Estimating the number of reachable positions in Minishogi

TL;DR

This work addresses the state-space complexity of Minishogi by estimating the number of positions reachable from the initial position using a sampling-based approach. It builds a pipeline that generates a large set of pseudo-legal positions, filters them through a legality test based on backward reachability, and uses random sampling to infer the total count of reachable positions. The study reports a high-precision estimate of about $2.38\times 10^{18}$ reachable Minishogi positions (95% CI: $2.376\times 10^{18}$ to $2.379\times 10^{18}$), suggesting strong solving is computationally challenging for this game. The methodology, including candidate-position enumeration and backward-reachability testing, is transferable to other chess-like games and is complemented by an open-source implementation.

Abstract

To investigate the feasibility of strongly solving Minishogi (Gogo Shogi), it is necessary to know the number of its reachable positions from the initial position. However, there currently remains a significant gap between the lower and upper bounds of the value, since checking the legality of a Minishogi position is difficult. In this paper, the authors estimate the number of reachable positions by generating candidate positions using uniform random sampling and measuring the proportion of those reachable by a series of legal moves from the initial position. The experimental results reveal that the number of reachable Minishogi positions is approximately $2.38\times 10^{18}$.

Figures (3)

• Figure 1: The initial position of Minishogi
• Figure 2: Unreachable positions.
• Figure 3: The unreachable position that can be traced back 8 plies