## Conway's Problem and the commutation of languages (2001)

Venue: | Bulletin of EATCS |

Citations: | 8 - 5 self |

### BibTeX

@ARTICLE{Karhumäki01conway'sproblem,

author = {Juhani Karhumäki and Ion Petre},

title = {Conway's Problem and the commutation of languages},

journal = {Bulletin of EATCS},

year = {2001},

volume = {74},

pages = {171--177}

}

### OpenURL

### Abstract

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

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(Show Context)
Citation Context ...ords we refer to [6], [30], and [31]. For details on the notion of centralizer and the commutation of languages we refer to [25], [26], and [33]. For basic notions on Automata Theory we refer to [3], =-=[17]-=-, or [36]. In this paper we denote additively the union of two sets: L + R stands for L ∪ R. Also, for a set S, we denote by 2 S the set of all its subsets. Throughout the paper, Σ denotes a finite al... |

489 | Formal Languages
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Citation Context ...efer to [6], [30], and [31]. For details on the notion of centralizer and the commutation of languages we refer to [25], [26], and [33]. For basic notions on Automata Theory we refer to [3], [17], or =-=[36]-=-. In this paper we denote additively the union of two sets: L + R stands for L ∪ R. Also, for a set S, we denote by 2 S the set of all its subsets. Throughout the paper, Σ denotes a finite alphabet, Σ... |

277 |
Regular Algebra and Finite Machines
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Citation Context ...l known for free monoids, i.e., for finite words over a given alphabet, almost nothing is known for sets of words, although the problem was considered in the last few decades in various contexts, see =-=[11]-=-, [13], [29], [37]. Two natural and apparently very difficult combinatorial problems have crystalized in the mean time: one proposed (somewhat in passing) by Conway in 1971, [11], and another one rais... |

237 | Transductions and Context-Free Languages
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(Show Context)
Citation Context ... on Words we refer to [6], [30], and [31]. For details on the notion of centralizer and the commutation of languages we refer to [25], [26], and [33]. For basic notions on Automata Theory we refer to =-=[3]-=-, [17], or [36]. In this paper we denote additively the union of two sets: L + R stands for L ∪ R. Also, for a set S, we denote by 2 S the set of all its subsets. Throughout the paper, Σ denotes a fin... |

107 | Combinatorics of words
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(Show Context)
Citation Context ...cuss the perspectives as they appear today. 2 Preliminaries We recall here several notions and results needed throughout the paper. For basic notions and results of Combinatorics on Words we refer to =-=[6]-=-, [30], and [31]. For details on the notion of centralizer and the commutation of languages we refer to [25], [26], and [33]. For basic notions on Automata Theory we refer to [3], [17], or [36]. In th... |

67 |
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Citation Context ...If L4 = {a n b n | n ≥ 1}, then C(L) = L + 4 . Note that the notion of the centralizer of a set is usually defined in Algebra with respect to element-wise commutation, see, e.g., [2], [5], [9], [14], =-=[18]-=-. Thus, for a group (alternatively, for a semigroup, a ring, or a semiring) G and a subset S of G, the centralizer of S in G is the set {x ∈ G | xs = sx, ∀s ∈ S}. In this respect, the centralizer of a... |

61 |
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Citation Context ...3) ⊂ Σ + . (iv) If L4 = {a n b n | n ≥ 1}, then C(L) = L + 4 . Note that the notion of the centralizer of a set is usually defined in Algebra with respect to element-wise commutation, see, e.g., [2], =-=[5]-=-, [9], [14], [18]. Thus, for a group (alternatively, for a semigroup, a ring, or a semiring) G and a subset S of G, the centralizer of S in G is the set {x ∈ G | xs = sx, ∀s ∈ S}. In this respect, the... |

43 |
Free Monoids and Languages
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Citation Context ...onoids, i.e., for finite words over a given alphabet, almost nothing is known for sets of words, although the problem was considered in the last few decades in various contexts, see [11], [13], [29], =-=[37]-=-. Two natural and apparently very difficult combinatorial problems have crystalized in the mean time: one proposed (somewhat in passing) by Conway in 1971, [11], and another one raised by Ratoandroman... |

22 |
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Citation Context ... two formal power series in noncommuting variables, with coefficients in a commutative field may be characterized in a similar way, as in free monoids – these results are due to Bergman and Cohn, see =-=[2]-=-, [9], and [10]. The property conjectured above for codes is called sometimes the BTC-property – the acronym stands for Bergmantype of characterization for the commutation of two sets of words. The ab... |

16 |
Algebraic combinatorics on words (Cambridge
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Citation Context ...ctives as they appear today. 2 Preliminaries We recall here several notions and results needed throughout the paper. For basic notions and results of Combinatorics on Words we refer to [6], [30], and =-=[31]-=-. For details on the notion of centralizer and the commutation of languages we refer to [25], [26], and [33]. For basic notions on Automata Theory we refer to [3], [17], or [36]. In this paper we deno... |

16 | On the decomposition of finite languages
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Citation Context ...is a particular instance of such a system. 3.2 Periodic sets It is well-known that two words commute if and only if they have the same primitive root. Based on this, it is not difficult to prove, see =-=[32]-=-, that a set of words X commutes with a word u if and only if X ⊆ ρ(u ∗ ), where ρ(u) denotes the primitive root of u. Thus, for a word u, the centralizer of {u} is ρ(u) ∗ . If instead of a singleton,... |

12 |
Language Equations
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Citation Context ...free monoids, i.e., for finite words over a given alphabet, almost nothing is known for sets of words, although the problem was considered in the last few decades in various contexts, see [11], [13], =-=[29]-=-, [37]. Two natural and apparently very difficult combinatorial problems have crystalized in the mean time: one proposed (somewhat in passing) by Conway in 1971, [11], and another one raised by Ratoan... |

9 | Combinatorics on words — A tutorial
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- 2003
(Show Context)
Citation Context ...antype of characterization for the commutation of two sets of words. The above problems recently received some well-deserved attention and a number of different approaches have been investigated, see =-=[4]-=-, [19], [26], [33] for some presentations. We survey in this paper all known results, presenting in each case the most simple proofs known at this point. We also present a number of open problems and ... |

9 |
Centralisateurs dans les corps libres
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Citation Context ...wer series in noncommuting variables, with coefficients in a commutative field may be characterized in a similar way, as in free monoids – these results are due to Bergman and Cohn, see [2], [9], and =-=[10]-=-. The property conjectured above for codes is called sometimes the BTC-property – the acronym stands for Bergmantype of characterization for the commutation of two sets of words. The above problems re... |

9 | Codes et motifs - Ratoandromanana - 1989 |

8 |
The commutation of finite sets: a challenging problem, Theoret
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(Show Context)
Citation Context ...er for periodic sets of words. 3.3 Binary sets The centralizer in the binary case turns out to be equally easy to describe as for periodic sets. Various proofs have been proposed for this result, see =-=[8]-=-, [26], [27], but they all rely essentially on a reduction to branching languages. The next result ([27]) shows that we can always reduce Conway’s problem to the so-called branching sets of words. We ... |

8 | Factorization in noncommuting power series rings - Cohn - 1962 |

7 | On Fatou properties of rational languages - Choffrut, Karhumäki - 2000 |

6 |
Conway's Problem for three word sets, Theoret
- Karhumaki, Petre
- 2002
(Show Context)
Citation Context ...ed throughout the paper. For basic notions and results of Combinatorics on Words we refer to [6], [30], and [31]. For details on the notion of centralizer and the commutation of languages we refer to =-=[25]-=-, [26], and [33]. For basic notions on Automata Theory we refer to [3], [17], or [36]. In this paper we denote additively the union of two sets: L + R stands for L ∪ R. Also, for a set S, we denote by... |

5 | Decision questions concerning semilinearity, morphisms and commutation of languages
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- 2001
(Show Context)
Citation Context ...able question on commuting with finite sets of words and discuss the complexity of a few other decidable cases. The intriguing nature of commutation is marvellously shown by the following result from =-=[15]-=-. Theorem 18 ([15]). It is undecidable whether a given two element set of words and a given context-free language commute. The problem is decidable however for deterministic context-free languages. 11... |

5 |
The equivalence problem of finite substitutions on ab*c, with applications
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(Show Context)
Citation Context ...oblem is decidable however for deterministic context-free languages. 11sAnother problem showing the surprising difficulty of problems on finite sets of words is the following undecidability result of =-=[23]-=-. Theorem 19 ([23]). It is undecidable whether two finite substitutions φ, ψ : {a, b, c} + → Σ + are equivalent on the language ab ∗ c, i.e., whether or not the identity φ(ab n c) = ψ(ab n c) of finit... |

5 |
The branching point approach to Conway's problem
- Karhumäki, Petre
- 2002
(Show Context)
Citation Context ...ed, observe that for any set of words L, there is a unique maximal element of the set {S ⊆ Σ ∗ | LS = SL}. We call this element the monoid centralizer of L and we denote it as Cm(L). We refer to [24]-=-=[27]-=- for details. Clearly, for any L, we have C(L) ∪ {1} ⊆ Cm(L). However, this is a strict inclusion in general. The reason for this is that Cm(L) \ {1} does not commute with L for all sets of words L. E... |

4 | Challenges of commutation: an advertisement
- Karhumaki
- 2001
(Show Context)
Citation Context ...e of characterization for the commutation of two sets of words. The above problems recently received some well-deserved attention and a number of different approaches have been investigated, see [4], =-=[19]-=-, [26], [33] for some presentations. We survey in this paper all known results, presenting in each case the most simple proofs known at this point. We also present a number of open problems and discus... |

4 | Commutation Problems on Set of Words and Formal Power Series
- Petre
(Show Context)
Citation Context ...erization for the commutation of two sets of words. The above problems recently received some well-deserved attention and a number of different approaches have been investigated, see [4], [19], [26], =-=[33]-=- for some presentations. We survey in this paper all known results, presenting in each case the most simple proofs known at this point. We also present a number of open problems and discuss the perspe... |

3 |
Motifs et bases de langages
- Autebert, Boasson, et al.
- 1989
(Show Context)
Citation Context ...lthough they cannot be characterized as in free monoids. (i,[8]) The sets X = {a, a 3 , b, ab, ba, aba} and Y = X ∪ {a 2 } commute, but they cannot be expressed as unions of powers of a same set. (ii,=-=[1]-=-) The sets X = {aa, ab, ba, bb, aaa} and Y = {a, b, aa, ab, ba, bb, aaa} commute, but they cannot be expressed in terms of another set of words. 9s(iii,[8]) The sets X = {a, ab, ba, bb} and Y = X ∪ X ... |

3 | On the centralizer of a finite set
- Karhumäki, Petre
- 2000
(Show Context)
Citation Context ... Indeed, observe that for any set of words L, there is a unique maximal element of the set {S ⊆ Σ ∗ | LS = SL}. We call this element the monoid centralizer of L and we denote it as Cm(L). We refer to =-=[24]-=--[27] for details. Clearly, for any L, we have C(L) ∪ {1} ⊆ Cm(L). However, this is a strict inclusion in general. The reason for this is that Cm(L) \ {1} does not commute with L for all sets of words... |

1 | Fixed point approach to commutation of languages - Culik, Salmela, et al. - 2004 |

1 |
personal communication
- Hirvensalo
(Show Context)
Citation Context ...mputed – typically rather easily – and in most cases it coincides with X + or Σ + . On the other hand, no efficient general method to compute the centralizer in general is known. As shown in [12] and =-=[16]-=-, the centralizer can be elegantly defined as the maximal fixed point of a mapping, but this might lead to infinite iterations. The centralizer or in fact, its complement, can also be computed by “exh... |

1 | Some open problems on combinatorics of words and related areas - Karhumäki - 2000 |

1 |
The commutation with codes and ternary sets of words, preliminary version
- Karhumäki, Latteux, et al.
- 2003
(Show Context)
Citation Context ...This was indeed proved in the same paper for ternary codes using some involved combinatorial arguments. We present here a simple proof that solves the above mentioned conjecture. This results is from =-=[21]-=-. Theorem 10. For any non-periodic ternary language F ⊆ Σ + , C(F ) = F + . Proof. We can assume by Lemma 7 that F is branching. Thus, let F = {u, u ′ , v}, where pref 1(v) �∈ {pref 1(u), pref 1(u ′ )... |

1 |
The commutation with prefix codes, manuscript
- Karhumäki, Latteux, et al.
- 2003
(Show Context)
Citation Context ...was achieved by Ratoandromanana [34], in the case of prefix codes, using ingenious (and involved) techniques on codes and prefix sets. A simpler proof, using only elementary techniques is obtained in =-=[22]-=-. Conjecture 1, however, remained open in its general form. Recall that a language X is said to satisfy the BTC-property if the commutation with X can be characterized as in the statement of Conjectur... |

1 |
On the complexity of decidable cases of the commutation problem of languages, preliminary version
- Karhumäki, Plandowski, et al.
(Show Context)
Citation Context ...substitutions φ, ψ : {a, b, c} + → Σ + are equivalent on the language ab ∗ c, i.e., whether or not the identity φ(ab n c) = ψ(ab n c) of finite languages holds for all n ≥ 0. The following results of =-=[28]-=- shed more light on the complexity of language commutation. Theorem 20 ([28]). (i) Let X be a finite language given by an acyclic nondeterministic finite automaton and let Y = Σ. Testing the commutati... |

1 | On the lattice of prefix codes, Theoret Comp Sci 4270 - Restivo, Silva - 2002 |