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Two-player Domino games

Benjamin Hellouin de Menibus, Rémi Pallen

TL;DR

This work studies a two-player Domino game on the infinite grid $\mathbb{Z}^d$, where $A$ aims to realize a pattern from a finite target set $\mathcal{F}$ and $B$ seeks to prevent it. It shows the Domino game problem on $\mathbb{Z}^d$ for $d>1$ is $\Sigma^0_1$-complete (hence undecidable) by a reduction from the classical Domino problem and proves that if $A$ wins, the win occurs in bounded time. A bounded-time variant is shown decidable, and a non-alternating variant with turn-order words $s$ (balanced) is characterized: the set of $s$ with $A$-winning is a subshift and the decision problem is decidable in many cases depending on the frequency $f_A(s)$, with a complete partition of several regimes. The paper connects symbolic dynamics and computability to game-theoretic tiling, providing insights into why determining winners in infinite-grid games is intricate, and it poses several open questions about the complexity of strategies and pass-less variants.

Abstract

We introduce a 2-player game played on an infinite grid, initially empty, where each player in turn chooses a vertex and colours it. The first player aims to create some pattern from a target set, while the second player aims to prevent it. We study the problem of deciding which player wins, and prove that it is undecidable. We also consider a variant where the turn order is not alternating but given by a balanced word, and we characterise the decidable and undecidable cases.

Two-player Domino games

TL;DR

This work studies a two-player Domino game on the infinite grid , where aims to realize a pattern from a finite target set and seeks to prevent it. It shows the Domino game problem on for is -complete (hence undecidable) by a reduction from the classical Domino problem and proves that if wins, the win occurs in bounded time. A bounded-time variant is shown decidable, and a non-alternating variant with turn-order words (balanced) is characterized: the set of with -winning is a subshift and the decision problem is decidable in many cases depending on the frequency , with a complete partition of several regimes. The paper connects symbolic dynamics and computability to game-theoretic tiling, providing insights into why determining winners in infinite-grid games is intricate, and it poses several open questions about the complexity of strategies and pass-less variants.

Abstract

We introduce a 2-player game played on an infinite grid, initially empty, where each player in turn chooses a vertex and colours it. The first player aims to create some pattern from a target set, while the second player aims to prevent it. We study the problem of deciding which player wins, and prove that it is undecidable. We also consider a variant where the turn order is not alternating but given by a balanced word, and we characterise the decidable and undecidable cases.
Paper Structure (20 sections, 13 theorems, 3 equations, 1 figure)

This paper contains 20 sections, 13 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Subshifts
  4. Computability
  5. The Domino game
  6. Game theory
  7. Complexity of the Domino game problem
  8. Membership
  9. Hardness
  10. Assume that $X_{\mathcal{F}} = \emptyset$.
  11. Assume that $X_{\mathcal{F}}\neq \emptyset$
  12. Games with bounded time
  13. Non-alternating play
  14. Case $f_A(s)=0$.
  15. Case $f_A(s)>\frac{1}{2}$.
  16. ...and 5 more sections

Key Result

theorem thmcountertheorem

The Domino game problem on $\mathbb{Z}^d$, $d>1$, is $\Sigma^0_1$-complete, and in particular undecidable.

Figures (1)

  • Figure 1: A pattern $p$ on alphabet $\{0,1,2,3,\blacksquare\}$, where $0,1,2,3$ are represented by colors $\textcolor{brown}{\blacksquare}^0,\textcolor{white!60!blue}{\blacksquare}^1,\textcolor{white!40!red}{\blacksquare}^2,\textcolor{white!40!green}{\blacksquare}^3$, respectively. $\iota_{i}(p)=\{\textcolor{brown}{\blacksquare}^0, \ \textcolor{white!40!blue}{\blacksquare}^1\}$ and $\iota_{i+e_1}(p)=\{ \textcolor{white!40!red}{\blacksquare}^2\}$.

Theorems & Definitions (22)

  • definition thmcounterdefinition: The Domino game problem
  • theorem thmcountertheorem
  • proposition thmcounterproposition
  • lemma thmcounterlemma
  • proof
  • corollary thmcountercorollary
  • proposition thmcounterproposition
  • proof
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • ...and 12 more