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Return words in Sturmian and episturmian words

Jacques Justin, Laurent Vuillon (2010)

RAIRO - Theoretical Informatics and Applications

Considering each occurrence of a word w in a recurrent infinite word, we define the set of return words of w to be the set of all distinct words beginning with an occurrence of w and ending exactly just before the next occurrence of w in the infinite word. We give a simpler proof of the recent result (of the second author) that an infinite word is Sturmian if and only if each of its factors has exactly two return words in it. Then, considering episturmian infinite words, which are a natural generalization...

Reverse mathematics of some topics from algorithmic graph theory

Peter Clote, Jeffry Hirst (1998)

Fundamenta Mathematicae

This paper analyzes the proof-theoretic strength of an infinite version of several theorems from algorithmic graph theory. In particular, theorems on reachability matrices, shortest path matrices, topological sorting, and minimal spanning trees are considered.

Rewriting on cyclic structures: Equivalence between the operational and the categorical description

Andrea Corradini, Fabio Gadducci (2010)

RAIRO - Theoretical Informatics and Applications

We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2-theories. We show that this presentation is equivalent to the well-accepted operational definition proposed by Barendregt et al. – but for the case of circular redexes , for which we propose (and justify formally) a different treatment. The categorical framework allows us to model in a concise way also automatic garbage collection and rules for sharing/unsharing and...

Root clustering of words

Gerhard Lischke (2014)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Six kinds of both of primitivity and periodicity of words, introduced by Ito and Lischke [M. Ito and G. Lischke, Math. Log. Quart. 53 (2007) 91–106; Corrigendum in Math. Log. Quart. 53 (2007) 642–643], give rise to defining six kinds of roots of a nonempty word. For 1 ≤ k ≤ 6, a k-root word is a word which has exactly k different roots, and a k-cluster is a set of k-root words u where the roots of u fulfil a given prefix relationship. We show that out of the 89 different clusters that can be considered...

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