Lambda-Search in Game Trees — with Application to Go

  • Thomas Thomsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2063)


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).


binary tree search threat-sequences null-moves proof-number search abstract game-knowledge Go block tactics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allis, L.V: Searching for Solutions in Games and Artificial Intelligence. PhD Thesis, University of Limburg, Maastricht (1994)Google Scholar
  2. 2.
    Allis, L.V., M. van der Meulen, and H.J. van den Herik: Proof-Number Search. Artificial Intelligence, Vol. 66, ISSN 0004-3702 (1994) 91–124zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Benson, D.B.: Life in the Game of Go. Information Sciences, vol. 10 (1976) 17–29MathSciNetGoogle Scholar
  4. 4.
    Cazenave, T.: Abstract Proof Search. In I. Frank and T.A. Marsland (eds.): Computers and Games 2000,Lecture Notes in Computer Science, Springer-Verlag (2001)Google Scholar
  5. 5.
    Heinz, E.A.: Adaptive null-move pruning. ICCA Journal, Vol. 22, No. 3 (1999) 123–132Google Scholar
  6. 6.
    Kano, Y: Graded Go Problems for Beginners, Volume Four, The Nihon Ki-in, Tokyo, Japan (1990)Google Scholar
  7. 7.
    Kierulf, A.: Smart Go Board: Algorithms for the Tactical Calculator. Diploma thesis (unpublished), ETH Zürich (1985)Google Scholar
  8. 8.
    Müller, M.: Computer Go as a Sum of Local Games: An Application of Combinatorial Game Theory. PhD thesis, ETH Zürich, 1995. Diss. ETH Nr. 11.006 (1996)Google Scholar
  9. 9.
    Müller, M.: Playing it safe: Recognizing secure territories in computer Go by using static rules and search. In H. Matsubara (ed.): Game Programming Workshop in Japan’ 97, Computer Shogi Association, Tokyo, Japan (1997) 80–86Google Scholar
  10. 10.
    Müller, M.: Race to capture: Analyzing semeai in Go. In Game Programming Workshop in Japan’ 99, volume 99 (14) of IPSJ Symposium Series (1999) 61–68.Google Scholar
  11. 11.
    Thomsen, T.: Material at (2000)
  12. 12.
    Wolf, T.: About problems in generalizing a tsumego program to open positions, 3rd Game Programming Workshop in Japan, Hakone (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Thomas Thomsen
    • 1
  1. 1.Stockholmsgade 11, 4th.CopenhagenDenmark

Personalised recommendations