5 research outputs found
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
Regular matching problems for infinite trees
We investigate regular matching problems. The classical reference is Conway's
textbook "Regular algebra and finite machines". Some of his results can be
stated as follows. Let and be
regular languages where is a set of constants and is a set of
variables. Substituting every by a regular subset of
yields a regular set . A substitution
solves a matching problem "?" if .
There are finitely many maximal solutions ; they are effectively
computable and is regular for all ; and every solution is
included in a maximal one. Also, in the case of words
"?" is decidable.
Apart from the last property, we generalize these results to infinite trees.
We define a notion of choice function which for any tree over
and position of a variable selects at most one tree
; next, we define as the limit of a
Cauchy sequence; and the union over all yields .
Since our definition coincides with that for IO substitutions, we write
instead of .
Our main result is the decidability of
"?" if is regular and belongs
to a class of tree languages closed under intersection with regular sets. Such
a special case pops up if is context-free. Note that
"?" is undecidable, in general in that case.
However, the decidability of "?" if both
and are regular remains open because, in contrast to word languages, the
homomorphic image of a regular tree language is not always regular if
is regular for all .Comment: 18 pages. This replacement eliminates a false claim from the previous
arXiv version of this paper: Item 4 of Theorem 1 did not hold for # = {=
Decidability of membership problems for flat rational subsets of GL(2, Q) and singular matrices
This work relates numerical problems on matrices over the rationals to symbolic algorithms on words and finite automata. Using exact algebraic algorithms and symbolic computation, we prove new decidability results for 2 Ă 2 matrices over Q. Namely, we introduce a notion of flat rational sets: if M is a monoid and N †M is its submonoid, then flat rational sets of M relative to N are finite unions of the form L0g1 L1 ··· gtLt where all Lis are rational subsets of N and gi â M. We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of GL(2, Q) over GL(2, Z) is decidable. We also show a dichotomy for nontrivial group extension of GL(2, Z) in GL(2, Q): if G is a f.g. group such that GL(2, Z) < G †GL(2, Q), then either G â
GL(2, Z) à Zk, for some k ℠1, or G contains an extension of the Baumslag-Solitar group BS(1, q), with q ℠2, of infinite index. It turns out that in the first case the membership problem for G is decidable but the equality problem for rational subsets of G is undecidable. In the second case, decidability of the membership problem is open for every such G. In the last section we prove new decidability results for flat rational sets that contain singular matrices. In particular, we show that the membership problem is decidable for flat rational subsets of M(2, Q) relative to the submonoid that is generated by the matrices from M(2, Z) with determinants 0, ± 1 and the central rational matrices
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