Reachability Analysis with State-Compatible Automata

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.

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.

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

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.

On a case study, we present a new approach for verifying cryptographic protocols, based on rewriting and on tree automata techniques. Protocols are operationally described using Term Rewriting Systems and the initial set of communication requests is described by a tree automaton. Starting from these two representations, we automatically compute an over-approximation of the set of exchanged messages (also recognized by a tree automaton). Then, proving classical properties like confidentiality or authentication can be done by automatically showing that the intersection between the approximation and a set of prohibited behaviors is the empty set. Furthermore, this method enjoys a simple and powerful way to describe intruder work, the ability to consider an unbounded number of parties, an unbounded number of interleaved sessions, and a theoretical property ensuring safeness of the approximation.KeywordsMutual AuthenticationCritical PairCryptographic ProtocolTree AutomatonTree 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.

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

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.

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 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. >

In this paper we address the problem of the automatic generation of visual languages from a sample set of visual sentences. We present an improvement of the inference module of the VLG system which was originally conceived for the generation of iconic languages [11]. With this extension any kind of visual languages, like diagrams and forms, can be considered. To this aim, we present an inference algorithm for the class of Boundary SR grammars. These grammars are a subclass of the SR grammars with the interesting property of confluence, which extends the concept of context-freeness to the case of nonlinear grammars. Moreover, in spite of the simplicity and naturalness of the formalism, the generative power of this class is sufficient to specify interesting visual languages. The inference algorithm exploits an elegant characterization of Boundary SR languages in terms of tree and string languages. More precisely, we show that a visual language is a Boundary SR language if and only if it can be defined as a regular tree language and a set of properly associated string languages. Based on this result, the problem of identifying structural properties in a diagrammatic visual sentence is brought back to the detection of structural properties in tree and string languages. The main advantage coming from the use of a grammatical inference technique in visual language specification is that the designer only needs to specify a set of visual sentences that he/she feels to sufficiently exemplify the intended target language.

We investigate regular tree languages exact learning from positive examples and membership queries. Input data are trees of the language to infer. The learner computes new trees from the inputs and asks to the oracle whether or not they belong to the language. From the answers, the learner may ask further membership queries until he finds the correct grammar that generates the target language. This paradigm was introduced by Angluin in the seminal work [1] for the case of regular word language. Neither negative examples, equivalence queries nor counter examples are allowed in this paradigm. We describe an efficient algorithm which is polynomial in the size of the examples for learning the whole class of regular tree languages. The convergence is insured when the set of examples contains a representative sample of the language to guess. A finite subset e of a regular tree language £ is representative for L if every transition of the minimal tree automaton for L is used at least once for the derivation of an element of the set e.

Tree-automatic scattered linear orders

Abstracting cryptographic protocols with tree automata

We study query rewriting using views (QRV) for XML. Our queries and views are regular tree languages (RTLs) represented by tree automata over marked alphabets, where the markers serve as “node selectors”. We formally define query rewriting using views for RTLs and give an automata-based algorithm to compute the maximally contained rewriting. The formalism we use is equal in power with Monadic Second Order (MSO) logic, and our algorithm for computing QRV is the first to target this expressive class. Furthermore we prove a tight lower bound, thus showing that our algorithm is optimal. Another strength of our automata-based approach is that we are able to cast computing QRV into executing a sequence of intuitive operations on automata, thus rendering our approach practical as it can be easily implemented utilizing off-the-shelf automata toolboxes. Finally, we generalize our framework to account for more complex queries in the spirit of the FOR clause in XQuery. For this generalization as well, we give an optimal algorithm for computing the maximally contained rewriting of queries using views.KeywordsXMLView-Based RewritingTree Automata

Closure of Tree Automata Languages under Innermost Rewriting