Optimal Depth-First Strategies for And-Or Trees

1. Yummy i#1 Sweet i#2 Milk Fruit Cereal + M + Yummy S + + nl - C smcf i#1 Sweet p=0.3c=1 S C + + + S M i#2 F Milk p=0.8c=1 M cost success probability F - Fruit p=0.2c=1 Cereal p=0.7c=1 - - - Optimal Depth-First Strategies for And-Or Trees Russell Greiner*, Ryan Hayward and Michael Molloy University of Alberta University of Toronto *greiner@cs.ualberta.ca Yummy  Sweet v[Milk & (Fruit vCereal)] What to test? Which order? … Yummy! ? ?  Cost = $1 Prob = 80% ?  Cost = $3 Prob = 70% … • … which strategy is best? • correct • minimize expected costC[]! • Expected Cost of subtree rooted in  is … Given of each test, A Strategyspecified when to perform which tests… C[] = c() + Pr(+)  C[+] + Pr( -)  C[-] … Why not… Depth-First Algorithm For depth 1… Yummy Yummy smcf p=0.1824c=2.428 Yummy … X1 X2 Xm i#1 Sweet p=0.3c=1 Sweet p=0.3c=1 • DFA( AndOr tree ): strategy • Order leaf nodes (in each “penultimate subtree”) • by P(+Xi)/c(Xi) for Or-nodes; P(-Xi)/c(Xi) for And-node • Compute Probability P, ExpectedCost C of this subtree • Replace subtree with single MegaNode, • w/ prob P, cost C • Recur… i#1 NMCF p=0.608c=2.04 Sweet p=0.3c=1 Order s.t. Milk p=0.8c=1 NCF p=0.76c=1.3 i#2 Milk p=0.8c=1 S before M+C+F S C Fruit p=0.2c=1 … Cereal p=0.7c=1 X1 X2 Xm M before C+F M F C Order s.t. C C before F M F [Simon/Kadane, AIJ, 1975] F • Applications: • What will Baby eat? • Efficient medical diagnosis • Mining for gold [Simon/Kadane, 75] “Satisficing search” • Competing on Game show [Garey, 73] • Performing inference in simple expert system [Smith, 89]

2. Yummy + + i#1 S C Sweet + F M i#2 - - Cereal Milk Fruit Lab p=0.95c=5.1 Milk p=0.8c=1 AndOr trees read-k formulae optimal near-optimal linear -- • Theorem: • DFA is optimal for • depth-1 trees • depth-2 trees (-DNF, - CNF). Theorem: DFA is SUBoptimal for depth-3 trees. Theorem:  >0, DFA on unit-cost tests can be n1- worse than optimal!! Results Tree (not DAG) Independent tests Arbitrary costs + But , non-DFA … smcf M + DFA returns: nl • Note for unit-cost tests: • Max possible expected cost is n. • Max possible expected cost is 1. • So n times worse is worst possible!! • Why? • DFA forces siblings to be considered together, • so bad (low p/c) nodes can hamper good (high p/c) siblings. S - C … + + S M F - C[ smcf ] = 2.428 > 2.392 = C[nl ] !! … A1 Am … … - B1 Bm Z1 Zm Results, wrt Preconditions… • Precondition Model: • Each intermediate node is a probabilistic test --- with • its success probability and cost.. • Linear Strategy: • A strategy is linearif it performs the tests in fixed linear order, • skipping any test that will not help answer the question, given known info. Theorem: DFA produces a LINEAR strategy.. • DFA DFA but … • uses [Smith’89] for each “leaf subtree Note: • if internal tests have cost=0 and probability=1 • then DFA= DFA Yummy smcf • Laboratory test for Fruit, Cereal • Cost to SEND TO LAB is 5.1 • Only 95% chance mail will succeed + + Linear! S C + i#1 • Theorem • DFA is optimal for • 0-alternation trees • 1-alternation trees F Sweet p=0.3c=1 M • Theorem: • >0,  and-or tree whose optimal linear strategy costs n1/3- worse than optimal!! - - + M nl • Corollary: • DFA is SUBoptimal for depth-3 trees. •  >0, DFA  on unit-cost tests can be n1- worse than optimal!! + S - C Non-Linear: Sometimes sometimes + + M before S Fruit p=0.2c=1 Cereal p=0.7c=1 S M S before M F - - • Future work: • Complexity of computing strategy for ? •  poly time algorithm? • Empirical studies, on real-world tasks This work was partially funded by various grants from NSERC