A survey on the local divisor technique

© 2015 Elsevier B.V. Local divisors allow a powerful induction scheme on the size of a monoid. We survey this technique by giving several examples of this proof method. These applications include linear temporal logic, rational expressions with Kleene stars restricted to prefix codes with bounded synchronization delay, Church-Rosser congruential languages, and Simon's Factorization Forest Theorem. We also introduce the notion of a localizable language class as a new abstract concept which unifies some of the proofs for the results above

A Survey on the Local Divisor Technique

Local divisors allow a powerful induction scheme on the size of a monoid. We survey this technique by giving several examples of this proof method. These applications include linear temporal logic, rational expressions with Kleene stars restricted to prefix codes with bounded synchronization delay, Church-Rosser congruential languages, and Simon's Factorization Forest Theorem. We also introduce the notion of localizable language class as a new abstract concept which unifies some of the proofs for the results above

Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions

What is the common link, if there is any, between Church-Rosser systems, prefix codes with bounded synchronization delay, and local Rees extensions? The first obvious answer is that each of these notions relates to topics of interest for WORDS: Church-Rosser systems are certain rewriting systems over words, codes are given by sets of words which form a basis of a free submonoid in the free monoid of all words (over a given alphabet) and local Rees extensions provide structural insight into regular languages over words. So, it seems to be a legitimate title for an extended abstract presented at the conference WORDS 2017. However, this work is more ambitious, it outlines some less obvious but much more interesting link between these topics. This link is based on a structure theory of finite monoids with varieties of groups and the concept of local divisors playing a prominent role. Parts of this work appeared in a similar form in conference proceedings where proofs and further material can be found.Comment: Extended abstract of an invited talk given at WORDS 201

Characterizing classes of regular languages using prefix codes of bounded synchronization delay

In this paper we continue a classical work of Sch\"utzenberger on codes with bounded synchronization delay. He was interested to characterize those regular languages where the groups in the syntactic monoid belong to a variety $H$. He allowed operations on the language side which are union, intersection, concatenation and modified Kleene-star involving a mapping of a prefix code of bounded synchronization delay to a group $G\in H$, but no complementation. In our notation this leads to the language classes $SD_G(A^\infty)$ and $SD_H(A^\infty$). Our main result shows that $SD_H(A^\infty)$ always corresponds to the languages having syntactic monoids where all subgroups are in $H$. Sch\"utzenberger showed this for a variety $H$ if $H$ contains Abelian groups, only. Our method shows the general result for all $H$ directly on finite and infinite words. Furthermore, we introduce the notion of local Rees products which refers to a simple type of classical Rees extensions. We give a decomposition of a monoid in terms of its groups and local Rees products. This gives a somewhat similar, but simpler decomposition than in Rhodes' synthesis theorem. Moreover, we need a singly exponential number of operations, only. Finally, our decomposition yields an answer to a question in a recent paper of Almeida and Kl\'ima about varieties that are closed under Rees products

The many facets of string transducers

Regular word transductions extend the robust notion of regular languages from a qualitative to a quantitative reasoning. They were already considered in early papers of formal language theory, but turned out to be much more challenging. The last decade brought considerable research around various transducer models, aiming to achieve similar robustness as for automata and languages. In this paper we survey some older and more recent results on string transducers. We present classical connections between automata, logic and algebra extended to transducers, some genuine definability questions, and review approaches to the equivalence problem