## Equations on Trees (1996)

Venue: | Proceedings MFCS'96. Lecture Notes in Computer Science n.1113 |

Citations: | 4 - 4 self |

### BibTeX

@INPROCEEDINGS{Mantaci96equationson,

author = {Sabrina Mantaci and Antonio Restivo},

title = {Equations on Trees},

booktitle = {Proceedings MFCS'96. Lecture Notes in Computer Science n.1113},

year = {1996},

pages = {443--456},

publisher = {Springer Verlag}

}

### OpenURL

### Abstract

. We introduce the notion of equation on trees, generalizing the corresponding notion for words, and we develop the first steps of a theory of tree equations. The main result of the paper states that, if a pair of trees is the solution of a tree equation with two indeterminates, then the two trees are both powers of the same tree. As an application, we show that a tree can be expressed in a unique way as a power of a primitive tree. This extends a basic result of combinatorics on words to trees. Some open problems are finally proposed. 1 Introduction In this paper we are mainly concerned with k-ary trees whose vertices are labeled by letters of an alphabet A. We look at a labeled k-ary tree as a generalization of a word, in the sense that words correspond to the particular case of k = 1, i.e. to unary trees. a a b c b a b a c b A binary tree A word Given that combinatorics on words is a well developed theory, with several applications to computer science, it appears to be a natural ...

### Citations

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Algebraic combinatorics on words
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- 2002
(Show Context)
Citation Context ...nd suggestions for further researches are finally proposed. 2 Basic definitions We recall here few notions and notations concerning words and free monoids; we refer for unexplained ones to [1] and to =-=[7]-=-. Some of the basic definitions and notations concerning trees are from [10]. Free monoids and semigroups generated by a (not necessairely finite) alphabet A are denoted by A and A + = A n ffflg respe... |

159 |
Problem of Solvability of Equations in Free Semigroup, Math. Sbornik 103
- Makanin
- 1977
(Show Context)
Citation Context ...ew here developed is very different from that considered, for instance, in [2]. A very general problem in the theory of tree equations is to decide the existence of a solution of a given equation. In =-=[9]-=- Makanin solved this problem in the special case of word equations, i.e. in the case of unary trees. We do not know whether this result can be extended to general k-ary trees. In section 4 we investig... |

107 |
Equations in free groups
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(Show Context)
Citation Context ...rees is solution of such an equation, then the two trees are both powers of the same tree. In the particular case of unary trees this corresponds to a classical result in the theory of word equations =-=[8]-=-. As an application, we show in section 5 that a tree can be expressed in a unique way as a power of a primitive tree. This extends a basic result of combinatorics on words. Some open problems and sug... |

40 |
Minimal and complete word unification
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(Show Context)
Citation Context ...define a tree equation as a pair of ordered trees. A general problem in this theory is to decide the existence of a solution of a given equation. This problem has been solved by Makanin [9] (cf. also =-=[5]-=-) in the special case of word equations, i.e. in the case of unary trees. It is not known whether this result can be extended to general k-ary trees. Another related research line is to develop method... |

28 |
The algebraic structure of rooted trees
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- 1978
(Show Context)
Citation Context ...eneralizing the corresponding notion for words, and we give the first elements of a theory of tree equations. The point of view here developed is very different from that considered, for instance, in =-=[2]-=-. A very general problem in the theory of tree equations is to decide the existence of a solution of a given equation. In [9] Makanin solved this problem in the special case of word equations, i.e. in... |

21 |
A combinatorial problem in the theory of free monoids
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Citation Context ...e We now prove that all solutions of tree equations in two indeterminates have a particular structure. This result generalizes the analogous result for equations on words with two indeterminates (cf. =-=[6], [7], [8]) -=-Theorem 13. Let (oe; i) be the solution of a tree equation on two indeterminates. Then there exists a tree �� such that oe; i 2 �� # , that is, oe and i are powers of the same tree ��. Pro... |

6 | A periodicity theorem for trees
- Giammarresi, Mantaci, et al.
- 1994
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Citation Context ...o trees. It can be considered a special part of the theory of tree automata and languages (cf. [11], [12]). Recent contributions in this direction have been given by the authors and al. in [3] and in =-=[4]-=-. In section 2 of this paper we introduce the basic notions of combinatorics on trees: subtree, prefix, suffix, concatenation, power, primitiveness, morphism. The theory of word equations constitutes ... |

3 |
Logical aspects in the theory of tree languages
- Thomas
- 1984
(Show Context)
Citation Context ...is approach was initiated by M. Nivat in [10], where in particular he extended the notion of code to trees. It can be considered a special part of the theory of tree automata and languages (cf. [11], =-=[12]-=-). Recent contributions in this direction have been given by the authors and al. in [3] and in [4]. In section 2 of this paper we introduce the basic notions of combinatorics on trees: subtree, prefix... |

2 |
Tree automata|an informal survey
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Citation Context ...es. This approach was initiated by M. Nivat in [10], where in particular he extended the notion of code to trees. It can be considered a special part of the theory of tree automata and languages (cf. =-=[11]-=-, [12]). Recent contributions in this direction have been given by the authors and al. in [3] and in [4]. In section 2 of this paper we introduce the basic notions of combinatorics on trees: subtree, ... |

1 | Periodicities on trees
- Giammarresi, Mantaci, et al.
- 1995
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Citation Context ...n of code to trees. It can be considered a special part of the theory of tree automata and languages (cf. [11], [12]). Recent contributions in this direction have been given by the authors and al. in =-=[3]-=- and in [4]. In section 2 of this paper we introduce the basic notions of combinatorics on trees: subtree, prefix, suffix, concatenation, power, primitiveness, morphism. The theory of word equations c... |

1 |
Binary tree codes". Tree automata and languages; pagg
- Nivat
- 1992
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Citation Context ... developed theory, with several applications to computer science, it appears to be a natural task to generalize results and methods of this theory to trees. This approach was initiated by M. Nivat in =-=[10]-=-, where in particular he extended the notion of code to trees. It can be considered a special part of the theory of tree automata and languages (cf. [11], [12]). Recent contributions in this direction... |