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Ackermannian and PrimitiveRecursive Bounds with Dickson’s Lemma
"... Dickson’s Lemma is a simple yet powerful tool widely used in decidability proofs, especially when dealing with counters or related data structures in algorithmics, verification and modelchecking, constraint solving, logic, etc. While Dickson’s Lemma is wellknown, most computer scientists are not ..."
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Dickson’s Lemma is a simple yet powerful tool widely used in decidability proofs, especially when dealing with counters or related data structures in algorithmics, verification and modelchecking, constraint solving, logic, etc. While Dickson’s Lemma is wellknown, most computer scientists are not aware of the complexity upper bounds that are entailed by its use. This is mainly because, on this issue, the existing literature is not very accessible. We propose a new analysis of the length of bad sequences over (N k, ≤), improving on earlier results and providing upper bounds that are essentially tight. This analysis is complemented by a “user guide” explaining through practical examples how to easily derive complexity upper bounds from Dickson’s Lemma.
LAVER’S RESULTS AND LOWDIMENSIONAL TOPOLOGY
"... Abstract. In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in lowdimensional topology, namely the existence of what is now known as the Laver tables and the wellfoundedness of the standard ordering of positive brai ..."
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Abstract. In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in lowdimensional topology, namely the existence of what is now known as the Laver tables and the wellfoundedness of the standard ordering of positive braids. Here we present these results and discuss the way they could be used in topological applications. Richard Laver established two remarkable results that might lead to significant applications in lowdimensional topology, namely the existence of a series of finite structures satisfying the leftselfdistributive law, now known as the Laver tables, and the wellfoundedness of the standard ordering of Artin’s positive braids. In this text, we shall explain the precise meaning of these results and discuss their (past or future) applications in topology. In one word, the current situation is that, although the depth of Laver’s results is not questionable, few topological applications have been found. However, the example of braid groups orderability shows that, once initial obstructions are solved, topological applications of algebraic
MSc in Logic
, 2011
"... Finding the phase transition for Friedman’s long finite sequences ..."
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Periodicity and Repetition in Combinatorics on Words
, 2004
"... This thesis concerns combinatorics on words. I present many results in this area, united by the common themes of periodicity and repetition. Most of these results have already appeared in journal or conference articles. Chapter 2  Chapter 5 contain the most significant contribution of this thesis ..."
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This thesis concerns combinatorics on words. I present many results in this area, united by the common themes of periodicity and repetition. Most of these results have already appeared in journal or conference articles. Chapter 2  Chapter 5 contain the most significant contribution of this thesis in the area of combinatorics on words. Below we give a brief synopsis of each chapter. Chapter 1 introduces the subject area in general and some background information. Chapter 2 and Chapter 3 grew out of attempts to prove the Decreasing Length Conjecture (DLC). The DLC states that if # is a morphism over an alphabet of size n then for any word w, there exists 0 (w). The DLC was proved by S. Cautis and S. Yazdani in Periodicity, morphisms, and matrices in Theoret. Comput. Sci.