AGENTIC-SYSTEMSApril 9, 20261 min read14 views

High-Precision Estimation of the State-Space Complexity of Shogi via the Monte Carlo Method

Researchers have utilized Monte Carlo sampling and a novel reverse search method to estimate Shogi's legal positions at 6.55 x 10^68, significantly refining previous bounds.

Computer Science > Artificial Intelligence

Title:High-Precision Estimation of the State-Space Complexity of Shogi via the Monte Carlo Method

View PDF HTML (experimental)Abstract:Determining the state-space complexity of the game of Shogi (Japanese Chess) has been a challenging problem, with previous combinatorial estimates leaving a gap of five orders of magnitude ($10^{64}$ to $10^{69}$). This large gap arises from the difficulty of distinguishing Shogi positions legally reachable from the initial position among the vast number of valid board configurations. In this paper, we present a high-precision statistical estimation of the number of reachable positions in Shogi. Our method combines Monte Carlo sampling with a novel reachability test that utilizes a reverse search toward a set of "King-King only" (KK) positions, rather than a single-target backward search to the single initial position. This approach significantly reduces the search effort for determining unreachability. Based on a sample of 5 billion positions, we estimated the number of legal positions in Shogi to be $6.55 \times 10^{68}$ (to three significant digits) with a $3\sigma$ confidence level, substantially improving upon previously known bounds. We also applied this method to Mini Shogi, determining its complexity to be approximately $2.38 \times 10^{18}$.

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