropriate definition of recognizability. It has been conjectured that recognizability implies definability over partial k -trees; but a proof was previously known only for k ≤ 2 . This paper proves the conjecture, and hence the equivalence of definability and recognizability, over partial 3-trees and k -connected partial k -trees. We use techniques that may lead to a proof of this equivalence over all partial k -trees.

We present a tool for checking the sufficient completeness of left-linear, order-sorted equational specifications modulo associativity, commutativity, and identity. Our tool treats this problem as an equational tree automata decision problem using the tree automata library CETA, which we also introduce in this paper. CETA implements a semi-algorithm for checking the emptiness of a class of tree automata that are closed under Boolean operations and an equational theory containing associativity, commutativity and identity axioms. Though sufficient completeness for this class of specifications is undecidable in general, our tool is a decision procedure for subcases known to be decidable, and has specialized techniques that are effective in practice for the general case.KeywordsTree AutomatonInductive TheoremEmptiness ProblemConstructor TermRegular Tree LanguageThese 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.

Tree automata with equality constraints modulo equational theories