Termination of Cycle Rewriting

Powerful methods and algorithms are known to learn regular languages. Aiming at extending them to more complex grammars, we choose to change the way we represent these languages. Among the formalisms that allow to define classes of languages, the one of string-rewriting systems (SRS) has outstanding properties. Indeed, SRS are expressive enough to define, in a uniform way, a noteworthy and non trivial class of languages that contains all the regular languages, { a n b n : n ≥ 0 }, {w∈{a,b}*:|w| a =|w| b }, the parenthesis languages of Dyck, the language of Lukasewitz, and many others. Moreover, SRS constitute an efficient (often linear) parsing device for strings, and are thus promising and challenging candidates in forthcoming applications of Grammatical Inference. In this paper, we pioneer the problem of their learnability. We propose a novel and sound algorithm which allows to identify them in polynomial time. We illustrate the execution of our algorithm throughout a large amount of examples and finally raise some open questions and research directions.

In the framework of self-stabilizing systems, the convergence proof is generally done by exhibiting a measure that strictly decreases until a legitimate configuration is reached. The discovery of such a mea- sure is very specific and requires a deep understanding of the studied transition system. In contrast we propose here a simple method for prov- ing convergence, which regards self-stabilizing systems as string rewrite systems, and adapts a procedure initially designed by Dershowitz for proving termination of string rewrite systems.

Non-Archimedean words have been introduced as a new type of infinite words which can be investigated through classical methods in combinatorics on words due to a length function. The length function, however, takes values in the additive group of polynomials ℤ[t] (and not, as traditionally, in ℕ), which yields various new properties. Non-Archimedean words allow to solve a number of interesting algorithmic problems in geometric and algorithmic group theory. There is also a connection to logic and the first-order theory in free groups (Tarski Problems). In the present paper we provide a general method to use infinite words over a discretely ordered abelian group as a tool to investigate certain group extensions for an arbitrary group G. The central object is a group E (A, G) which is defined in terms of a non-terminating, but confluent rewriting system. The groupG as well as some natural HNN-extensions of G embed into E (A, G) (and still "behave like" G), which makes it interesting to study its algorithmic properties. In order to show that every group G embeds into E (A, G) we combine methods from combinatorics on words, string rewriting and group theory.

Reductions in tree replacement systems

We present a new technique to prove termination of Term Rewriting Systems, with full automation. A crucial task in this context is to find suitable well-founded orderings. A popular approach consists in interpreting terms into a domain equipped with an adequate well-founded ordering. In addition to the usual interpretations: natural numbers or polynomials over integer/rational numbers, the recently introduced matrix based interpretations have proved to be very efficient regarding termination of string rewriting and of term rewriting. In this spirit we propose to interpret terms as polynomials over integer matrices. Designed for term rewriting, our generalisation subsumes previous approaches allowing for more orderings without increasing the search space. Thus it performs better than the original version. Another advantage is that, interpreting terms to actual polynomials of matrices, it opens the way to matrix non linear interpretations. This result is implemented in the CiME3 rewriting toolkit.

A canonical automaton for one-rule length-preserving string rewrite systems

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsAnne M. BurnsAnne Burns received a Ph.D. in mathematics from SUNY at Stony Brook in 1976. Since that time she has been a member of the Department of Mathematics at the C.W. Post campus of Long Island University. Her current research interests are in discrete dynamical systems, applications of computer graphics to mathematics and the sciences, and using mathematics to describe nature.

Algorithms for solving uniform decision problems for algebraic structures crucially depend on the chosen finite presentations for the structures under consideration. Rewriting techniques have been used very successfully to solve uniform decision problems, when the presentations considered involve finite, noetherian, and (λ)-confluent rewriting systems. Whenever the class of algebraic structures considered is closed under the operation of taking finitely generated substructures, then the algorithms for solving the uniform decision problems can be applied to the substructures as well. However, since these algorithms depend on the form of the presentations, this involves the task of constructing a presentation of a certain form for a substructure given a presentation of this form for the structure itself and a finite set of generating elements for the substructure. This problem, which has received a lot of attention in algebra, is here investigated from an algorithmic point of view. The structures considered are the following two classes of groups, which have been studied extensively before: the polycyclic groups and the context-free groups. Finitely generated context-free groups can be presented by finite, monadic, and λ-confluent string-rewriting systems. Due to their nice algorithmic properties these systems provide a way to effectively solve many decision problems for context-free groups. Since finitely generated subgroups of context-free groups are again contextfree, they can be presented in the same way. Here we describe a process that, from a finite, monadic, and λ-confluent string-rewriting system presenting a context-free groupG and a finite subsetU ofG, determines a presentation of this form for the subgroup 〈U〉 ofG that is generated byU. For finitely presented polycyclic groups we obtain an analogous result, when we use finite confluent PCP2-presentations to describe these groups.