The HOM Problem is EXPTIME-Complete

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.

We present effective criteria for first-order definability of regular tree languages. It is known that over words the absence of modulo counting (the “noncounting property”) characterizes the expressive power of first-order logic (McNaughton, Schutzenberger), whereas non-counting regular tree languages exist which are not first-order definable. We present new conditions on regular tree languages (more precisely, on tree automata) which imply nondefinability in first-order logic. One method is based on tree homomorphisms which allow to deduce nondefinability of one tree language from nondefinability of another tree language. Additionly we introduce a structural property of tree automata (the socalled Λ-∨-patterns) which also causes tree languages to be undefinable in first-order logic. Finally, it is shown that this notion does not yet give a complete characterization of first-order logic over trees. The proofs rely on the method of Ehrenfeucht-Fraisse games.

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 of Tree Automata Languages under Innermost Rewriting

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. >

The generative capacity of combinatory categorial grammars as acceptors of tree languages is investigated. It is demonstrated that the such obtained tree languages can also be generated by simple monadic context-free tree grammars. However, the subclass of pure combinatory categorial grammars cannot even accept all regular tree languages. Additionally, the tree languages accepted by combinatory categorial grammars with limited rule degrees are characterized: If only application rules are allowed, then they can accept only a proper subset of the regular tree languages, whereas they can accept exactly the regular tree languages once first degree composition rules are permitted.

Regular tree languages are a popular device for reachability analysis over term rewrite systems, with many applications like analysis of cryptographic protocols, or confluence and termination analysis. At the heart of this approach lies tree automata completion, first introduced by Genet for left-linear rewrite systems. Korp and Middeldorp introduced so-called quasi-deterministic automata to extend the technique to non-left-linear systems. In this paper, we introduce the simpler notion of quasi-compatible automata, which are slightly more general than quasi-deterministic, compatible automata. This notion also allows us to decide whether a regular tree language is closed under rewriting, a problem which was not known to be decidable before. Several of our results have been formalized in the theorem prover Isabelle/HOL. This allows to certify automatically generated non-confluence and termination proofs that are using tree automata techniques.

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

Regular tree expressions are a formalism for describing regular tree languages, which can be accepted by a finite tree automaton as a standard model of computation. It was proved that the class of regular tree languages is a proper subclass of tree languages whose linear notations can be accepted by deterministic string pushdown automata. In this paper, we present a new algorithm for transforming regular tree expressions to equivalent real-time height-deterministic pushdown automata that accept the trees in their postfix notation.

We study the problem of learning regular tree languages from text. We show that the framework of function distinguishability as introduced in our ALT 2000 paper is generalizable from the case of string languages towards tree languages, hence providing a large source of identifiable classes of regular tree languages. Each of these classes can be characterized in various ways. Moreover, we present a generic inference algorithm with polynomial update time and prove its correctness. In this way, we generalize previous works of Angluin, Sakakibara and ourselves. Moreover, we show that this way all regular tree languages can be identified approximately.

Hypersubstitutions map operation symbols to terms of the corresponding arity. Any hypersubstitution can be extended to a mapping defined on the set Wτ(X) of all terms of type τ. If σ : {fi | i ∈ I} → Wτ(X) is a hypersubstitution and [Formula: see text] its canonical extension, then the set [Formula: see text] is a tree transformation where the original language and the image language are of the same type. Tree transformations of the type Tσ can be produced by tree transducers. Here Tσ is the graph of the function [Formula: see text]. Since the set of all hypersubstitutions of type r forms a semigroup with respect to the multiplication [Formula: see text], semigroup properties influence the properties of tree transformations of the form Tσ. For instance, if σ is idempotent, the relation Tσ is transitive (see [1]). Non-deterministic tree transducers produce tree transformations which are not graphs of some functions. If such tree transformations have the form Tσ, then σ is no longer a function. Therefore, there is some interest to study non-deterministic hypersubstitutions. That means, there are operation symbols which have not only one term of the corresponding arity as image, but a set of such terms. To define the extensions of non-deterministic hypersubstitutions, we have to extent the superposition operations for terms to a superposition defined on sets of terms. Let [Formula: see text] be the power set of the set of all n-ary terms of type τ. Then we define a superposition operation [Formula: see text] and get a heterogeneous algebra [Formula: see text] (ℕ+ is the set of all positive natural numbers), which is called the power clone of type τ. We prove that the algebra [Formula: see text] satisfies the well-known clone axioms (C1), (C2), (C3), where (C1) is the superassociative law (see e.g. [5], [4]). It turns out that the extensions of non-deterministic hypersubstitutions are precisely those endomorphisms of the heterogeneous algebra [Formula: see text] which preserve unions of families of sets. As a consequence, to study tree transformations of the form Tσ, where σ is a non-deterministic hypersubstitution, one can use the structural properties of non-deterministic hypersubstitutions. Sets of terms of type τ are tree languages in the sense of [3] and the operations [Formula: see text] are operations on tree languages. In [3] also another kind of superposition of tree languages is introduced which generalizes the usual complex product of subsets of the universe of a semigroup. We show that the extensions of non-deterministic hypersubstitutions are not endomorphisms with respect to this kind of superposition.

We consider the framework of regular tree model checking where sets of configurations of a system are represented by regular tree languages and its dynamics is modeled by a term rewriting system (or a regular tree transducer). We focus on the computation of the reachability set R*(L) where R is a regular tree transducer and L is a regular tree language. The construction of this set is not possible in general. Therefore, we present a general acceleration technique, called regular tree widening which allows to speed up the convergence of iterative fixpoint computations in regular tree model checking. This technique can be applied uniformly to various kinds of transformations. We show the application of our framework to different analysis contexts: verification of parameterized tree networks and data-flow analysis of multithreaded programs. Parametrized networks are modeled by relabeling tree transducers, and multithreaded programs are modeled by term rewriting rules encoding transformations on control structures. We prove that our widening technique can emulate many existing algorithms for special classes of transformations and we show that it can deal with transformations beyond the scope of these algorithms.

Given a term rewriting system R and a normalizable term t, a redex is needed if in any reduction sequence of t to a normal for m, this redex will be contracted. Roughly, R is sequential if there is an optimal reduction strategy in which only needed redexes are contracted. More generally, G. Huet and J.-J. Levy (1991) define the sequentiality of a predicate P on partially evaluated terms. We show that the sequentiality of P is definable in SkS, the second order monadic logic with k: successors, provided P is definable in SkS. We derive several known an new consequences of this remark: strong sequentiality, as defined by Huet and Levy, of a left linear (possibly overlapping) rewrite system is decidable; NV sequentiality, as defined by M. Oyamaguchi (1993), is decidable, even in the case of overlapping rewrite systems; sequentiality of any linear shallow rewrite system is decidable. Then we describe a direct construction of an automaton recognizing the set of terms that have needed redexes, which again, yields immediate consequences: strong sequentiality of possibly overlapping linear rewrite systems is decidable in EXPTIME; for strongly sequential rewrite systems, needed redexes can be read directly on the automaton.

Test Sets for the Universal and Existential Closure of Regular Tree Languages