Sturmian Trees

Berstel, Jean; Boasson, Luc; Carton, Olivier; Fagnot, Isabelle

doi:10.1007/s00224-009-9228-0

Cited by 14 publications

(41 citation statements)

References 6 publications

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“…In 1997 M. Nivat conjectured that an analogue of these statements might hold in higher dimensions, specifically for labelings of the integer lattice in Z 2 by a finite alphabet; see for example [15,16,26] for the precise statement, some positive progress, and references. Extensions of these statements to labeled trees were provided in [11,12]. See also [18,20].…”

Section: Introductionmentioning

confidence: 99%

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Tree shift topological entropy

Petersen

Salama

2018

Theoretical Computer Science

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Section: Introductionmentioning

confidence: 99%

“…Acknowledgments. We thank Professors Jung-Chao Ban, Chih-Hung Chang, and Nic Ormes for discussions on this topic, Professor Olivier Carton and his colleagues for alerting us to the papers [11,12], and the referees for helpful comments..…”

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confidence: 99%

Tree shift topological entropy

Petersen

Salama

2018

Theoretical Computer Science

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“…In [5] we proved that for unary cyclic automata associated to finite Fibonacci words there is a unique execution with running time Θ(n log n). Such a result is extended in [3].…”

Section: Introductionmentioning

confidence: 73%

“…Indeed at the first step the waiting set contains the pairs ( [1,2], 0) and ([1, 2], 1). If we choose the first pair, the class [1,2] is split in [1] and [2], if we choose the second one, the class [3,4,5] is split in [3,4] and [5].…”

Section: The Algorithmmentioning

confidence: 99%

Hopcroft's algorithm and tree-like automata

Castiglione

Restivo

Sciortino

2011

RAIRO-Theor. Inf. Appl.

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Minimizing a deterministic finite automata (DFA) is a very important problem in theory of automata and formal languages. Hopcroft's algorithm represents the fastest known solution to the such a problem. In this paper we analyze the behavior of this algorithm on a family binary automata, called tree-like automata, associated to binary labeled trees constructed by words. We prove that all the executions of the algorithm on tree-like automata associated to trees, constructed by standard words, have running time with the same asymptotic growth rate. In particular, we provide a lower and upper bound for the running time of the algorithm expressed in terms of combinatorial properties of the trees. We consider also tree-like automata associated to trees constructed by de Brujin words, and we prove that a queue implementation of the waiting set gives a Θ(n log n) execution while a stack implementation produces a linear execution. Such a result confirms the conjecture given in [A. Paun, M. Paun and A. Rodríguez-Patón. Theoret. Comput. Sci. 410 (2009) 2424-2430.] formulated for a family of unary automata and, in addition, gives a positive answer also for the binary case.

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“…With this motivation in mind, we consider unrooted trees, rather than rooted binary trees which were studied in computer science ( [4], [5], [9]). Theory of subword complexity and Sturmian colorings developed in this article is quite different from…”

Section: Introductionmentioning

confidence: 99%

Subword complexity and Sturmian colorings of regular trees

Kim¹,

Lim²

2013

Ergod. Th. Dynam. Sys.

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Abstract. In this article, we study subword complexity of colorings of regular trees. We characterize colorings of bounded subword complexity and study Sturmian colorings, which are colorings of minimal unbounded subword complexity.We classify Sturmian colorings using their type sets. We show that any Sturmian coloring is a lifting of a coloring on a quotient graph of the tree which is a geodesic or a ray with loops possibly attached, thus a lifting of an "infinte word". We further give a complete characterization of the quotient graph for eventually periodic ones.

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Sturmian Trees

Cited by 14 publications

References 6 publications

Tree shift topological entropy

Tree shift topological entropy

Hopcroft's algorithm and tree-like automata

Subword complexity and Sturmian colorings of regular trees

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