A Memory Efficient Retrograde Algorithm and Its Application to Chinese Chess Endgames

Summary

ABSTRACT. We present an improved, memory efficient retrograde algorithm we developed during our research on solving Chinese chess endgames. This domain-independent retrograde algorithm, along with a carefully designed domain-specific indexing function, has enabled us to solve many interesting Chinese chess endgame on standard consumer class hardware.

We also report some of the most interesting results here. Some of these are real surprises for human Chinese chess experts. For example, the aegp-aaee1 ending is a theoretical win, not as previously believed, a draw. Human analysis for this endgame over many years by top players has been proved to be wrong.

1. Introduction

Endgame databases have several benefits. First, the knowledge they provide about the game is perfect knowledge. Second, the databases, because of their complete knowledge about certain domains, are a useful background for Artificial Intelligence research, especially in machine learning. Third, the databases often provide knowledge beyond that achieved by humans, (and increasingly, beyond that achievable by humans). Many endgame databases in many games have been constructed since Ströhlein's pioneering work (Ströhlein, 1970). In Chess, Thompson (1986) generated almost all 5-men chess endgame databases and made them widely available in CD format. Databases construction also enable Gasser (1996) to solve the game of Nine Men's Morris. Perhaps the most impressive endgame database construction so far is Schaeffer's work (Schaeffer et al., 1994) on Checkers. His program created all 8 men endgame databases and comprised more than 440 billion (4.4 x 10¹¹) positions. The databases played a very important role in Chinook's success.

We started our work on constructing Chinese chess endgame databases back in 1992. We have constructed many Chinese chess endgame databases since. We wanted our program to solve as many endgames as possible using only moderate hardware we had access to. So a major effort was made to improve the retrograde algorithm as well as examine ways to reduce the size of the database.

Armed with careful analysis to reduce the databases’ size and our fast, memory efficient retrograde algorithm, we were able to solve one class of very interesting Chinese chess endgames. In this class of endgames, one side has no attacking pieces left but have various defending pieces, while the other side has various attacking pieces.