Dealing with Infinite Loops, Underestimation, and Overestimation of Depth-First Proof-Number Search

Depth-first proof-number search (df-pn) is powerful AND/OR tree search to solve positions in games. However, df-pn has a notorious problem of infinite loops when applied to domains with repetitions. Df-pn(r) cures it by ignoring proof and disproof numbers that may lead to infinite loops. This paper points out that df-pn(r) has a serious issue of underestimating proof and disproof numbers, while it also suffers from the overestimation problem occurring in directed acyclic graph. It then presents two practical solutions to these problems. While bypassing infinite loops, the threshold controlling algorithm (TCA) solves the underestimation problem by increasing the thresholds of df-pn. The source node detection algorithm (SNDA) detects the cause of overestimation and modifies the computation of proof and disproof numbers. Both TCA and SNDA are implemented on top of df-pn to solve tsume-shogi (checkmating problem in Japanese chess). Results show that df-pn with TCA and SNDA is far superior to df-pn(r). Our tsume-shogi solver is able to solve several difficult positions previously unsolved by any other solvers