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Mind the gap: A real-valued distance on combinatorial games

Kyle Burke, Michael Fisher, Craig Tennenhouse

TL;DR

This work defines a real-valued distance $wd$ on the space $\mathcal{C}$ of short combinatorial games in canonical form to quantify proximity between game positions under play. It proves that $wd$ is a metric and extends naturally to loopy games by considering infinite DAG representations, enabling analysis of Cauchy sequences and their limit points, including various loopy targets such as $over$, $on$, and $upon$. The paper investigates the closure $\overline{\mathcal{C}}$, showing it partitions loopy games non-trivially and exploring which loopy games can be approached by $wd$-convergent sequences (e.g., Bach's Carousel is not a limit point; several non-stoppers are). It further compares $wd$ to alternative metrics and outlines open questions about plum trees, non-stopper-sided loopy games, and the role of infinite-DAG sequences in expanding the closure.

Abstract

We define a real-valued distance metric $wd$ on the space $\mathcal{C}$ of short combinatorial games in canonical form. We demonstrate the existence of Cauchy sequences informed by sidling sequences, find limit points, and investigate the closure $\overline{\mathcal{C}}$, which is shown to partition the set of loopy games in a non-trivial way. Stoppers, enders, and non-stopper-sided loopy games are explored, as well as the topological properties of $(\mathcal{C},wd)$.

Mind the gap: A real-valued distance on combinatorial games

TL;DR

This work defines a real-valued distance on the space of short combinatorial games in canonical form to quantify proximity between game positions under play. It proves that is a metric and extends naturally to loopy games by considering infinite DAG representations, enabling analysis of Cauchy sequences and their limit points, including various loopy targets such as , , and . The paper investigates the closure , showing it partitions loopy games non-trivially and exploring which loopy games can be approached by -convergent sequences (e.g., Bach's Carousel is not a limit point; several non-stoppers are). It further compares to alternative metrics and outlines open questions about plum trees, non-stopper-sided loopy games, and the role of infinite-DAG sequences in expanding the closure.

Abstract

We define a real-valued distance metric on the space of short combinatorial games in canonical form. We demonstrate the existence of Cauchy sequences informed by sidling sequences, find limit points, and investigate the closure , which is shown to partition the set of loopy games in a non-trivial way. Stoppers, enders, and non-stopper-sided loopy games are explored, as well as the topological properties of .
Paper Structure (8 sections, 4 theorems, 26 equations, 10 figures)

This paper contains 8 sections, 4 theorems, 26 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Sequences in metric spaces
  3. Diminishing weight distance
  4. Breaking away from sidling sequences
  5. Non plum trees
  6. Topological properties
  7. Other distance metrics
  8. Conclusion and remaining questions

Key Result

Theorem 3.2

Diminished weight distance is a distance metric on $\mathcal{C}$.

Figures (10)

  • Figure 1: Digraph representations of the games $\frac{1}{8}$, $\frac{1}{16}$, and $\frac{1}{256}$
  • Figure 2: Digraph representations of $\Uparrow\!*$, $\uparrow\!4*$, and $\uparrow\!5$
  • Figure 3: The games $\uparrow, \uparrow^{[2]}, \uparrow^{[3]}, \uparrow^{[4]}$
  • Figure 4: The loopy game $\chi$ and the $2^{nd}, 3^{rd}$, and $4^{th}$ terms of an approximation sequence
  • Figure 5: The game $\psi$ and a term of an approximation sequence. Some nodes are deidentified for clarity.
  • ...and 5 more figures

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Theorem 3.2
  • ...and 9 more