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The branching point approach to Conway's problem
 LNCS
, 2002
"... Abstract. A word u is a branching point for a set of words X if there are two different letters a and b such that both ua and ub can be extended to words in X+. A branching point u is critical for X if u 6 ∈ X+. Using these notions, we give an elementary solution for Conway’s Problem in the case of ..."
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Abstract. A word u is a branching point for a set of words X if there are two different letters a and b such that both ua and ub can be extended to words in X+. A branching point u is critical for X if u 6 ∈ X+. Using these notions, we give an elementary solution for Conway’s Problem in the case of finite biprefixes. We also discuss a possible extension of this approach towards a complete solution for Conway’s Problem. 1
Challenges of Commutation  An Advertisement
, 2001
"... We consider a few problems connected to the commutation of languages, in particular finite ones. The goal is to emphasize the challenging nature of such simply formulated problems. In doing so we give a survey of results achieved during the last few years, restate several open problems and illustr ..."
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Cited by 5 (3 self)
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We consider a few problems connected to the commutation of languages, in particular finite ones. The goal is to emphasize the challenging nature of such simply formulated problems. In doing so we give a survey of results achieved during the last few years, restate several open problems and illustrate some approaches to attack such problems by two simple constructions.
Commutation Problems on Sets of Words and Formal Power Series
, 2002
"... We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generaliza ..."
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We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generalizations of the semilinear languages, and we study their behaviour under rational operations, morphisms, Hadamard product, and difference. Turning to commutation on sets of words, we then study the notions of centralizer of a language  the largest set commuting with a language , of root and of primitive root of a set of words. We answer a question raised by Conway more than thirty years ago  asking whether or not the centralizer of any rational language is rational  in the case of periodic, binary, and ternary sets of words, as well as for rational ccodes, the most general results on this problem. We also prove that any code has a unique primitive root and that two codes commute if and only if they have the same primitive root, thus solving two conjectures of Ratoandromanana, 1989. Moreover, we prove that the commutation with an ccode X can be characterized similarly as in free monoids: a language commutes with X if and only if it is a union of powers of the primitive root of X.
Commutation with codes
 Theor. Comput. Sci
, 2005
"... The centralizer of a set of words X is the largest set of words C(X) commuting with X: XC(X) = C(X)X. It has been a long standing open question due to Conway, 1971, whether the centralizer of any rational set is rational. While the answer turned out to be negative in general, see Kunc 2004, we prov ..."
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The centralizer of a set of words X is the largest set of words C(X) commuting with X: XC(X) = C(X)X. It has been a long standing open question due to Conway, 1971, whether the centralizer of any rational set is rational. While the answer turned out to be negative in general, see Kunc 2004, we prove here that the situation is different for codes: the centralizer of any rational code is rational and if the code is finite, then the centralizer is finitely generated. This result has been previously proved only for binary and ternary sets of words in a series of papers by the authors and for prefix codes in an ingenious paper by Ratoandromanana 1989 – many of the techniques we use in this paper follow her ideas. We also give in this paper an elementary proof for the prefix case. Key words: Codes, Commutation, Centralizer, Conway’s problem, Prefix codes. 1
On Conjugacy of Languages
, 2001
"... We say that two languages X and Y are conjugated if they satisfy the conjugacy equation XZ = ZY for some language Z. We study several problems associated to this equation. For example, we characterize all sets which are conjugated via a twoelement bipre x set Z, as well as all twoelement sets whic ..."
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We say that two languages X and Y are conjugated if they satisfy the conjugacy equation XZ = ZY for some language Z. We study several problems associated to this equation. For example, we characterize all sets which are conjugated via a twoelement bipre x set Z, as well as all twoelement sets which are conjugates.
Unique Decipherability in the Monoid of Languages: an Application of Rational Relations
, 2008
"... We attack the problem of deciding whether a finite collection of finite languages is a code, that is, possesses the unique decipherability property in the monoid of finite languages. We investigate a few subcases where the theory of rational relations can be employed to solve the problem. The case ..."
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We attack the problem of deciding whether a finite collection of finite languages is a code, that is, possesses the unique decipherability property in the monoid of finite languages. We investigate a few subcases where the theory of rational relations can be employed to solve the problem. The case of unary languages is one of them and as a consequence, we show how to decide for two given finite subsets of nonnegative integers, whether they are the nth root of a common set, for some n ≥ 1. We also show that it is decidable whether a finite collection of finite languages is a Parikh code, in the sense that whenever two products of these sets are commutatively equivalent, so are the sequences defining these products. Finally, we consider a nonunary special case where all finite sets consist of words containing exactly one occurrence of the specific letter.
Theoretical Informatics and Applications Will be set by the publisher ON CONJUGACY OF LANGUAGES ∗
"... Abstract. We say that two languages X and Y are conjugates if they satisfy the conjugacy equation XZ = ZY for some language Z. We study several problems associated to this equation. For example, we characterize all sets which are conjugated via a twoelement biprefix set Z, as well as all twoelemen ..."
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Abstract. We say that two languages X and Y are conjugates if they satisfy the conjugacy equation XZ = ZY for some language Z. We study several problems associated to this equation. For example, we characterize all sets which are conjugated via a twoelement biprefix set Z, as well as all twoelement sets which are conjugates. Mathematics Subject Classification. 68R15, 68Q70. 1.
On the existence of prime decompositions www.elsevier.com/locate/tcs
"... We investigate factorizations of regular languages in terms of prime languages. A language is said to be strongly prime decomposable if any way of factorizing it yields a prime decomposition in a finite number of steps. We give a characterization of the strongly prime decomposable regular languages ..."
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We investigate factorizations of regular languages in terms of prime languages. A language is said to be strongly prime decomposable if any way of factorizing it yields a prime decomposition in a finite number of steps. We give a characterization of the strongly prime decomposable regular languages and using the characterization we show that every regular language over a unary alphabet has a prime decomposition. We show that there exist nonregular unary languages that do not have prime decompositions. We also consider infinite factorizations of unary languages. c ○ 2007 Elsevier B.V. All rights reserved.