---
title: Sidling Sequences in Combinatorial Games
url: https://www.emergentmind.com/topics/sidling-sequences
type: topic
---

# Sidling Sequences in Combinatorial Games

Sidling sequences are a canonical construct in the metric topology of combinatorial games. Arising from the theory of diminishing-weight edit distances on canonical short games, they formalize the process of approaching loopy games via chains of well-ordered finite approximants. Sidling sequences underpin the analytic closure of the canonical game space, provide Cauchy foundations respecting the $wd$-metric, and enable rigorous classification of loopy and infinite games according to their metric proximity to short, canonical forms.

## 1. The Diminishing-Weight Metric and Canonical Game Space

Consider the space $\mathcal{C}$ of short two-player combinatorial games in canonical form—excluding dominated or reversible options and repeated positions—each with a directed acyclic game graph $D(G)$. The diminishing-weight distance $wd$ is determined via edit sequences on $D(G)$: for any $G, H \in \mathcal{C}$,

$$
wd(G,H) = \min_{\substack{\text{edit sequences}\ D(G)\rightarrow D(H)}} \sum_{e\in\text{sequence}} \left(\frac{1}{2}\right)^{d(e)}
$$

where $d(e)$ is the graph-theoretic distance from the source of $D(G)$ to the tail of edge $e$. Each edit either adds or removes an edge (color-preserving by player), possibly dropping isolated vertices. The metric structure $(\mathcal{C}, wd)$ is proven in [2601.10574].

## 2. Formal Definition of Sidling Sequences

Fix a loopy combinatorial game $\gamma$ (i.e., one with cycles or an infinite DAG). A sidling sequence *from above* is a (possibly infinite) sequence of short games $(a_n)$ satisfying
- $\gamma \le a_1$, and
- $a_{n+1}$ is the simplest canonical game with $\gamma \le a_{n+1} < a_n$.

Dually, *sidling from below* yields $(b_n)$ with
- $\gamma \ge b_1$
- $\gamma \ge b_{n+1} > b_n$

Typically, the initial term is a trivial bound (such as a number or *) and each step takes the simplest dyadic rational or fuzzy game strictly between the predecessor and $\gamma$. Both $(a_n)$ and $(b_n)$ converge to $\gamma$ under the $wd$-metric and are termed *canonical sidling sequences* [2601.10574].

## 3. Explicit Construction: The “Over” Example

Let $\over = \{0\,|\,\over\}$ denote a loopy infinitesimal game. Siding constructions are as follows:

- **From Above:**  
  - $a_1 = 1$ (since $\over \le \text{on} = \{0\,|\,\text{on}\} = 1$).
  - Recurrence: $a_{n+1} = \{0 | a_n\}$, giving the sequence $(1, \frac{1}{2}, \frac{1}{4}, \ldots) = (2^{-n})$.
- **From Below:**  
  - $b_1 = * = \{0|0\}$.
  - Recurrence: $b_{n+1} = \{0|b_n\}$, yielding $( *, \uparrow, \uparrow + \uparrow^2, \uparrow + \uparrow^2 + \uparrow^3, \ldots )$.

Both sequences strictly bound $\over$ and approach it in $wd$ [2601.10574]. Their construction generalizes: for any loopy $\gamma$, one generates $(a_n)$ and $(b_n)$ as Cauchy sequences, converging in the metric topology to $\gamma$.

## 4. Cauchy Sequence Properties and Metric Closure

The convergence of sidling sequences is explicit. For $(a_n) = (2^{-n})$, embedding $D(a_m)$ inside $D(a_n)$ allows computation:

$$
wd(a_n, a_m) \le \sum_{i=m}^{n-1} \left(2^{-i} + 2^{-(i+1)}\right) = 3 \cdot 2^{-m} - 3 \cdot 2^{-n}
$$

As $m \to \infty$, $wd(a_n, a_m) \to 0$, confirming Cauchyness. A parallel calculation applies to $(b_n)$. The mutual convergence $wd(a_n, b_n) = (1/2)^{n+1} \to 0$ ensures both sequences share the same limit in the metric completion [2601.10574].

The closure $\overline{\mathcal{C}}$, defined as the set of all such Cauchy limits, includes loopy/infinite games reachable at finite $wd$-distance. For example, $\over \in \overline{\mathcal{C}}$. Other limit points include “on,” “upon,” and further loopy stoppers as exhibited in [2601.10574]. Notably, certain loopy configurations, such as Bach’s Carousel and the games "tis", "tisn," are inaccessible via sidling and hence lie outside $\overline{\mathcal{C}}$.

## 5. Proof Sketches for Sidling Convergence and Discreteness

Direct estimation yields:
- For $\over$, $wd(a_n, \over) \le \sum_{i=n}^{\infty} (2^{-i} + 2^{-(i+1)}) = 3\cdot 2^{-n} \to 0$.
- For any $G \in \mathcal{C}$, distinct short games satisfy $wd(G, H) > 2^{-d}$, where $d$ is the larger birthday, establishing metric discreteness of canonical games.

Thus, metric closure via sidling sequences strictly extends $\mathcal{C}$ by countably many loopy or infinite DAG games, precisely those approachable in $wd$ from canonical short games [2601.10574].

## 6. Topological and Algorithmic Significance

Sidling sequences formalize the analytic process of “squeezing” dyadic or infinitesimal canonical forms toward a loopy boundary in $\overline{\mathcal{C}}$. This enables:
- Metrizable topology on combinatorial games, unifying finite and infinite forms.
- Rigorous topological classification of loopy games (stoppers, non-stoppers, and their sides).
- Quantitative metric-based similarity search in large-scale combinatorial game theory (CGT) databases.
- Application frameworks for automated game-solving and heuristic optimization, where small $wd$-perturbations provide systematic tuning.

Sidling sequences thus provide both a concrete construction tool and the analytic infrastructure for the real-valued topology in combinatorial games [2601.10574].

Source: https://www.emergentmind.com/topics/sidling-sequences