Dfpn Depthfirst Proof Number Search Studied by Ayumu

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 • 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*

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

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

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

Proof-Number Search (3/3) Algorithm 2, 2 1. Traverse from the root to a leaf by choosing – Node with smallest (dis)proof number at (AND)OR node 2. Expand a leaf node 3. Update (Backup) pn & dn to the root 4. Continue 1 -3 until proven or disproven 3, 1 1, 1 2, 1 1, 2 pn, dn OR node pn, dn AND node

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

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

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

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

Sketch of Seo’s Algorithm • Node Selection: – OR node: c 1 – AND node: as you like • Updating Threshold: – OR: PN = min(pn(c 2) + 1, PN) – AND: PN = PN + pn (c 1) – (pn(c 2) + … + 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 – c 1, …, ck: children of n where pn(c 1)<=…<=pn(ck) – PN: threshold

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

Sketch of df-pn • Node Selection: – Always c 1 • Updating Threshold: – OR: – PN’= min(pn(c 2) + 1, PN) – DN’= DN – (dn(c 2) + … + dn(ck)) – AND: – PN’= PN – (pn(c 2) + … + pn(ck)) – DN’= min(dn(c 2) + 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 – c 1, …, ck: children of n where pn(c 1)<=…<=pn(ck) at OR node n, dn(c 1)<=…<=dn(ck) at AND node n PN: threshold of pn DN: threshold of dn

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

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