Tree Automata, (Dis-)Equality Constraints and Term Rewriting: What\u27s New?

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

Collapsible Pushdown Automata and Recursion Schemes

International audienceWe consider recursion schemes (not assumed to be homogeneously typed, and hence not necessarily safe) and use them as generators of (possibly infinite) ranked trees. A recursion scheme is essentially a finite typed {deterministic term} rewriting system that generates, when one applies the rewriting rules ad infinitum, an infinite tree, called its value tree. A fundamental question is to provide an equivalent description of the trees generated by recursion schemes by a class of machines. In this paper we answer this open question by introducing collapsible pushdown automata (CPDA), which are an extension of deterministic (higher-order) pushdown automata. A CPDA generates a tree as follows. One considers its transition graph, unfolds it and contracts its silent transitions, which leads to an infinite tree which is finally node labelled thanks to a map from the set of control states of the CPDA to a ranked alphabet. Our contribution is to prove that these two models, higher-order recursion schemes and collapsible pushdown automata, are equi-expressive for generating infinite ranked trees. This is achieved by giving an effective transformations in both directions

Proving Looping and Non-Looping Non-Termination by Finite Automata

A new technique is presented to prove non-termination of term rewriting. The basic idea is to find a non-empty regular language of terms that is closed under rewriting and does not contain normal forms. It is automated by representing the language by a tree automaton with a fixed number of states, and expressing the mentioned requirements in a SAT formula. Satisfiability of this formula implies non-termination. Our approach succeeds for many examples where all earlier techniques fail, for instance for the S-rule from combinatory logic

Towards Static Analysis of Functional Programs using Tree Automata Completion

This paper presents the first step of a wider research effort to apply tree automata completion to the static analysis of functional programs. Tree Automata Completion is a family of techniques for computing or approximating the set of terms reachable by a rewriting relation. The completion algorithm we focus on is parameterized by a set E of equations controlling the precision of the approximation and influencing its termination. For completion to be used as a static analysis, the first step is to guarantee its termination. In this work, we thus give a sufficient condition on E and T(F) for completion algorithm to always terminate. In the particular setting of functional programs, this condition can be relaxed into a condition on E and T(C) (terms built on the set of constructors) that is closer to what is done in the field of static analysis, where abstractions are performed on data.Comment: Proceedings of WRLA'14. 201

Verifying Temporal Regular Properties of Abstractions of Term Rewriting Systems

The tree automaton completion is an algorithm used for proving safety properties of systems that can be modeled by a term rewriting system. This representation and verification technique works well for proving properties of infinite systems like cryptographic protocols or more recently on Java Bytecode programs. This algorithm computes a tree automaton which represents a (regular) over approximation of the set of reachable terms by rewriting initial terms. This approach is limited by the lack of information about rewriting relation between terms. Actually, terms in relation by rewriting are in the same equivalence class: there are recognized by the same state in the tree automaton. Our objective is to produce an automaton embedding an abstraction of the rewriting relation sufficient to prove temporal properties of the term rewriting system. We propose to extend the algorithm to produce an automaton having more equivalence classes to distinguish a term or a subterm from its successors w.r.t. rewriting. While ground transitions are used to recognize equivalence classes of terms, epsilon-transitions represent the rewriting relation between terms. From the completed automaton, it is possible to automatically build a Kripke structure abstracting the rewriting sequence. States of the Kripke structure are states of the tree automaton and the transition relation is given by the set of epsilon-transitions. States of the Kripke structure are labelled by the set of terms recognized using ground transitions. On this Kripke structure, we define the Regular Linear Temporal Logic (R-LTL) for expressing properties. Such properties can then be checked using standard model checking algorithms. The only difference between LTL and R-LTL is that predicates are replaced by regular sets of acceptable terms

Non-termination using Regular Languages

We describe a method for proving non-looping non-termination, that is, of term rewriting systems that do not admit looping reductions. As certificates of non-termination, we employ regular (tree) automata.Comment: Published at International Workshop on Termination 201

Unified Analysis of Collapsible and Ordered Pushdown Automata via Term Rewriting

We model collapsible and ordered pushdown systems with term rewriting, by encoding higher-order stacks and multiple stacks into trees. We show a uniform inverse preservation of recognizability result for the resulting class of term rewriting systems, which is obtained by extending the classic saturation-based approach. This result subsumes and unifies similar analyses on collapsible and ordered pushdown systems. Despite the rich literature on inverse preservation of recognizability for term rewrite systems, our result does not seem to follow from any previous study.Comment: in Proc. of FRE

Rigid Tree Automata and Applications

International audienceWe 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 some inequality and 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 an RTA language under a restricted family of term rewriting systems, whereas this closure is not an 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, in particular for the verification of communicating processes with several local non rewritable memories, like security protocols. Finally, a comparison of RTA with several classes of tree automata with local and global equality tests, with dag automata and Horn clause formalisms is also provided