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 $2.38 \times 10^{18}$.

Bibliographic and Citation Tools

Code, Data and Media Associated with this Article

Demos

Recommenders and Search Tools

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.