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. Proof-Number Search (3/3) Algorithm
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
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. Sketch of Seo’s Algorithm
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

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 Notations: n: node searching 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 , 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