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Df-pn: Depth-first Proof Number Search

Df-pn: Depth-first Proof Number Search. Studied by Ayumu Nagai Surveryed by Akihiro Kishimoto. Outline of Presentation. Motivations Proof Number Search Seo’s Algorithm Df-pn Conclusions. Motivations. Why do we use depth-first search? Dfs uses less memory than bfs in general

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Df-pn: Depth-first Proof Number Search

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  1. Df-pn: Depth-first Proof Number Search Studied by Ayumu Nagai Surveryed by Akihiro Kishimoto

  2. Outline of Presentation • Motivations • Proof Number Search • Seo’s Algorithm • Df-pn • Conclusions

  3. Motivations • Why do we use depth-first search? • Dfs uses less memory than bfs in general • O(d) v.s. O(b^d) • Has low overhead to manage nodes • C.f. A* needs priority queue, IDA* just needs stack • Can behave like bfs with TT • C.f. IDA* and A*, SSS* and MT-SSS*, Seo and AO*

  4. Why df-pn? • Proof Number Search: best-first search for AND/OR trees • Df-pn: depth-first search • Same behavior as pn-search • Less expansion of interior nodes Generally better performance than pn-search

  5. Proof-Number Search(1/3):Proof Numbers • Pn- search [Allis et.al.:94] • Proof numbers • Minimum number of leaf nodes to be expanded to prove • Better to select the child with small proof number for proving trees • Example of proof numbers 1 3 1 Leaf nodes 1 1 1 1 pn OR node pn AND node

  6. Proof Number Search(2/3):Disproof Numbers • Pn- search [Allis et.al.:94] • Disproof numbers • Minimum number of leaf nodes to be expanded to disprove • Better to select the child with small disproof number for disproving trees • Example of disproof numbers 1 3 1 1 1 1 1 dn OR node dn AND node

  7. Traverse from the root to a leaf by choosing Node with smallest (dis)proof number at (AND)OR node Expand a leaf node Update (Backup) pn & dn to the root Continue 1-3 until proven or disproven Proof-Number Search (3/3)Algorithm 2,2 3,1 2,1 1,1 1,1 1,1 1,1 1,2 pn,dn OR node pn,dn AND node

  8. Seo’s algorithm(1/4) • Proof number as a threshold • Transposition table • Multiple Iterative Deepening

  9. Seo’s algorithm(2/4):Depth-first search • Threshold for proof number • Iterative deepening Behavior of Seo’s algorithm PN=1 PN=2 PN=3

  10. Seo’s algorithm(3/4):Transposition table • Iterative deepening • Re-expansion of the nodes • Save proof numbers of the nodes previously expanded in TT

  11. Seo’s algorithm(4/4):Multiple iterative deepening • Overestimation of proof numbers at AND nodes • When an OR node is proved • Iterative deepening at all OR nodes Behavior of Multiple Iterative Deepening PN=13 12 5 7 Proved PN=8 PN=9 AND node OR node PN=10

  12. Node Selection: OR node: c1 AND node: as you like Updating Threshold: OR: PN = min(pn(c2) + 1, PN) AND: PN = PN + pn (c1) – (pn(c2) + … + pn(ck)) Returning Condition: PN <= pn(n) Others: Iterative deepening at OR node as far as PN > pn(n) Notations: n: node searching pn(n): proof number at n c1,…,ck: children of n where pn(c1)<=…<=pn(ck) PN: threshold Sketch of Seo’s Algorithm

  13. Df-pn • If you understand Seo’s algorithm, df-pn is easy to understand! • has two thresholds for proof & disproof numbers • C.f. Seo had only one threshold • Iterative deepening at both OR and AND nodes • Need to avoid overestimations of proof and disproof numbers • C.f. Seo avoided only the case of proof numbers

  14. Sketch of df-pn • Node Selection: • Always c1 • Updating Threshold: • OR: • PN’= min(pn(c2) + 1, PN) • DN’= DN – (dn(c2) + … + dn(ck)) • AND: • PN’= PN – (pn(c2) + … + pn(ck)) • DN’= min(dn(c2) + 1, DN) • Returning Condition : • PN <= pn(n) || DN <= dn(n) • Others: • Iterative deepening as far as PN > pn(n) && DN > dn(n) • Notations: • n: node searching • pn(n): proof number at n • dn(n): disproof number at n • c1,…,ck: children of n where pn(c1)<=…<=pn(ck) at OR node n, • dn(c1)<=…<=dn(ck) at AND node n • PN: threshold of pn • DN: threshold of dn

  15. Conclusions • I believe enhanced AND/OR tree search algorithms will improve the ability of tsumego solvers!

  16. References • Masahiro Seo: The C* Algorithm for AND/OR Tree Search and its Application to a Tsume-Shogi Program, M.Sc. Thesis, Department of Information Science, University of Tokyo, 1995 • Ayumu Nagai and Hiroshi Imai: Proof for the Equivalence Between Some Best-First Algorithms and Depth-First Algorithms for AND/OR Trees, KOREA-JAPAN Joint Workshop on Algorithms and Computation,pp. 163-170, 1999 • Ayumu Nagai: Df-pn Algorithm for Searching AND/OR Trees and its Applications, Ph.D. Thesis, Department of Computing Science, University of Tokyo, 2002

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