Decidability for Left-Linear Growing Term Rewriting Systems

A term rewriting system is called growing if each variable occurring on both the left-hand side and the right-hand side of a rewrite rule occurs at depth zero or one in the left-hand side. Jacquemard showed that the reachability and the sequentiality of linear (i.e., left-right-linear) growing term rewriting systems are decidable. In this paper we show that Jacquemard's result can be extended to left-linear growing rewriting systems that may have right-nonlinear rewrite rules. This implies that the reachability and the joinability of some class of right-linear term rewriting systems are decidable, which improves the results for right-ground term rewriting systems by Oyamaguchi. Our result extends the class of left-linear term rewriting systems having a decidable call-by-need normalizing strategy. Moreover, we prove that the termination property is decidable for almost orthogonal growing term rewriting systems.     

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 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.     

Decidability of the confluence of finite ground term rewrite systems and of other related term rewrite systems

The aim of this paper is to propose an algorithm to decide the confluence of finite ground term rewrite systems. Actually a more general class of possibly infinite ground term rewrite systems is studied. It is well known that the confluence is not decidable for general term rewrite systems, but this paper proves it is for ground term rewrite systems following a conjecture made by Huet and Oppen in their survey. The result is also applied to the confluence of left-linear and right-ground term rewrite systems. We also sketch an algorithm for checking this property. This algorithm is based on tree automata and tree transducers. Here, we regard them as rewrite systems and specialists in automata theory would translate that easily in their language.     

Approximations for Strategies and Termination

The theorem of Huet and Lévy 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 illustrate, based on the framework introduced in [6], how the use of approximations and their associated tree automata results allows one to obtain decidable conditions in a simple and elegant way.

We further show how the very same ideas can be used to improve [18] the dependency pair method of Arts and Giesl [1] for proving termination of rewrite systems automatically. More precisely, we show how approximations and tree automata techniques provide a better estimation of the dependency graph. This graph determines the ordering constraints that have to be solved in order to conclude termination. Furthermore, we present a new estimation of the dependency graph that does not rely on computationally expensive tree automata techniques.     

Match-bounds revisited

The use of automata techniques to prove the termination of string rewrite systems and left-linear term rewrite systems is advocated by Geser et al. in a recent sequence of papers. We extend their work to non-left-linear rewrite systems. The key to this extension is the introduction of so-called raise rules and the use of tree automata that are not quite deterministic. Furthermore, to increase the applicability of the method we show how it can be incorporated into the dependency pair framework. To achieve this we introduce two new enrichments which take the special properties of dependency pair problems into account.     

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 [F. Jacquemard. Decidable approximations of term rewriting systems. In Proc. 7th RTA, volume 1103 of LNCS, pages 362–376, 1996; I. Durand and A. Middeldorp. Decidable call by need computations in term rewriting (extended abstract). In Proc. 14th CADE, volume 1249 of LNAI, pages 4–18, 1997; Irène Durand and Aart Middeldorp. On the complexity of deciding call-by-need. Technical Report 1196–98, LaBRI, 1998; T. Nagaya and Y. Toyama. Decidability for left-linear growing term rewriting systems. Information and Computation, 178(2):499–514, 2002] and many useful operations on terms, term rewrite systems and tree automata.     

Decidable call-by-need computations in term rewriting

The theorem of Huet and Lévy 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.     

On Symbolic Verification of Weakly Extended PAD

We consider the verification problem of a class of infinite-state systems called wPAD. These systems can be used to model programs with (possibly recursive) procedure calls and dynamic creation of parallel processes. They correspond to PAD models extended with an acyclic finite-state control unit, where PAD models can be seen as combinations of prefix rewrite systems (pushdown systems) with context-free multiset rewrite systems (synchronization-free Petri nets). Recently, we have presented symbolic reachability techniques for the class of PAD based on the use of a class of unranked tree automata. In this paper, we generalize our previous work to the class wPAD which is strictly larger than PAD. This generalization brings a positive answer to an open question on decidability of the model checking problem for wPAD against EF logic. Moreover, we show how symbolic reachability analysis of wPAD can be used in (under) approximate analysis of Synchronized PAD, a (Turing) powerful model for multithreaded programs (with unrestricted synchronization between parallel processes). This leads to a pragmatic approach for detecting the presence of erroneous behaviors in these models based on the bounded reachability paradigm where the notion of bound considered here is the number of synchronization actions.     

右线性文法与有限自动机等价性的一个新证明

迄今为止,左、右线性文法与有限自动机的等价性都是通过相互模拟构造来证明的。文章首先引入字母表上的右线性方程组及其最小解的概念,证明了最小解的存在性与有效可解性,描述了最小解的结构;其次通过右线性方程组及其最小解,证明了右线性文法与有限自动机的等价性。完全类似地,可以引入字母表上的左线性方程组及其最小解,并且证明左线性文...     

Tree automata with equality constraints modulo equational theories

This paper presents new classes of tree automata combining automata with equality test and automata modulo equational theories. We believe that these classes have a good potential for application in e.g. software verification. These tree automata are obtained by extending the standard Horn clause representations with equational conditions and rewrite systems. We show in particular that a generalized membership problem (extending the emptiness problem) is decidable by proving that the saturation of tree automata presentations with suitable paramodulation strategies terminates. Alternatively our results can be viewed as new decidable classes of first-order formula.     

On sufficient-completeness and related properties of term rewriting systems

Summary The decidability of the sufficient completeness property of equational specifications satisfying certain conditions is shown. In addition, the decidability of the related concept of quasi-reducibility of a term with respect to a set of rules is proved. Other results about irreducible ground terms of a term rewriting system also follow from a key technical lemma used in these decidability proofs; this technical lemma states that there is a finite bound on the substitutions of ground terms that need to be considered in order to check for a given term, whether the result obtained by any substitution of ground terms into the term is irreducible. These results are first shown for untyped systems and are subsequently extended to typed systems.Partially supported by the National Science Foundation Grant no. DCR-8408461     

On LR(k) grammars and languages

Many different definitions for LR(k) grammars exist in the literature. One of these definitions is chosen and many important implications are drawn from it. In particular, the LR(k) characterization theorem provides valuable information about chains of derivations. The LR(0) languages are then characterized by acceptance by deterministic pushdown automata with a special termination condition, by a condition on the strings in the language, and set theoretically. Important closure properties of the LR(0) languages and a related class of languages are then examined. These are used to examine some decidability questions relating to the class of LR languages. One of these questions is shown to be equivalent to the equality problem for deterministic pushdown automata.A survey of other LR(k) definitions is given and the exact differences are characterized. On the basis of this analysis, justification for the choice of definition used here is provided.     

ACTAS : A System Design for Associative and Commutative Tree Automata Theory

ACTAS is an integrated system for manipulating associative and commutative tree automata (AC-tree automata for short), that has various functions such as for Boolean operations of AC-tree automata, computing rewrite descendants, and solving emptiness and membership problems. In order to deal with high-complexity problems in reasonable time, over- and under-approximation algorithms are also equipped. Such functionality enables us automated verification of safety property in infinite state models, that is helpful in the domain of, e.g. network security, in particular, for security problems of cryptographic protocols allowing an equational property. In runtime of model construction, a tool support for analysis of state space expansion is provided. The intermediate status of the computation is displayed in numerical data table, and also the line graphs are generated. Besides, a graphical user interface of the system provides us a user-friendly environment for handy use.     

Tree automata with one memory set constraints and cryptographic protocols

We introduce a class of tree automata that perform tests on a memory that is updated using function symbol application and projection. The language emptiness problem for this class of tree automata is shown to be in DEXPTIME.We also introduce a class of set constraints with equality tests and prove its decidability by completion techniques and a reduction to tree automata with one memory.Finally, we show how to apply these results to cryptographic protocols. We introduce a class of cryptographic protocols and show the decidability of secrecy for an arbitrary number of agents and an arbitrary number of (concurrent or successive) sessions, provided that only a bounded number of new data is generated. The hypothesis on the protocol (a restricted copying ability) is shown to be necessary: without this hypothesis, we prove that secrecy is undecidable, even for protocols without nonces.     

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

Finite test sets are a useful tool for deciding the membership problem for the universal closure of a given tree language, that is, for deciding whether a term has all its ground instances in the given language. A uniform test set for the universal closure must serve the following purpose: In order to decide membership of a term, it is sufficient to check whether all its test set instances belong to the underlying language. A possible application, and our main motivation, is ground reducibility, an essential concept for many approaches to inductive reasoning. Ground reducibility modulo some rewrite system is membership in the universal closure of the set of reducible ground terms. Here, test sets always exist, and several algorithmic approaches are known. The resulting sets, however, are often unnecessarily large. In this paper we consider regular languages and linear closure operators. We prove that universal as well as existential closure, defined analogously, preserve regularity. By relating test sets to tree automata and to appropriate congruence relations, we show how to characterize, how to compute, and how to minimize ground and non-ground test sets. In particular, optimal solutions now replace previous ad hoc approximations for the ground reducibility problem.     

Narrowing-based simulation of term rewriting systems with extra variables and its Termination Proof

Term rewriting systems (TRSs) extended by allowing to contain extra variables in their rewrite rules are called EV-TRSs. They are ill-natured since every one-step reduction by their rules with extra variables is infinitely branching and they are not terminating. To solve these problems, this paper shows that narrowing can simulate reduction sequences of EV-TRSs as narrowing sequences starting from ground terms. We prove the soundness of ground narrowing sequences for the reduction sequences. We prove the completeness for the case of right-linear systems, and also for the case that any redex reduced in the reduction sequence is not introduced by means of extra variables. Moreover, we give a method to prove the termination of the simulation, extending the dependency pair method to prove termination of TRSs, into that of narrowing on EV-TRSs starting from ground terms. We show that the method is useful for right-linear or constructor systems.     

Some Undecidable Approximations of TRSs

In this paper we study the decidability of reachability, normalisation, and neededness in n-shallow and n-growing TRSs. In an n-growing TRS, a variable that occurs both on the left- and right-hand side of a rewrite rule must be at depth n on the left-hand side and at depth greater than n on the right-hand side. In an n-shallow TRS, a variable that occurs both on the left- and right-hand side of a rewrite rule must be at depth n on both sides.The n-growing and n-shallow TRSs are generalisations of the growing and shallow TRSs as introduced by Jacquemard and Comon. For both shallow and growing TRSs reachability, normalisation, and (in the orthogonal case) neededness are decidable. However, as we show, these results do not generalise to n-growing and n-shallow TRSs. Consequently, no algorithm exists that performs a needed reduction strategy in n-growing or n-shallow TRSs.     

Real-Time Verification Techniques for Untimed Systems

We show that verification techniques for timed automata based on the Alur and Dill region-graph construction can be applied to much more general kinds of systems, including asynchronous untimed systems over unbounded integer variables. We follow this approach in proving that the model-checking problem for the n-process Bakery algorithm is decidable, for any fixed n. We believe this is the first decidability proof for this problem to appear in the literature.     

Transducer-based analysis of cryptographic protocols

Cryptographic protocols can be divided into (1) protocols where the protocol steps are simple from a computational point of view and can thus be modeled by simple means, for instance, by single rewrite rules—we call these protocols non-looping—and (2) protocols, such as group protocols, where the protocol steps are complex and typically involve an iterative or recursive computation—we call them recursive. While much is known on the decidability of security for non-looping protocols, only little is known for recursive protocols. In this paper, we prove decidability of security (with respect to the standard Dolev–Yao intruder) for a core class of recursive protocols and undecidability for several extensions. The key ingredient of our protocol model is specifically designed tree transducers which work over infinite signatures and have the ability to generate new constants (which allow us to mimic key generation). The decidability result is based on an automata-theoretic construction which involves a new notion of regularity, designed to work well with the infinite signatures we use.     

PSL构造双向交换自动机及非确定自动机的方法

PSL(property specification language)是一种用于描述并行系统的属性规约语言,包括线性时序逻辑FL(foundation language)和分支时序逻辑OBE(optional branching extension)两部分.由于OBE就是CTL(computation tree logic),并且具有时钟声明的公式很容易改写成非时钟公式,因此重点研究了非时钟FL逻辑.为便于进行模型检验,每个FL公式必须转化成为一种可验证形式,通常是自动机(非确定自动机).构造非确定自动机的过程主要是通过中间构建交换自动机来实现.详细给出了由非时钟FL构造双向交换自动机的构造规则.构造规则的核心逻辑不仅仅局限于是在LTL(linear temporal logic)基础上的正规表达式,而且全面而充分地考虑了各种FL操作算子的可能性.并且给出了将双向交换自动机转化为非确定自动机的一种方法.最后,编写了将PSL转化为上述自动机的实现工具.FL双向交换自动机的构造规则计算复杂度仅是FL公式长度的线性表达式,验证了构造规则的正确性.在此基础上,证明了双向交换自动机与其转化的等价的非确定自动机接受的语言相同.上述工作对解决复杂并行系统建模和模型验证问题具有重要的理论意义和应用价值.