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

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

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

Regular languages are Church-Rosser congruential

© 2015 ACM 0004-5411/2015/10-ART32 15.00. This article shows a general result about finite monoids and weight reducing string rewriting systems. As a consequence it proves a long standing conjecture in formal language theory: All regular languages are Church-Rosser congruential. The class of Church-Rosser congruential languages was introduced by McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser congruential if there exists a finite, confluent, and length-reducing semi-Thue system S such that L is a finite union of congruence classes modulo S. It was known that there are deterministic linear context-free languages which are not Church- Rosser congruential, but the conjecture was that all regular languages are of this form. The article offers a stronger statement: A language is regular if and only if it is strongly Church-Rosser congruential. It is the journal version of the conference abstract which was presented at ICALP 2012

Transfinite reductions in orthogonal term rewriting systems

Strongly convergent reduction is the fundamental notion of reduction in infinitary orthogonal term rewriting systems (OTRSs). For these we prove the Transfinite Parallel Moves Lemma and the Compressing Lemma. Strongness is necessary as shown by counterexamples. Normal forms, which we allow to be infinite, are unique, in contrast to ω-normal forms. Strongly converging fair reductions result in normal forms. In general OTRSs the infinite Church-Rosser Property fails for strongly converging reductions. However for Böhm reduction (as in Lambda Calculus, subterms without head normal forms may be replaced by ⊥) the infinite Church-Rosser property does hold. The infinite Church-Rosser Property for non-unifiable OTRSs follows. The top-terminating OTRSs of Dershowitz c.s. are examples of non-unifiable OTRSs

Small overlap monoids II: automatic structures and normal forms

We show that any finite monoid or semigroup presentation satisfying the small overlap condition C(4) has word problem which is a deterministic rational relation. It follows that the set of lexicographically minimal words forms a regular language of normal forms, and that these normal forms can be computed in linear time. We also deduce that C(4) monoids and semigroups are rational (in the sense of Sakarovitch), asynchronous automatic, and word hyperbolic (in the sense of Duncan and Gilman). From this it follows that C(4) monoids satisfy analogues of Kleene's theorem, and admit decision algorithms for the rational subset and finitely generated submonoid membership problems. We also prove some automata-theoretic results which may be of independent interest.Comment: 17 page