Abstract
Variants of best-first search algorithms and their expansions have continuously been introduced to solve challenging problems. The probability-based proof number search (PPNS) is a best-first search algorithm that can be used to solve positions in AND/OR game tree structures. It combines information from explored (based on winning status) and unexplored (through Monte Carlo simulation) nodes from a game tree using an indicator called the probability-based proof number (PPN). In this study, PPNS is employed to solve randomly generated positions in Connect Four and Othello, in which the results are compared with the two well-known best-first search algorithms (proof number search (PNS) and Monte Carlo proof number search). Adopting a simple improvement parameter in PPNS reduces the number of nodes that need to be explored by up to 57%. Moreover, further observation showed the varying importance of information from explored and unexplored nodes in which PPNS relies critically on the combination of such information in earlier stages of the Othello game. Discussion and insights from these findings are provided where the potential future works are briefly described.
Highlights
A best-first search algorithm was initially introduced as a method to find the game-theoretic value using a specific technique to progress towards a game tree framework
Note that proof number search (PNS) is bounded by the amount of memory, while probability-based proof number search (PPNS) and MCPNS are bounded by time
This observation implies that PPNS was not able to thoroughly combine the information from both parts of the tree, which can be demonstrated from the probability-based proof number (PPN) value of the root for such a position (Figure 5a)
Summary
A best-first search algorithm was initially introduced as a method to find the game-theoretic value using a specific technique to progress towards a game tree framework. One of the prominent best-first search algorithms in such a context is the proof number search (PNS) [1]. PNS utilizes two variables, proof and disproof numbers (pn and dn for short, respectively), as the search indicators to find the best possible options. A node is chosen to provide the best possible choice (called the most proving node (MPN)) and subsequently expanded for further search [3,4,5]. Several drawbacks observed from its implementation include the usage of a large amount of memory space, overly long solutions, and the see-saw effect (The see-saw effect is observed when the search goes back and forth between several sub-trees, preventing it from progressing to a deeper branch) [3,4]
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