Scientific conference in universities or research centers (Scientific conferences in universities or research centers)
From combinatorial games to shape-symmetric morphisms
2017

#### Files

##### Full Text
Rigo.pdf
Author preprint (1.96 MB)
Beamer of the 3 courses
##### Annexes
Rigo-lecture3.pdf
Publisher postprint (1.41 MB)
Updated lecture for the third lecture

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#### Details

Keywords :
Combinatorial game; words; multidimensional; formal language; numeration system; shape-symmetry
Abstract :
[en] The general aim of these lectures is to present some interplay between combinatorial game theory (CGT) and combinatorics on (multidimensional) words. In the first introductory lecture, we present some basic concepts from combinatorial game theory (positions of a game, Nim-sum, Sprague-Grundy function, Wythoff's game, ...). We also review some concepts from combinatorics on words. We thus introduce the well-known k-automatic sequences and review some of their characterizations (in terms of morphisms, finiteness of their k-kernel,...). These sequences take values in a finite set but the Sprague-Grundy function of a game, such as Nim of Wythoff, is usually unbounded. This provides a motivation to introduce k-regular sequences (in the sense of Allouche and Shallit) whose k-kernel is not finite, but finitely generated. In the second lecture, games played on several piles of token naturally give rise to a multidimensional setting. Thus, we reconsider k-automatic and k-regular sequences in this extended framework. In particular, determining the structure of the bidimensional array encoding the (loosing) P-positions of the Wythoff's game is a long-standing and challenging problem in CGT. Wythoff's game is linked to non-standard numeration system: P-positions can be determined by writing those in the Fibonacci system. Next, we present the concept of shape-symmetric morphism: instead of iterating a morphism where images of letters are (hyper-)cubes of a fixed length k, one can generalize the procedure to images of various parallelepipedic shape. The shape-symmetry condition introduced twenty years ago by Maes permits to have well-defined fixed point. In the last lecture, we move to generalized numeration systems: abstract numeration systems (built on a regular language) and link them to morphic (multidimensional) words. In particular, pictures obtained by shape-symmetric morphisms coincide with automatic sequences associated with an abstract numeration system. We conclude these lectures with some work in progress about games with a finite rule-set. This permits us to discuss a bit Presburger definable sets.
Disciplines :
Mathematics
Author, co-author :
Rigo, Michel  ;  Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Language :
English
Title :
From combinatorial games to shape-symmetric morphisms
Publication date :
November 2017
Event name :
Research school: Tiling Dynamical System
Event organizer :
Jean-Morlet chair: Shigeki Akiyama, Pierre Arnoux
Event date :
from 20-11-2017 to 24-11-2017
Audience :
International
Available on ORBi :
since 20 November 2017

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