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The power of commuting with finite sets of words
 In Proc. STACS’05, Springer LNCS 3404
, 2005
"... We construct a finite language L such that the largest language commuting with L is not recursively enumerable. This gives a negative answer to the question raised by Conway in 1971 and also strongly disproves Conway’s conjecture on contextfreeness of maximal solutions of systems of semilinear ine ..."
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Cited by 19 (1 self)
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We construct a finite language L such that the largest language commuting with L is not recursively enumerable. This gives a negative answer to the question raised by Conway in 1971 and also strongly disproves Conway’s conjecture on contextfreeness of maximal solutions of systems of semilinear inequalities. 1
Strict language inequalities and their decision problems
 Mathematical Foundations of Computer Science (MFCS 2005
, 2005
"... Abstract. Systems of language equations of the form {ϕ(X1,..., Xn) = ∅, ψ(X1,..., Xn) � = ∅} are studied, where ϕ, ψ may contain settheoretic operations and concatenation; they can be equivalently represented as strict inequalities ξ(X1,..., Xn) ⊂ L0. It is proved that the problem whether such an ..."
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Cited by 5 (3 self)
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Abstract. Systems of language equations of the form {ϕ(X1,..., Xn) = ∅, ψ(X1,..., Xn) � = ∅} are studied, where ϕ, ψ may contain settheoretic operations and concatenation; they can be equivalently represented as strict inequalities ξ(X1,..., Xn) ⊂ L0. It is proved that the problem whether such an inequality has a solution is Σ2complete, the problem whether it has a unique solution is in (Σ3 ∩Π3)\(Σ2 ∪Π2), the existence of a regular solution is a Σ1complete problem, while testing whether there are finitely many solutions is Σ3complete. The class of languages representable by their unique solutions is exactly the class of recursive sets, though a decision procedure cannot be algorithmically constructed out of an inequality, even if a proof of solution uniqueness is attached. 1
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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Cited by 4 (3 self)
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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.
Simple language equations
 Bull. Eur. Assoc. Theor. Comput. Sci. EATCS
, 2005
"... Abstract. We survey results, both positive and negative, on regularity of maximal solutions of systems of implicit language equations and inequalities. These results concern inequalities with constant righthand sides, onesided linear inequalities, inequalities with restrictions on constants, and c ..."
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Cited by 3 (1 self)
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Abstract. We survey results, both positive and negative, on regularity of maximal solutions of systems of implicit language equations and inequalities. These results concern inequalities with constant righthand sides, onesided linear inequalities, inequalities with restrictions on constants, and commutation equations and inequalities. In addition, we present some of these results in a generalized form in order to underline common principles. 1.
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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Cited by 2 (1 self)
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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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Cited by 1 (1 self)
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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.