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By Igor Potapov

This booklet constitutes the court cases of the nineteenth foreign convention on advancements in Language conception, DLT 2015, held in Liverpool, united kingdom. The 31 papers offered including five invited talks have been rigorously reviewed and chosen from fifty four submissions. Its scope is particularly basic and comprises, between others, the subsequent issues and components: combinatorial and algebraic homes of phrases and languages, grammars, acceptors and transducers for strings, bushes, graphs, arrays, algebraic theories for automata and languages, codes, effective textual content algorithms, symbolic dynamics, selection difficulties, relationships to complexity concept and good judgment, photograph description and research, polyominoes and bidimensional styles, cryptography, concurrency, mobile automata, bio-inspired computing, and quantum computing.

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Extra resources for Developments in Language Theory: 19th International Conference, DLT 2015, Liverpool, UK, July 27-30, 2015, Proceedings.

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Consider the free abelian monoid (N, +)n with n 1, simply denoted by Nn . , n}, thus writing g(k) for the kth entry of an element g, and use a multiplicative notation: f g = h means ∀k (f (k) + g(k) = h(k)). , an }, where ai is defined by ai (k) = 1 for k = i, and 0 otherwise. Then An is a basis of Nn as an abelian monoid. It is straightforward to obtain a normal form nfLex for Nn with respect to An by fixing a linear ordering on An , for instance a1 < ··· < an , and, for g in Nn , defining nfLex (g) to be the lexicographically smallest word representing g.

In Sec. 1, we analyze two motivating examples of greedy normal forms, involving free abelian monoids, a toy case that already contains the main ideas, and braid monoids, a more complicated case. Extending these examples, we describe in Sec. 2 the mechanism of the Δ-normal form in the now classical framework of Garside monoids. Next, in Sec. 3, we explain how most of the results can be generalized and, at the same time, simplified, using the notion of an S-normal form derived from a Garside family.

Sp in Div(Δ), the following conditions are equivalent: (i) The sequence s1 | ··· |sp is Δ-normal. (ii) For every i < p, one has si = H(si si+1 ···sp ). (iii) For every i < p, one has si = H(si si+1 ). Proof. Assume that s1 | ··· |sp is Δ -normal. By definition, we have si Δ and si si si+ ···sp , whence si H(si si+1 ···sp ) by the definition of a left-gcd. Conversely, let s be the right-lcm of si and H(si si+1 ···sp ). As we have si Δ and H(si si+1 ···sp ) Δ, the definition of a right-lcm implies s Δ.

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