PN, PN2 and PN in Lines of Action

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PN, PN2 and PN in Lines of Action

• Mark Winands and Jos Uiterwijk

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Overview

• LOA
• PN Search
• PN2 Search
• PN Search
• Results
• Conclusions

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Board Set-up of LOA
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Movement of pieces
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A Final Position
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Position with a long forcing line
Black to move and to win in 21 plies. Mona vs.
YL, Game 4
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Problems in the endgame

• Evaluation functions for LOA are not good
  predictors in the case of forced wins.
• Most terminal positions still have more than 10
  pieces remaining on the board, which makes the
  game not suitable for endgame databases.
• Mate provers like proof number search are
  probably a better method for this issue.

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Proof-number (PN) Search

• Inspired by conspiracy-number search (McAllester
  1988).
• PN is invented by Allis (1994).
• Best-first search algorithm.
• Goal is to prove that there exists a win for the
  player to move in a certain position.
• Not a heuristic is used to select the
  most-proving node, but the shape of the tree and
  the value of the leaf nodes.
• Mostly (dis)proves faster than alpha-beta for a
  certain position.

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PN search
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Enhancements of PN

• Set the proof number and disproof number to 1 and
  n for a MAX node (and the reverse for a MIN
  node), where n is the number of legal moves.
  (Thus, using a mobility component.)
• Nodes once proved by PN search are stored in a
  transposition table (TT), which is used in the
  ?-? search. What is the effect of this TT in the
  PN search?

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PN2 search(Breuker, 1998)
Bounded by
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PN2 (2)

• Other enhancements of PN can also be used in PN2.
• Deleting also (dis)proven subtrees in the second
  level search. If there are a lot of (dis)proven
  subtrees, the second level search will search
  more nodes than usual (deeper). Possibly one of
  the reasons is that trees are regenerated less.
  Almost (dis)proven subtrees in the normal
  situation are now proved.
• Caching instead of deleting second-level search
  trees
• Leaving probably-unsolvable trees prematurely.

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PN (Seo et al., 2001)

• PN solves two problems of PN and PN2 (i.e,
  memory and sub optimal solutions)
• PN transforms the PN-search algorithm into an
  iterative- deepening depth-first approach
• PN is enhanced with methods as recursive
  iterative deepening and dynamic evaluation.

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Experimental set-up

• The proof number search algorithms and
  enhancements are tested on a test set of 116
  forced win positions
• The maximum number of nodes searched is
  50,000,000.
• The maximum number of nodes stored in the memory
  is 900,000.
• PN, PN2 and PN search are compared with an
  iterative alpha-beta search.

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Results (1)
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Results (2)
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Results (3)
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Results (4)
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Position with a long forcing line
Black to move and to win in 21 plies. Mona vs.
YL, Game 4
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PN is sometimes better
White to move and win in 13 plies. Fifth Annual
E-mail Tournament, Thordsen vs. Mona
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Conclusions

• PN, PN, PN2 search are quite good proving LOA
  positions.
• Even a poor implemented PN outperforms ?-? in
  LOA positions.
• Mobility improves the PN search.
• Deleting (dis)proved nodes in the second-level
  search improves the PN2 search.
• Caching second-level trees or leaving
  probably-unsolvable trees prematurely has not
  paid of yet

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Epilogue

• Currently PN2 search is combined with alpha-beta
  search in MIA II.
• PN2 search is done for some fixed time after some
  moves in the players own time. During this
  search proven position are stored in
  transposition table (PN-TT).
• If PN2 proves the position the (sub) optimal move
  is played.
• Otherwise the normal alpha beta search is applied
  using the PN-TT.
• Experiments reveal that this strengthens the
  program.