Irène Durand

Automata for the verification of monadic second-order graph properties

The model-checking problem for monadic second-order logic on graphs is fixed-parameter tractable with respect to tree-width and clique-width. The proof constructs finite automata from monadic second-order sentences. These automata recognize the terms over fixed finite signatures that define graphs satisfying the given sentences. However, this construction produces automata of hyper-exponential sizes, and is thus impossible to use in practice in many cases. To overcome this difficulty, we propose to specify the transitions of automata by programs instead of tables. Such automata are called fly-automata. By using them, we can check certain monadic second-order graph properties with limited quantifier alternation depth, that are nevertheless interesting for Graph Theory. We give explicit constructions of automata relative to graphs of bounded clique-width, and we report on experiments.

Decidable call-by-need computations in term rewriting

The theorem of Huet and Levy stating that for orthogonal rewrite systems (i) every reducible term contains a needed redex and (ii) repeated contraction of needed redexes results in a normal form if the term under consideration has a normal form, forms the basis of all results on optimal normalizing strategies for orthogonal rewrite systems. However, needed redexes are not computable in general. In the paper we show how the use of approximations and elementary tree automata techniques allows one to obtain decidable conditions in a simple and elegant way. Surprisingly, by avoiding complicated concepts like index and sequentiality we are able to cover much larger classes of rewrite systems. We also study modularity aspects of the classes in our hierarchy. It turns out that none of the classes is preserved under signature extension. By imposing various conditions we recover the preservation under signature extension. By imposing some more conditions we are able to strengthen the signature extension results to modularity for disjoint and constructor-sharing combinations.

Bottom-up rewriting is inverse recognizability preserving

For the whole class of linear term rewriting systems, we define bottom-up rewriting which is a restriction of the usual notion of rewriting. We show that bottom-up rewriting effectively inverse-preserves recognizability and analyze the complexity of the underlying construction. The Bottom-Up class (BU) is, by definition, the set of linear systems for which every derivation can be replaced by a bottom-up derivation. Membership to BU turns out to be undecidable; we are thus lead to define more restricted classes: the classes SBU(k), k ∈ N of Strongly Bottom-Up(k) systems for which we show that membership is decidable. We define the class of Strongly Bottom-Up systems by SBU = Uk∈N SBU(k). We give a polynomial sufficient condition for a system to be in SBU. The class SBU contains (strictly) several classes of systems which were already known to inverse preserve recognizability.

Autowrite: A Tool for Term Rewrite Systems and Tree Automata

Autowrite is an experimental software tool written in Common Lisp Oriented System (CLOS) which handles term rewrite systems and bottom-up tree automata. A graphical interface written using McCLIM, (the free implementation of the CLIM specification) frees the user of any Lisp knowledge. Software and documentation can be found at http://dept-info.labri.u-bordeaux.fr/~idurand/autowrite. Autowrite was initially designed to check call-by-need properties of term rewrite systems. For this purpose, it implements the tree automata constructions used in [J96,DM97,DM98,NT02] and many useful operations on terms, term rewrite systems and tree automata.

Autowrite: A Tool for Checking Properties of Term Rewriting Systems

Huet and Levy [6] showed that for the class of orthogonal term rewriting systems (TRSs) every term not in normal form contains a needed redex (i.e., a redex contracted in every normalizing rewrite sequence) and that repeated contraction of needed redexes results in a normal form if it exists. However, neededness is in general undecidable. In order to obtain a decidable approximation to neededness Huet and Levy introduced the subclass of strongly sequential TRSs and showed that strong sequentiality is a decidable property of orthogonal TRSs.

Left-linear Bounded TRSs are Inverse Recognizability Preserving

Bounded rewriting for linear term rewriting systems has been defined in (I. Durand, G. Senizergues, M. Sylvestre. Termination of linear bounded term rewriting systems. Proceedings of the 21st International Conference on Rewriting Techniques and Applications) as a restriction of the usual notion of rewriting. We extend here this notion to the whole class of left-linear term rewriting systems, and we show that bounded rewriting is eectively inverse-recognizability preserving. The bounded class (BO) is, by definition, the set of left-linear systems for which every derivation can be replaced by a bottom-up derivation. The class BO contains (strictly) several classes of systems which were already known to be inverse-recognizability preserving: the left-linear growing systems, and the inverse right-linear finite-path overlapping systems.

Fly-automata, their properties and applications

We address the concrete problem of implementing huge bottom-up term automata. Such automata arise from the verification of Monadic Second Order propositions on graphs of bounded tree-width or clique-width. This applies to graphs of bounded tree-width because bounded tree-width implies bounded clique-width. An automaton which has so many transitions that they cannot be stored in a transition table is represented be a fly-automaton in which the transition function is represented by a finite set of meta-rules. Fly-automata have been implemented inside the Autowrite software and experiments have been run in the domain of graph model checking.

On the Modularity of Deciding Call-by-Need

In a recent paper we introduced a new framework for the study of call by need computations. Using elementary tree automata techniques and ground tree transducers we obtained simple decidability proofs for a hierarchy of classes of rewrite systems that are much larger than earlier classes defined using the complicated sequentiality concept. In this paper we study the modularity of membership in the new hierarchy. Surprisingly, it turns out that none of the classes in the hierarchy is preserved under signature extension. By imposing various conditions we recover the preservation under signature extension. By imposing some more conditions we are able to strengthen the signature extension results to modularity for disjoint and constructor-sharing combinations.

Model-Checking by Infinite Fly-Automata

We present logic based methods for constructing XP and FPT graph algorithms, parameterized by tree-width or clique-width. We will use fly-automata introduced in a previous article. They make it possible to check properties that are not monadic second-order expressible because their states may include counters, so that their set of states may be infinite. We equip these automata with output functions, so that they can compute values associated with terms or graphs. We present tools for constructing easily algorithms by combining predefined automata for basic functions and properties.

Termination of linear bounded term rewriting systems

For the whole class of linear term rewriting systems and for each integer k, we define k-bounded rewriting as a restriction of the usual notion of rewriting. We show that the k-bounded uniform termination, the k-bounded termination, the inverse k-bounded uniform, and the inverse k-bounded problems are decidable. The k-bounded class (BO(k)) is, by definition, the set of linear systems for which every derivation can be replaced by a k-bounded derivation. In general, for BO(k) systems, the uniform (respectively inverse uniform) k-bounded termination problem is not equivalent to the uniform (resp. inverse uniform) termination problem, and the k-bounded (respectively inverse k-bounded) termination problem is not equivalent to the termination (respectively inverse termination) problem. This leads us to define more restricted classes for which these problems are equivalent: the classes BOLP(k) of k-bounded systems that have the length preservation property. By definition, a system is BOLP(k) if every derivation of length n can be replaced by a k-bounded derivation of length n. We define the class BOLP of bounded systems that have the length preservation property as the union of all the BOLP(k) classes. The class BOLP contains (strictly) several already known classes of systems: the inverse left-basic semi-Thue systems, the linear growing term rewriting systems, the inverse Linear-Finite-Path-Ordering systems, the strongly bottom-up systems.