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14
Finite semigroups and recognizable languages: an introduction
, 2002
"... This paper is an attempt to share with a larger audience some modern developments in the theory of finite automata. It is written for the mathematician who has a background in semigroup theory but knows next to nothing on automata and languages. No proofs are given, but the main results are illustra ..."
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Cited by 29 (9 self)
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This paper is an attempt to share with a larger audience some modern developments in the theory of finite automata. It is written for the mathematician who has a background in semigroup theory but knows next to nothing on automata and languages. No proofs are given, but the main results are illustrated by several examples and counterexamples. What is the topic of this theory? It deals with languages, automata and semigroups, although recent developments have shown interesting connections with model theory in logic, symbolic dynamics and topology. Historically, in their attempt to formalize natural languages, linguists such as Chomsky gave a mathematical definition of natural concepts such as words, languages or grammars: given a finite set A, a word on A is simply an element of the free monoid on A, and a language is a set of words. But since scientists are fond of classifications of all sorts, language theory didn’t escape to this mania. Chomsky established a first hierarchy, based on his formal grammars. In this paper, we are interested in the recognizable languages, which form the lower level of the
Some Results on the Generalized StarHeight Problem
, 2001
"... We prove some results related to the generalized starheight problem. In this problem, as opposed to the restricted starheight problem, complementation is considered as a basic operator. We first show that the class of languages of starheight n is closed under certain operations (left and right qu ..."
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Cited by 7 (5 self)
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We prove some results related to the generalized starheight problem. In this problem, as opposed to the restricted starheight problem, complementation is considered as a basic operator. We first show that the class of languages of starheight n is closed under certain operations (left and right quotients, inverse alphabetic morphisms, injective starfree substitutions). It is known that languages recognized by a commutative group are of starheight 1. We extend this result to nilpotent groups of class 2 and to the groups that divide a semidirect product of a commutative group by (Z=2Z) n . In the same direction, we show that one of the languages that was conjectured to be of star height 2 during the past ten years, is in fact of star height 1. Next we show that if a rational language L is recognized by a monoid of the variety generated by wreath products of the form M (G N ), where M and N are aperiodic monoids, and G is a commutative group, then L is of starheight...
ALGORITHMIC PROBLEMS IN VARIETIES
, 1994
"... Mankind always sets itself only such problems as it can solve. Karl Marx, The Introduction to "A Critique of Political Economy". ..."
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Mankind always sets itself only such problems as it can solve. Karl Marx, The Introduction to &quot;A Critique of Political Economy&quot;.
Classifying Regular Languages by a Split Game
, 2006
"... In this paper, we introduce a variant of the EhrenfeuchtFraïssé game from logic which is useful for analyzing the expressive power of classes of generalized regular expressions. An extension of the game to generalized ωregular expressions is also established. To gain insight into how the game can ..."
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In this paper, we introduce a variant of the EhrenfeuchtFraïssé game from logic which is useful for analyzing the expressive power of classes of generalized regular expressions. An extension of the game to generalized ωregular expressions is also established. To gain insight into how the game can be applied to attack the longstanding generalized star height 2 problem, we propose and solve a related but easier problem, the omega power problem. Namely we show that omega powers, together with boolean combinations and concatenations, are not sufficient to express the class of ωregular languages.
Free Burnside Semigroups
, 1999
"... This paper surveys the area of Free Burnside Semigroups. The theory of these semigroups, as is the case for groups, is far from being completely known. For semigroups, the most impressive results were obtained in the last 10 years. ..."
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This paper surveys the area of Free Burnside Semigroups. The theory of these semigroups, as is the case for groups, is far from being completely known. For semigroups, the most impressive results were obtained in the last 10 years.
Semigroups Satisfying x m+n = x n
"... We summarize recent results on semigroups satisfying the identity x m+n = x n , for n # 0andm # 1, and some rewrite techniques that havecontributed to their investigation. 1 Introduction Ninetyyears ago, Burnside #1902# posed the question whether every group satisfying the identity x m ..."
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We summarize recent results on semigroups satisfying the identity x m+n = x n , for n # 0andm # 1, and some rewrite techniques that havecontributed to their investigation. 1 Introduction Ninetyyears ago, Burnside #1902# posed the question whether every group satisfying the identity x m =1,andhaving a #nite number of generators, is #nite. In 1969, Brzozowski #see the list of open questions in #Brzozowski, 1980## conjectured that the congruence classes on words generated by x n+1 = x n , are all regular sets. Recently, McCammond #1991# extended this conjecture to all semigroups satisfying x m+n = x n and investigated the decidability of their word problems. These conjectures have been the topic of recent research, whichwe summarize here. Consider the set A # of #nite words over some #nite alphabet A containing at least two letters, and suppose weidentify certain repetitious words. #The case jAj = 1 is patently uninteresting.# Speci#cally,aword of the form ux m+n ...