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Periodicities on Trees
, 1995
"... We introduce the notion of periodicity for kary labeled trees: roughly speaking, a tree is periodic if it can be obtained by a sequence of concatenations of a smaller tree plus a "remainder". The period is the shape of such smaller tree (i.e. the corresponding unlabeled tree). This defini ..."
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We introduce the notion of periodicity for kary labeled trees: roughly speaking, a tree is periodic if it can be obtained by a sequence of concatenations of a smaller tree plus a "remainder". The period is the shape of such smaller tree (i.e. the corresponding unlabeled tree). This definition reduces to the classical one for string when restricted to the case of unary trees. Then, we define the greatest common divisor of two unlabeled trees and relate right congruences to unlabeled trees. This allows us to give a characterization of tree periodicity in terms of right congruences and then to prove a periodicity theorem for trees that is a generalization to trees of the Fine and Wilf's periodicity theorem for words. Keywords: Congruence, periodicity, labeled tree. Work partially supported by the ESPRIT II Basic Research Actions Program of the EC under Project ASMICS 2 (contract No. 6317) and in part by the Italian Ministry of Universities and Scientific Research MURST 40% Algoritmi, ...
Defect Theorems for Trees
 Proceedings of the &quot;8th International Conference Automata and Formal Languages&quot;, Salg'otarj'an (Hungary
, 1996
"... We generalize different notions of a rank of a set of words to sets of trees. We prove that almost all of those ranks can be used to formulate a defect theorem. However, as we show, the prefix rank forms an exception. ..."
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We generalize different notions of a rank of a set of words to sets of trees. We prove that almost all of those ranks can be used to formulate a defect theorem. However, as we show, the prefix rank forms an exception.
Codes and Equations on Trees
, 1998
"... The objective of this paper is to study, by new formal methods, the notion of tree code introduced by M. Nivat in [23]. In particular we introduce the notion of stability for sets of trees closed under concatenation. This allows us to give a characterization of tree codes which is very close to the ..."
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The objective of this paper is to study, by new formal methods, the notion of tree code introduced by M. Nivat in [23]. In particular we introduce the notion of stability for sets of trees closed under concatenation. This allows us to give a characterization of tree codes which is very close to the algebraic characterization of word codes in terms of free monoids. We further define the stable hull of a set of trees and derive a defect theorem for trees, which generalizes the analogous result for words. As a consequence we obtain some properties of tree codes having two elements. Moreover we propose a new algorithm to test whether a finite set of trees is a tree code. The running time of the algorithm is polynomial in the size of the input. We also introduce the notion of tree equation as a complementary point of view to tree codes. The main problem emerging in this approach is to decide the satisfiability of tree equations: it is a special case of second order unification, and it remains still open.
Tree codes and Equations
"... Contents 1 Prerequisites in Combinatorics on words 1 1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Codes and the defect theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Stable, left unitary and right unitary monoids . . . . . . . . . ..."
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Contents 1 Prerequisites in Combinatorics on words 1 1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Codes and the defect theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Stable, left unitary and right unitary monoids . . . . . . . . . . . . 4 1.2.2 The defect theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 An algorithm for the code problem . . . . . . . . . . . . . . . . . . 6 1.3 Word Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Generalities on trees 11 2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Operations between kary trees . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Tree codes 25 3.1 Factorizations on trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Stable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .