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Axel Thue's work on repetitions in words
 Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics
, 1992
"... The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched. ..."
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The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.
Coherence in monoidal track categories
"... Abstract – We introduce homotopical methods based on rewriting on higherdimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an asphericity problem for a track category and use rewriting ..."
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Abstract – We introduce homotopical methods based on rewriting on higherdimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an asphericity problem for a track category and use rewriting methods on polygraphs to solve it. The setting is generalized to more general coherence problems, seen as 3dimensional word problems in a track category. We prove general results that, in the case of braided monoidal categories, yield the coherence theorem for braided monoidal categories.
The origins of combinatorics on words
, 2007
"... We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early ..."
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We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early results were obtained as a byproduct of investigations on various combinatorial objects. For example, paths in graphs are encoded by words in a natural way, and conversely, the Cayley graph of a group or a semigroup encodes words by paths. We give in this text an account of this twosided interaction.
Thue Systems for Pattern Recognition
, 2003
"... This report presents a synoptic overview of Thue Systems. Thue Systems were introduced in the early 1900s by the Norwegian mathematician and logician Axel Thue. In this report the author suggests ways in which such systems can be used in pattern recognition. ..."
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This report presents a synoptic overview of Thue Systems. Thue Systems were introduced in the early 1900s by the Norwegian mathematician and logician Axel Thue. In this report the author suggests ways in which such systems can be used in pattern recognition.
HIGHERDIMENSIONAL NORMALISATION STRATEGIES FOR ACYCLICITY
"... Abstract – We introduce acyclic track polygraphs, a notion of complete categorical cellular models for small categories: they are polygraphs containing generators, with additional invertible cells for relations and higherdimensional globular syzygies. We give a rewriting method to realise such a mo ..."
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Abstract – We introduce acyclic track polygraphs, a notion of complete categorical cellular models for small categories: they are polygraphs containing generators, with additional invertible cells for relations and higherdimensional globular syzygies. We give a rewriting method to realise such a model by proving that a convergent presentation canonically extends to an acyclic track polygraph. For that, we introduce normalising strategies, defined as homotopically coherent ways to relate each cell of a track polygraph to its normal form, and we prove that acyclicity is equivalent to the existence of a normalisation strategy. Using track polygraphs, we extend to every dimension the homotopical finiteness condition of finite derivation type, introduced by Squier in string rewriting theory, and we prove that it implies a new homological finiteness condition that we introduce here. The proof is based on normalisation strategies and relates acyclic track polygraphs to free abelian resolutions of the small categories they present.
CS5236 – Advanced Automata Theory
"... Advanced Automata Theory is a lecture which will first review the basics of formal languages and automata theory and then give insight into specific topics from the theory of automata theory. In computer science, automata are an important tool for many theoretical results and various types of automa ..."
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Advanced Automata Theory is a lecture which will first review the basics of formal languages and automata theory and then give insight into specific topics from the theory of automata theory. In computer science, automata are an important tool for many theoretical results and various types of automata have been used to characterise complexity classes. The lecture will give a deeper understanding of automata theory, describe the Chomsky hierarchy and introduce to various advanced topics like automatic structures, automata on infinite words, automata on trees and the learnability of classes of regular languages from queries and from positive data.