How to Correctly Prune Tropical Trees

  • Jean-Vincent Loddo
  • Luca Saiu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6167)


We present tropical games, a generalization of combinatorial min-max games based on tropical algebras. Our model breaks the traditional symmetry of rational zero-sum games where players have exactly opposed goals (min vs. max), is more widely applicable than min-max and also supports a form of pruning, despite it being less effective than α − β. Actually, min-max games may be seen as particular cases where both the game and its dual are tropical: when the dual of a tropical game is also tropical, the power of α − β is completely recovered. We formally develop the model and prove that the tropical pruning strategy is correct, then conclude by showing how the problem of approximated parsing can be modeled as a tropical game, profiting from pruning.


combinatorial game search alpha-beta pruning rational game tropical algebra tropical game term rewriting logic parsing 


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  1. 1.
    Hart, T.P., Edwards, D.J.: The tree prune (TP) algorithm. In: Artificial Intelligence Project Memo 30. Massachusetts Institute of Technology, Cambridge (1961)Google Scholar
  2. 2.
    Knuth, D.E., Moore, R.W.: An analysis of alpha-beta pruning. Artificial Intelligence 6, 293–326 (1975)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Loddo, J.V.: Généralisation des Jeux Combinatoires et Applications aux Langages Logiques. PhD thesis, Université Paris VII (2002)Google Scholar
  4. 4.
    Loddo, J.V., Cosmo, R.D.: Playing logic programs with the alpha-beta algorithm. In: Parigot, M., Voronkov, A. (eds.) LPAR 2000. LNCS (LNAI), vol. 1955, pp. 207–224. Springer, Heidelberg (2000)Google Scholar
  5. 5.
    Ginsberg, M.L., Jaffray, A.: Alpha-beta pruning under partial orders. In: Games of No Chance II (2001)Google Scholar
  6. 6.
    Klop, J.W., de Vrijer, R.: First-order term rewriting systems. In: Terese (ed.) Term Rewriting Systems, pp. 24–59. Cambridge University Press, Cambridge (2003)Google Scholar
  7. 7.
    Huet, G.: Confluent reductions: Abstract properties and applications to term rewriting systems. J. ACM 27(4), 797–821 (1980)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Klop, J.W., Oostrom, V.V., de Vrijer, R.: Orthogonality. In: Terese (ed.) Term Rewriting Systems, pp. 88–148. Cambridge University Press, Cambridge (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jean-Vincent Loddo
    • 1
  • Luca Saiu
    • 1
  1. 1.Laboratoire d’Informatique de l’Université Paris Nord - UMR 7030 Université Paris 13 - CNRSVilletaneuse

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