A New Proof-Number Calculation Technique for Proof-Number Search

We propose a new simple calculation technique for proof numbers in Proof-Number Search. Search algorithms based on (dis)proof numbers are known to be effective for solving problems such as tsumego, tsume-shogi, and checkers. Usually, the Proof-Number Search expands child nodes with the smallest (dis)proof number because such nodes are expected to be the easiest to (dis)prove the node. However, when many unpromising child nodes exist, (dis)proof numbers are not always a suitable measure for move ordering because unpromising nodes temporarily increase the (dis)proof numbers. For such cases, we propose the use of only some child nodes (instead of all child nodes) for calculating (dis)proof numbers. We call this technique Dynamic Widening. We combined Dynamic Widening with the Depth-first Proof-Number Search (df-pn) algorithm and tested its performance on capturing problems of Go on 19x19 boards. Our results show that the approach is more than 30 times faster than normal df-pn when we generate almost all legal moves (about 300 moves on average). The required time for processing remained approximately four times as long as that of df-pn using heuristic pruning (about 20 moves on average), but the correctness of the search result is guaranteed.