Lambda-Search in Game Trees – With Application to Go

Abstract

Abstract. This paper proposes a new method for searching two-valued (binary) game trees in games like chess or Go. Lambda-search uses null-moves together with different orders of threat-sequences (so-called lambda-trees), focusing the search on threats and threat-aversions, but still guaranteeing to find the minimax value (provided that the game-rules allow passing or zugzwang is not a motive). Using negligible working memory in itself, the method seems able to offer a large relative reduction in search space over standard alpha-beta comparable to the relative reduction in search space of alpha-beta over minimax, among other things depending upon how non-uniform the search tree is. Lambda-search is compared to other resembling approaches, such as null-move pruning and proof-number search, and it is explained how the concept and context of different orders of lambda-trees may ease and inspire the implementation of abstract game-specific knowledge. This is illustrated on open-space Go block tactics, distinguishing between different orders of ladders, and offering some possible grounding work regarding an abstract formalization of the concept of relevancy-zones (zones outside of which added stones of any colour cannot change the status of the given problem).

Job-Level Proof Number Search

“…In this section, we review the results from [27] and [35] as follows. NCTU6 included a solver that was able to find victory by continuous four (VCF), a common term for winning strategies in the Renju community.…”

Solving games

“…This does not mean that the winning threatsequence only occurs if the opponent passes: a winning threat sequence will occur whenever the opponent does not counter it. If no winning threat-sequence exists at the current turn, Lambda search is very useful to determine a relevant next move, and to find a winning threat-sequence in the next turn [18].…”

Conservative Relevance Zone Search in Connect6

“…The proving is an enormous work when the branching factor is huge; therefore, the concept of relevance zones [10][17] [22] is used to develop a search tree. Relevance zone can substantially reduce the search space; therefore, this subsection introduces relevance-zone search.…”