8 research outputs found
Star-Free Languages are Church-Rosser Congruential
The class of Church-Rosser congruential languages has been introduced by
McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser
congruential (belongs to CRCL), if there is a finite, confluent, and
length-reducing semi-Thue system S such that L is a finite union of congruence
classes modulo S. To date, it is still open whether every regular language is
in CRCL. In this paper, we show that every star-free language is in CRCL. In
fact, we prove a stronger statement: For every star-free language L there
exists a finite, confluent, and subword-reducing semi-Thue system S such that
the total number of congruence classes modulo S is finite and such that L is a
union of congruence classes modulo S. The construction turns out to be
effective
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
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 . 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 , but no complementation. In
our notation this leads to the language classes and
). Our main result shows that always
corresponds to the languages having syntactic monoids where all subgroups are
in . Sch\"utzenberger showed this for a variety if contains Abelian
groups, only. Our method shows the general result for all 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
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
Locally countable pseudovarieties
The purpose of this paper is to contribute to the theory of profinite semigroups by considering the special class consisting of those all of whose finitely generated closed subsemigroups are countable, which are said to be locally countable. We also call locally countable a pseudovariety V (of finite semigroups) for which all pro-V semigroups are locally countable. We investigate operations preserving local countability of pseudovarieties and show that, in contrast with local finiteness, several natural operations do not preserve it. We also investigate the relationship of a finitely generated profinite semigroup being countable with every element being expressible in terms of the generators using multiplication and the idempotent (omega) power. The two properties turn out to be equivalent if there are only countably many group elements, gathered in finitely many regular J -classes. We also show that the pseudovariety generated by all finite ordered monoids satisfying the inequality 1 6 x n is locally countable if and only if n = 1
Local divisors in formal languages
Regular languages are exactly the class of recognizable subsets of the free monoid. In particular, the syntactic monoid of a regular language is finite. This is the starting point of algebraic language theory. In this thesis, the algebraic connection between regular languages and monoids is studied using a certain monoid construction - local divisors.
Using the local divisor construction, we give a Rees decomposition of a monoid into smaller parts - the monoid is a Rees extension of a submonoid and a local divisor. Iterating this concept gives an iterated Rees decomposition of a monoid into groups appearing in the monoid. This decomposition is similar to the synthesis theorem of Rhodes and Allen. In particular, the Rees decomposition shows that closure of a variety V of finite monoids under Rees extensions is the variety HÌ
induced by the groups H contained in V.
Due to the connection between HÌ
and local divisors, we turn our attention to a language description of HÌ
. The language description is a continuation of classical work of SchĂŒtzenberger. He studied prefix codes of bounded synchronization delay and used those codes to give a language description of HÌ
in the case that the variety H of groups contains only abelian groups. We use the local divisor approach to generalize SchĂŒtzenberger's language description of HÌ
for all varieties H of finite groups. The main ingredient of this generalization is the concept of group-controlled stars. The group-controlled star is an operation on prefix codes of bounded synchronization delay which generalizes the usual Kleene star. The language class SDH(Aâ) is the smallest class which contains all finite languages and is closed under union, concatenation product and group-controlled stars for groups in H. We show that SDH(Aâ) is the language class corresponding to HÌ
. As a by-product of the proof we give another language characterization of HÌ
: the localizable closure LocH(Aâ) of H.
In the last part of this thesis, we deal with Church-Rosser congruential languages (CRCL). A language is Church-Rosser congruential if it is a finite union of congruence classes modulo a finite, confluent and length-reducing semi-Thue system. This yields a linear time algorithm for the membership problem of a fixed language in CRCL. A natural question, which was open for over 25 years, is whether all regular languages are in CRCL. We give an affirmative answer to this question by proving a stronger statement: for every regular language L and for every weight, there exists a finite, confluent and weight-reducing semi-Thue system S such that A*/S is finite and recognizes L. Lifting the result from the special case of length-reducing to weight-reducing allows the use of local divisors.
Next, we focus on Parikh-reducing Church-Rosser systems for regular languages. Instead of constructing a semi-Thue system for a fixed weight, a Parikh-reducing Church-Rosser system is weight-reducing for every weight. We construct such systems for all languages in AÌ
bÌ
, that is, for all languages such that the groups in the syntactic monoid are abelian. Additionally, small changes in the proof of this result also yield that for all languages L over a two letter alphabet there exists a Parikh-reducing Church-Rosser system S of finite index such that L is recognized by A*/S.
Lastly, we deal with the size of the monoid A*/S for the constructed systems S. We show that in the group case this size has an exponential lower bound and a triple exponential upper bound. The key observation is that one can restrict the alphabet used in the inductive construction. Using the same observation, one can lower the upper bound in the general monoid case from a non-primitive function without this optimization to a quadruple exponential upper bound.Die Klasse der regulĂ€ren Sprachen entspricht genau den erkennbaren Sprachen ĂŒber dem freien Monoid. Ăquivalent dazu ist die Klasse der Sprachen, deren syntaktisches Monoid endlich ist. Dies ist der Ausgangspunkt der algebraischen Sprachtheorie. In dieser Arbeit wird dieser algebraische Zusammenhang zwischen regulĂ€ren Sprachen und Monoiden mit Hilfe einer Monoid-Konstruktion untersucht: den lokalen Divisoren.
ZunĂ€chst werden lokale Divisoren benutzt um ein Monoid in kleinere Teile zu zerlegen. Die dabei verwendete Konstruktion ist Ă€hnlich zur Rees-Matrix-Halbgruppe und liefert eine Zerlegung eines Monoids als sogenannte Rees-Erweiterung eines echten Untermonoids und eines lokalen Divisors. Wiederholtes Anwenden dieses Sachverhalts fĂŒhrt dann auf eine Rees-Zerlegung, bei der die grundlegenden Bausteine Gruppen sind, die im ursprĂŒnglichen Monoid vorkommen. Diese Zerlegung ist Ă€hnlich zum Synthese-Theorem von Rhodes und Allen. Insbesondere liefert dies, dass der Abschluss einer VarietĂ€t V unter Rees-Erweiterungen die VarietĂ€t HÌ
ist, wobei H die VarietÀt der endlichen Gruppen ist, die in V vorkommen.
Aufgrund des Zusammenhangs zwischen lokalen Divisoren und den VarietĂ€ten HÌ
, werden als nĂ€chstes Sprachbeschreibungen der VarietĂ€ten HÌ
untersucht. Dabei wird die Arbeit von SchĂŒtzenberger ĂŒber Sprachcharakterisierungen mit Hilfe von PrĂ€fix-Codes mit beschrĂ€nkter Synchronisierungsverzögerung (englisch: bounded synchronization delay) fortgesetzt. SchĂŒtzenberger benutzte diese Codes um die VarietĂ€ten der Form HÌ
zu beschreiben, wobei V eine VarietĂ€t von endlichen abelschen Gruppen ist. Wir verallgemeinern seine Beschreibung um HÌ
fĂŒr alle VarietĂ€ten H von endlichen Gruppen zu charakterisieren. Das Hauptkonzept dieser Verallgemeinerung sind gruppen-kontrollierte Sterne. Dabei sind gruppen-kontrollierte Sterne Sprachoperationen, die auf PrĂ€fix-Codes mit beschrĂ€nkter Synchronisierungsverzögerung aufbauen und als Spezialfall fĂŒr die triviale Gruppe den Kleene-Stern liefern.
Die Sprachklasse SDH(Aâ) ist die kleinste Klasse von Sprachen, die alle endlichen Sprachen enthĂ€lt und abgeschlossen ist unter Vereinigung, Konkatenationsprodukt und gruppen-kontrollierten Sternen, wobei die Gruppen aus H sind. Wir zeigen, dass SDH(Aâ) die zu HÌ
zugehörige Sprachklasse ist. Als Nebenprodukt des Beweises dieser Sprachcharakterisierung geben wir eine weitere Charakterisierung von HÌ
an: der lokale Abschluss LocH(Aâ) von H.
Der letzte Abschnitt dieser Arbeit handelt von der Sprachklasse CRCL (Church-Rosser congruential languages). Eine Sprache ist in CRCL, falls sie eine endliche Vereinigung von Kongruenzklassen eines endlichen, konfluenten und lĂ€ngenreduzierenden Ersetzungssystems ist. Dies liefert direkt einen Linearzeit-Algorithmus fĂŒr das Wortproblem von Sprachen aus CRCL. Eine 25 Jahre lang offene Fragestellung war, ob alle regulĂ€ren Sprachen in CRCL enthalten sind. Wir beantworten diese Frage positiv, indem wir eine stĂ€rkere Aussage beweisen: FĂŒr alle regulĂ€ren Sprachen L und alle Gewichtsfunktionen gibt es ein endliches, konfluentes und gewichtsreduzierendes Ersetzungssystem S, fĂŒr das A*/S endlich ist und L erkennt. Durch das Erweitern der Aussage auf alle Gewichtsfunktionen erlaubt dies die Benutzung von lokalen Divisoren.
Als nĂ€chstes werden Parikh-reduzierende Church-Rosser-Ersetzungssysteme betrachtet. Diese reprĂ€sentieren eine Vertauschung der Quantorenreihenfolge: Ein Parikh-reduzierendes Ersetzungssystem ist gewichtsreduzierend fĂŒr alle Gewichtsfunktionen. Wir konstruieren solche Systeme fĂŒr alle Sprachen in der VarietĂ€t AÌ
bÌ
, d.h. fĂŒr alle Sprachen, in denen die im syntaktischem Monoid vorkommenden Gruppen abelsch sind. ZusĂ€tzlich liefert eine Abwandlung dieses Beweises dasselbe Resultat fĂŒr alle regulĂ€ren Sprachen ĂŒber einem zwei-elementigem Alphabet.
Als letztes beschĂ€ftigt sich die Arbeit mit AbschĂ€tzungen fĂŒr die GröĂe von A*/S fĂŒr die zuvor konstruierten Systeme S. Im Fall von Gruppensprachen ist die GröĂe von unten durch eine Exponentialfunktion und von oben durch eine dreifache Exponentialfunktion beschrĂ€nkt. FĂŒr die obere Schranke wird dabei eine Beobachtung benutzt, wie man das Alphabet in der Induktion beschrĂ€nken kann. Mit Hilfe dieser Beobachtung ist es ebenfalls möglich die obere Schranke im Monoid-Fall von einer nicht primitiven Funktion auf eine vierfach exponentielle Funktion zu reduzieren