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).

A Novel Approach to Solving Goal-Achieving Problems for Board Games

“…Relevance zones can be derived based on null moves, as in lambda search (Thomsen 2000;Wu and Lin 2010), and must-play regions (Hayward et al 2003;Hayward and Van Rijswijck 2006;Hayward 2009). For example, in Figure 1, if Black makes a null move on p a (in this case any move outside of z a , including passes), White simply plays at E2 to reach UCA, obtaining in the process an RZ, z b , like p b .…”

“…For RZ pruning, previous lambda-based search methods tried null moves first, then derived RZs as above (Thomsen 2000;Yoshizoe, Kishimoto, and Müller 2007;Wu and Lin 2010). However, the decision on when and how to play null moves is itself a heuristic that requires deliberate thought.…”

“…Comparison with Previous Null Move Methods Other than Hex, most traditional RZ approaches need to search null moves in advance to facilitate pruning, e.g. null-move pruning in chess (Donninger 1993;Heinz 1999;Campbell, Hoane Jr, and Hsu 2002) and lambda search (Thomsen 2000;Yoshizoe, Kishimoto, and Müller 2007;Wu and Lin 2010). For example, for the position in Figure 4 (a), if we want to make another region, say at G3, two consecutive null moves for Black need to be searched first (say, two passes) for White to play consecutively at F4 and G4, as in second order lambda search (Thomsen 2000).…”

A Novel Approach to Solving Goal-Achieving Problems for Board Games

“…This subsection presents some sufficient conditions for RZs such that replay is always legal, which in turn justifies the correctness for RZSTs. An important notion of RZs, which originates from threat-space search (Thomsen 2000;Wu and Lin 2010), is that π moves played outside the zone do not interfere with the winning strategy of π inside the zone, which is represented by RZSTs. Let a winning node n be the root of an RZST, and z be an RZ for n. From Property RZ-1, all positions p * with β(p * ) z = β(n) z are wins too.…”

Selective search in games of different complexity

“…The technique reduces the size of the search tree in such a way that it was possible to solve the game (Allis et al, 1996). Other examples of mate-solvers are Proof-Number Search (Allis et al, 1994) and λ-search (Thomson, 2000).…”