Skip to main content
SpringerLink
Log in

Proof-Number Search and Its Variants

  • Chapter
Oppositional Concepts in Computational Intelligence

Part of the book series: Studies in Computational Intelligence ((SCI,volume 155))

Summary

Proof-Number (PN) search is a best-first adversarial search algorithm especially suited for finding the game-theoretical value in game trees. The strategy of the algorithm may be described as developing the tree into the direction where the opposition characterised by value and branching factor is to expect to be the weakest. In this chapter we start by providing a short description of the original PN-search method, and two main successors of the original PN search, i.e., PN2 search and the depth-first variant Proof-number and Disproof-number Search (PDS). A comparison of the performance between PN, PN2, PDS, and αβ is given. It is shown that PN-search algorithms clearly outperform αβ in solving endgame positions in the game of Lines of Action (LOA). However, memory problems make the plain PN search a weaker solver for harder problems. PDS and PN2 are able to solve significantly more problems than PN and αβ. But PN2 is restricted by its working memory, and PDS is considerably slower than PN2. Next, we present a new proof-number search algorithm, called PDS-PN. It is a two-level search (like PN2), which performs at the first level a depth-first PDS, and at the second level a best-first PN search. Hence, PDS-PN selectively exploits the power of both PN2 and PDS. Experiments show that within an acceptable time frame PDS-PN is more effective for really hard endgame positions. Finally, we discuss the depth-first variant df-pn. As a follow up of the comparison of the four PN variants, we compare the algorithms PDS and df-pn. However, the hardware conditions of the comparison were different. Yet, experimental results provide promising prospects for df-pn. We conclude the article by seven observations, three conclusions, and four suggestions for future research.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allis, L.V.: Searching for Solutions in Games and Artificial Intelligence. PhD thesis, Rijksuniversiteit Limburg, Maastricht, The Netherlands (1994)

    Google Scholar 

  2. Allis, L.V., van der Meulen, M., van den Herik, H.J.: Proof-number search. Artificial Intelligence 66(1), 91–123 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berkey, D.D.: Calculus. Saunders College Publishing, New York (1988)

    MATH  Google Scholar 

  4. Berliner, H.J.: The B*-tree search algorithm: A best-first proof procedure. Artificial Intelligence 12(1), 23–40 (1979)

    Article  MathSciNet  Google Scholar 

  5. Breuker, D.M.: Memory versus Search in Games. PhD thesis, Universiteit Maastricht, Maastricht, The Netherlands (1998)

    Google Scholar 

  6. Breuker, D.M., Allis, L.V., van den Herik, H.J.: How to mate: Applying proof-number search. In: van den Herik, H.J., Herschberg, I.S., Uiterwijk, J.W.H.M. (eds.) Advances in Computer Chess, University of Limburg, Maastricht, The Netherlands, vol. 7, pp. 251–272 (1994)

    Google Scholar 

  7. Breuker, D.M., Uiterwijk, J.W.H.M., van den Herik, H.J.: Replacement schemes and two-level tables. ICCA Journal 19(3), 175–180 (1996)

    Google Scholar 

  8. Breuker, D.M., Uiterwijk, J.W.H.M., van den Herik, H.J.: The PN2-search algorithm. In: van den Herik, H.J., Monien, B. (eds.) Advances in Computer Games, IKAT, Universiteit Maastricht, Maastricht, The Netherlands, vol. 9, pp. 115–132 (2001)

    Google Scholar 

  9. Breuker, D.M., van den Herik, H.J., Uiterwijk, J.W.H.M., Allis, L.V.: A solution to the GHI problem for best-first search. Theoretical Computer Science 252(1-2), 121–149 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Campbell, M.: The graph-history interaction: On ignoring position history. In: 1985 Association for Computing Machinery Annual Conference, pp. 278–280 (1985)

    Google Scholar 

  11. Campbell, M., Hoane Jr., A.J., Hsu, F.-h.: Deep Blue. Artificial Intelligence 134(1-2), 57–83 (2002)

    Article  MATH  Google Scholar 

  12. Junghanns, A.: Are there practical alternatives to alpha-beta? ICCA Journal 21(1), 14–32 (1998)

    Google Scholar 

  13. Kishimoto, A.: Correct and Efficient Search Algorithms in the Presence of Repetitions. PhD thesis, University of Alberta, Edmonton, Canada (2005)

    Google Scholar 

  14. Kishimoto, A., Müller, M.: Df-pn in Go: An application to the one-eye problem. In: van den Herik, H.J., Iida, H., Heinz, E.A. (eds.) Advances in Computer Games 10: Many Games, Many Challenges, pp. 125–141. Kluwer Academic Publishers, Boston (2003)

    Google Scholar 

  15. Kishimoto, A., Müller, M.: A solution to the GHI problem for depth-first proof-number search. Information Sciences 175(4), 296–314 (2005)

    Article  Google Scholar 

  16. Knuth, D.E., Moore, R.W.: An analysis of alpha-beta pruning. Artificial Intelligence 6(4), 293–326 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  17. McAllester, D.A.: Conspiracy numbers for min-max search. Artificial Intelligence 35(1), 278–310 (1988)

    MathSciNet  Google Scholar 

  18. Nagai, A.: A new AND/OR tree search algorithm using proof number and disproof number. In: Proceedings of Complex Games Lab Workshop, ETL, Tsukuba, Japan, pp. 40–45 (1998)

    Google Scholar 

  19. Nagai, A.: A new depth-first-search algorithm for AND/OR trees. Master’s thesis, The University of Tokyo, Tokyo, Japan (1999)

    Google Scholar 

  20. Nagai, A.: Df-pn Algorithm for Searching AND/OR Trees and its Applications. PhD thesis, The University of Tokyo, Tokyo, Japan (2002)

    Google Scholar 

  21. Nagai, A., Imai, H.: Application of df-pn+ to Othello endgames. In: Proceedings of Game Programming Workshop in Japan 1999, Hakone, Japan, pp. 16–23 (1999)

    Google Scholar 

  22. Nalimov, E.V., Haworth, G.M.C., Heinz, E.A.: Space-efficient indexing of chess endgame tables. ICGA Journal 23(3), 148–162 (2000)

    Google Scholar 

  23. Palay, A.J.: Searching with Probabilities. PhD thesis, Carnegie Mellon University, Pittsburgh, PA, USA (1983); Also published by Pitman, Boston (1985)

    Google Scholar 

  24. Pawlewicz, J., Lew, Ł.: Improving depth-first pn-search: 1 + ε trick. In: van den Herik, H.J., Ciancarini, P., Donkers, H.H.L.M(J.) (eds.) CG 2006. LNCS, vol. 4630, pp. 160–171. Springer, Heidelberg (2007)

    Google Scholar 

  25. Plaat, A., Schaeffer, J., Pijls, W., de Bruin, A.: Best-first fixed-depth minimax algorithms. Artificial Intelligence 87(2), 255–293 (1996)

    Article  MathSciNet  Google Scholar 

  26. Sackson, S.: A Gamut of Games. Random House, New York (1969)

    Google Scholar 

  27. Saito, J.-T., Chaslot, G., Uiterwijk, J.W.H.M., van den Herik, H.J.: Monte-carlo proof-number search for computer Go. In: van den Herik, H.J., Ciancarini, P., Donkers, H.H.L.M(J.) (eds.) CG 2006. LNCS, vol. 4630, pp. 50–61. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  28. Sakuta, M.: Deterministic Solving of Problems with Uncertainty. PhD thesis, Shizuoka University, Hamamatsu, Japan (2001)

    Google Scholar 

  29. Sakuta, M., Iida, H.: The performance of PN*, PDS and PN search on 6×6 Othello and Tsume-Shogi. In: van den Herik, H.J., Monien, B. (eds.) Advances in Computer Games, Universiteit Maastricht, Maastricht, The Netherlands, vol. 9, pp. 203–222 (2001)

    Google Scholar 

  30. Schaeffer, J.: Conspiracy numbers. Artificial Intelligence 43(1), 67–84 (1990)

    Article  Google Scholar 

  31. Schaeffer, J.: Game over: Black to play and draw in Checkers. ICGA Journal 30(4), 187–197 (2007)

    MathSciNet  Google Scholar 

  32. Schaeffer, J., Burch, N., Björnsson, Y., Kishimoto, A., Müller, M., Lake, R., Lu, P., Sutphen, S.: Checkers is solved. Science 317(5844), 1518–1522 (2007)

    Article  MathSciNet  Google Scholar 

  33. Schaeffer, J., Lake, R.: Solving the game of Checkers. In: Nowakowski, R.J. (ed.) Games of No Chance, pp. 119–133. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  34. Seo, M., Iida, H., Uiterwijk, J.W.H.M.: The PN*-search algorithm: Application to Tsume-Shogi. Artificial Intelligence 129(1-2), 253–277 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  35. Winands, M.H.M., Uiterwijk, J.W.H.M., van den Herik, H.J.: PDS-PN: A new proof-number search algorithm: Application to Lines of Action. In: Schaeffer, J., Müller, M., Björnsson, Y. (eds.) CG 2002. LNCS, vol. 2883, pp. 170–185. Springer, Heidelberg (2003)

    Google Scholar 

  36. Winands, M.H.M., Uiterwijk, J.W.H.M., van den Herik, H.J.: An effective two-level proof-number search algorithm. Theoretical Computer Science 313(3), 511–525 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  37. Winands, M.H.M., van den Herik, H.J., Uiterwijk, J.W.H.M.: An evaluation function for Lines of Action. In: van den Herik, H.J., Iida, H., Heinz, E.A. (eds.) Advances in Computer Games 10: Many Games, Many Challenges, pp. 249–260. Kluwer Academic Publishers, Boston (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tilburg centre for Creative Computing (TiCC), Faculty of Humanities, Tilburg University, The Netherlands

    H. Jaap van den Herik

  2. Maastricht ICT Competence Centre (MICC), Faculty of Humanities and Sciences, Maastricht University, The Netherlands

    Mark H. M. Winands

Authors
  1. H. Jaap van den Herik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark H. M. Winands
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Hamid R. Tizhoosh Mario Ventresca

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

van den Herik, H.J., Winands, M.H.M. (2008). Proof-Number Search and Its Variants. In: Tizhoosh, H.R., Ventresca, M. (eds) Oppositional Concepts in Computational Intelligence. Studies in Computational Intelligence, vol 155. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70829-2_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-70829-2_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-70826-1

  • Online ISBN: 978-3-540-70829-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation