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From Combinatorial Games to Shape-Symmetric Morphisms

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Substitution and Tiling Dynamics: Introduction to Self-inducing Structures

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2273))

Abstract

Siegel suggests in his book on combinatorial games that quite simple games provide us with challenging problems: “No general formula is known for computing arbitrary Grundy values of Wythoff’s game. In general, they appear chaotic, though they exhibit a striking fractal-like pattern.”. This observation is the first motivation behind this chapter. We present some of the existing connections between combinatorial game theory and combinatorics on words. In particular, multidimensional infinite words can be seen as tilings of \(\mathbb {N}^d\). They naturally arise from subtraction games on d heaps of tokens. We review notions such as k-automatic, k-regular or shape-symmetric multidimensional words. The underlying general idea is to associate a finite automaton with a morphism.

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Notes

  1. 1.

    A video is available at http://library.cirm-math.fr/.

  2. 2.

    Consider a simple path v 0 → v 1 →⋯ → v r of maximal length r in a finite acyclic graph. Then v r has out-degree zero. Proceed by contradiction and assume that there is an edge starting from v r. Either it goes to one of the v i’s with i < r and it creates a cycle. Or, it goes to some other vertex and we may build a longer simple path. Both situations lead to a contradiction.

  3. 3.

    The following proof is inspired by the one found in Thomas S. Ferguson’s lecture notes on CGT.

  4. 4.

    A coding is a morphism from A ∗ to B ∗ where the image of every letter has length 1.

  5. 5.

    This means that every letter of the alphabet appears at least once in x.

  6. 6.

    When writing this chapter, a paper by Thijmen J. P. Krebs appeared on arXiv [55].

  7. 7.

    Compared with complete functions in Definition 5.4.8.

  8. 8.

    Integer are the only rational numbers that are Pisot numbers.

  9. 9.

    The first few values may be checked by hand.

  10. 10.

    One can relate this result to a theorem of Kummer. The p-adic valuation of \(\binom {m}{n}\) is the number of carries when adding n to m − n in base p. See, e.g., [77] and the references therein.

  11. 11.

    One could relax the assumption about regularity of the language on which the numeration system is built to encompass a larger framework. Nevertheless, most of the nice properties that we shall present (in particular, the equivalence with morphic words) do not hold without the regularity assumption.

  12. 12.

    A set \(X\subseteq \mathbb {N}\) is k-recognizable if \( \operatorname {\mathrm {rep}}_k(X)\subseteq \{0,\ldots ,k-1\}^*\) is recognized by a DFA or, equivalently, if the characteristic sequence of X is k-automatic.

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Acknowledgements

Even though it was some hard work preparing the lectures and this chapter, I am quite happy with the final result (with this kind of exercise, you always selfishly learn a lot and ask yourself new questions). I therefore warmly thank Professors Shigeki Akiyama and Pierre Arnoux for their invitation to contribute to this school. I would also like to thank Eric Duchêne for his constant help when collaborating on game related problems. I also had several colleagues reading drafts of this chapter: first M. Stipulanti and then, E. Duchêne, J. Leroy and A. Parreau. I thank them all for their feedback. Finally, I thank the anonymous reviewer for his/her careful reading and many suggestions.

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  1. University of Liège, Department of Mathematics, Liège, Belgium

    Michel Rigo

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  1. Michel Rigo
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  1. Institute of Mathematics, University of Tsukuba, Tsukuba, Japan

    Shigeki Akiyama

  2. Institut de Mathématiques de Marseille (I2M), Aix-Marseille University, Marseille Cedex 09, France

    Pierre Arnoux

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Rigo, M. (2020). From Combinatorial Games to Shape-Symmetric Morphisms. In: Akiyama, S., Arnoux, P. (eds) Substitution and Tiling Dynamics: Introduction to Self-inducing Structures. Lecture Notes in Mathematics, vol 2273. Springer, Cham. https://doi.org/10.1007/978-3-030-57666-0_5

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