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## The power of commuting with finite sets of words (2005)

Venue: | In Proc. STACS’05, Springer LNCS 3404 |

Citations: | 19 - 3 self |

### Citations

355 |
Regular Algebra and Finite Machines.
- Conway
- 1971
(Show Context)
Citation Context ...system consisting of two equations [20]. Known results on regularity of maximal solutions of systems of implicit language inequalities are surveyed in [14]. Such issues were first addressed by Conway =-=[8]-=-, who observed that inequalities of the form E ⊆ L, where E is a regular function of variables and L is a regular language, possess only finitely many maximal solutions, all of them are regular and co... |

353 |
Decidability of second-order theories and automata on infinite trees
- Rabin
- 1969
(Show Context)
Citation Context ...des are regular [16]. Moreover, if all constant languages occurring in the system are regular, then computability of the largest solution follows from Rabin’s results on MSO logic over infinite trees =-=[21]-=-; actually, the computation of the solution is an ExpTime-complete problem [1, 5, 2]. The problem of regularity of the largest language commuting with a given regular language was formulated already b... |

74 | The complexity of set constraints
- Aiken, Kozen, et al.
- 1993
(Show Context)
Citation Context ...em are regular, then computability of the largest solution follows from Rabin’s results on MSO logic over infinite trees [21]; actually, the computation of the solution is an ExpTime-complete problem =-=[1, 5, 2]-=-. The problem of regularity of the largest language commuting with a given regular language was formulated already by Conway [8, p. 55 and 124] in 1971. There are actually two variants of the problem ... |

30 |
Centralizers in free associative algebras,
- Bergman
- 1969
(Show Context)
Citation Context ...formal series in non-commuting variables over a field (in particular, for multisets ∗ Supported by the project MSM0021622409 of the Ministry of Education of the Czech Republic. 1sof words) by Bergman =-=[3]-=- and Cohn [7], respectively. But in the case of languages the situation is completely different. Systems of language equations and inequalities were studied mainly in connection with context-free lang... |

18 | Co-definite set constraints
- Charatonik, Podelski
- 1998
(Show Context)
Citation Context ...em are regular, then computability of the largest solution follows from Rabin’s results on MSO logic over infinite trees [21]; actually, the computation of the solution is an ExpTime-complete problem =-=[1, 5, 2]-=-. The problem of regularity of the largest language commuting with a given regular language was formulated already by Conway [8, p. 55 and 124] in 1971. There are actually two variants of the problem ... |

16 | Unification in a description logic with transitive closure of roles
- Baader, Küsters
(Show Context)
Citation Context ...em are regular, then computability of the largest solution follows from Rabin’s results on MSO logic over infinite trees [21]; actually, the computation of the solution is an ExpTime-complete problem =-=[1, 5, 2]-=-. The problem of regularity of the largest language commuting with a given regular language was formulated already by Conway [8, p. 55 and 124] in 1971. There are actually two variants of the problem ... |

14 |
Language Equations
- Leiss
- 1999
(Show Context)
Citation Context ...anguages can be characterized as components of smallest or largest solutions of explicit systems of right-linear equations. Regular solutions of more general systems were studied for example by Leiss =-=[17]-=-. For systems of implicit right-linear inequalities, i.e. inequalities of the form K ∪ K1X1 ∪ · · · ∪ KnXn ⊆ L ∪ L1X1 ∪ · · · ∪ LnXn , where K, K1, ..., Kn and L, L1, ..., Ln are constant languages, i... |

12 |
The commutation of a finite set: a challenging problem, Theoret
- Choffrut, Karhum\aki, et al.
(Show Context)
Citation Context ...) and C+(L) are in general different and no direct relation between them (except for the obvious inclusion C+(L) ⊆ C(L)) has been found yet. The problem was recently studied in several articles (e.g. =-=[22, 6, 9, 10]-=-), but affirmative answers were given only for regular codes [10] and at most ternary sets of words [9]. Moreover, it even remained an open problem whether the largest language commuting with a given ... |

11 | Two problems on commutation of languages, in
- Karhumäki, Petre
- 2004
(Show Context)
Citation Context ...se of a general result about systems of language equations, which is proved in [19]. A summary of known results concerning commutation of languages and some examples can be found in the recent survey =-=[12]-=-. In this paper we give the most negative possible answer to Conway’s problem by showing that there exists a finite language L such that C(L) is not recursively enumerable. More precisely, we show tha... |

11 |
Codes et motifs
- Ratoandromanana
- 1989
(Show Context)
Citation Context ...) and C+(L) are in general different and no direct relation between them (except for the obvious inclusion C+(L) ⊆ C(L)) has been found yet. The problem was recently studied in several articles (e.g. =-=[22, 6, 9, 10]-=-), but affirmative answers were given only for regular codes [10] and at most ternary sets of words [9]. Moreover, it even remained an open problem whether the largest language commuting with a given ... |

10 |
Factorization in non-commutative power series rings
- Cohn
- 1962
(Show Context)
Citation Context ... in non-commuting variables over a field (in particular, for multisets ∗ Supported by the project MSM0021622409 of the Ministry of Education of the Czech Republic. 1sof words) by Bergman [3] and Cohn =-=[7]-=-, respectively. But in the case of languages the situation is completely different. Systems of language equations and inequalities were studied mainly in connection with context-free languages since t... |

7 |
Regular solutions of language inequalities and well quasi-orders, in
- Kunc
(Show Context)
Citation Context ...e encoded into a solution of a commutation equation. This contrasts with the fact that the largest solution of the inequality XK ⊆ LX is regular provided the language L is regular, as demonstrated in =-=[13]-=-. On the other hand, negative results analogous to those for commutation equations were proved in [15] for systems consisting of two such inequalities. We formulate our results for the case of languag... |

6 | The branching point approach to Conway's problem
- Karhumäki, Petre
- 2002
(Show Context)
Citation Context ...lts for the case of languages C+(L) too, and further we show that for a regular language L the difference C(L)\C+(L) also does not have to be recursively enumerable, which answers a question posed in =-=[11]-=-. In addition, our results disprove Conway’s conjecture [8, p. 129] stating that every maximal solution of a system of so-called semi-linear inequalities is context-free. Before dealing with the main ... |

4 |
Commutation with codes, Theoret
- Karhumäki, Latteux, et al.
- 2005
(Show Context)
Citation Context ...) and C+(L) are in general different and no direct relation between them (except for the obvious inclusion C+(L) ⊆ C(L)) has been found yet. The problem was recently studied in several articles (e.g. =-=[22, 6, 9, 10]-=-), but affirmative answers were given only for regular codes [10] and at most ternary sets of words [9]. Moreover, it even remained an open problem whether the largest language commuting with a given ... |

4 |
On language inequalities XK ⊆ LX
- Kunc
- 2005
(Show Context)
Citation Context ...ution of the inequality XK ⊆ LX is regular provided the language L is regular, as demonstrated in [13]. On the other hand, negative results analogous to those for commutation equations were proved in =-=[15]-=- for systems consisting of two such inequalities. We formulate our results for the case of languages C+(L) too, and further we show that for a regular language L the difference C(L)\C+(L) also does no... |

3 | Simple Language Equations
- Kunc
(Show Context)
Citation Context ...it equation and precisely defined by an explicit system consisting of two equations [20]. Known results on regularity of maximal solutions of systems of implicit language inequalities are surveyed in =-=[14]-=-. Such issues were first addressed by Conway [8], who observed that inequalities of the form E ⊆ L, where E is a regular function of variables and L is a regular language, possess only finitely many m... |

3 |
Largest solutions of left-linear language inequalities
- Kunc
- 2005
(Show Context)
Citation Context ... · ∪ LnXn , where K, K1, ..., Kn and L, L1, ..., Ln are constant languages, it is known that their largest solutions are always regular provided all constant languages on right-hand sides are regular =-=[16]-=-. Moreover, if all constant languages occurring in the system are regular, then computability of the largest solution follows from Rabin’s results on MSO logic over infinite trees [21]; actually, the ... |

3 |
On computational universality in language equations
- Okhotin
- 2005
(Show Context)
Citation Context ...ing concatenation and all Boolean operations [19] and that each such language can be encoded even in a single explicit equation and precisely defined by an explicit system consisting of two equations =-=[20]-=-. Known results on regularity of maximal solutions of systems of implicit language inequalities are surveyed in [14]. Such issues were first addressed by Conway [8], who observed that inequalities of ... |

2 |
Commutation with Ternary Sets of Words
- Karhumäki, Latteux, et al.
(Show Context)
Citation Context |

2 |
Decision problems for language equations. Submitted for publication, available at http://www.cs.queensu.ca/home/okhotin/. Preliminary version
- Okhotin
- 2003
(Show Context)
Citation Context ...o-recursively enumerable) languages are exactly languages definable as unique (smallest, largest) solutions of systems of implicit language inequalities using concatenation and all Boolean operations =-=[19]-=- and that each such language can be encoded even in a single explicit equation and precisely defined by an explicit system consisting of two equations [20]. Known results on regularity of maximal solu... |

1 | On conjugacy of languages
- Cassaigne, Karhumäki, et al.
- 2001
(Show Context)
Citation Context ...cidable whether two starfree languages are conjugated via some language containing the empty word. This result is a step towards dealing with basic problems about conjugacy of languages formulated in =-=[4]-=-. Section 4 is devoted to the case of finite languages; by encoding the example from the beginning of Section 3 into finitely many words, we show that the language C(L) can be non-regular even for a f... |