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Conway's Problem and the commutation of languages
 Bulletin of EATCS
, 2001
"... We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The se ..."
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We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The second problem, proposed by Ratoandromanana in 1989, is asking for a characterization of those languages commuting with a given code – the conjecture is that the commutation with codes may be characterized as in free monoids. We present here simple proofs for the known results on these two problems. 1
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
The Commutation With Codes and Ternary Sets of Words
, 2002
"... We prove several results on the commutation of languages. First, we prove that the largest set commuting with a given code X , i.e., its centralizer C(X), is always (X) , where (X) is the primitive root of X . Using this result, we characterize the commutation with codes similarly as for words, p ..."
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We prove several results on the commutation of languages. First, we prove that the largest set commuting with a given code X , i.e., its centralizer C(X), is always (X) , where (X) is the primitive root of X . Using this result, we characterize the commutation with codes similarly as for words, polynomials, and formal power series: a language commutes with X if and only if it is a union of powers of (X). This solves a conjecture of Ratoandromanana, 1989, and also gives an armative answer to a special case of an intriguing problem raised by Conway in 1971. Second, we prove that for any nonperiodic ternary set of words F , and moreover, a language commutes with F if and only if it is a union of powers of F , results previously known only for ternary codes. A boundary point is thus established, as these results do not hold for languages with at least four words.
Playing with Conway’s Problem
"... The centralizer of a language is the maximal language commuting with it. The question, raised by Conway in 1971, whether the centralizer of a rational language is always rational, recently received a lot of attention. In Kunc 2005, a strong negative answer to this problem was given by showing that ..."
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The centralizer of a language is the maximal language commuting with it. The question, raised by Conway in 1971, whether the centralizer of a rational language is always rational, recently received a lot of attention. In Kunc 2005, a strong negative answer to this problem was given by showing that even complete corecursively enumerable centralizers exist for finite languages. Using a combinatorial game approach, we give here an incremental construction of rational languages embedding any recursive computation in their centralizers. In 1999, Choffrut et al. [1] renewed an old problem raised by Conway [2] in 1971: given a rational language, does its centralizer — the maximal language commuting with it — have to be rational? The property is known to hold for some particular families of languages. In the case of codes, Ratoandramanana [3] showed in 1989 that it holds for prefix codes, raising a restriction of Conway’s problem to codes which recently recieved a positive answer by Karhumäki et al. [4]. In the general case, until recently, the best known result, by Karhumäki and Petre [5], was that the centralizer of a recursive language has to be corecursively enumerable. This property may also be considered as a particular case of results of Okhotin [6] concerning the computational power of systems of equations on languages. For a complete survey on Conway’s problem, the reader may refer to [7–10]. In 2004, the community was thrilled by an announcement by Kunc [11] that a centralizer can actually be nonrecursive. This announcement was followed by a conference communication [12] in 2005 showing that finite languages exist whose centralizers are complete for corecursively enumerable languages 1. It includes a sketch of the proof for the special case of rational languages. While simpler than the proof ∗ Corresponding author.