Finding Search Heuristics

1. Finding Search Heuristics Henry Kautz

2. A* Graph Search for Any Admissible Heuristic if State[node] is not in closed OR g[node] < g[LookUp(State[node],closed)] then

3. Inconsistent but Admissible

4. Memory Efficient Versions of A* • Iterative Deepening A* (IDA*) • Just like IDS, except that: • Stopping condition: g(n)+h(n)=f(n) > Cutoff • Increment: Min{ f(n) | n was cutoff last iteration} • Optimal • Memory linear in cheapest path • Will re-expand nodes frequently

5. Memory-Bounded A* • Throw nodes out of Closed and/or Fringe when memory gets low • Dropping nodes from Closed can make extra work, but doesn’t hurt optimality • Dropping nodes from Fringe: • Choose worst node n from Fringe • Set f(Parent(n)) = Max { f(Parent(n)), Min{ f(n’) | n=Parent(n’) } • If Parent is not in Fringe, put it in Fringe • Many variations – MA*, SMA*, SMAG*, …

9. Properties of Heuristics • Let h1 and h2 be admissible heuristics • if for all s, h1(s)  h2(s), then • h1 dominates h2 • h1 is better than h2 • h3(s) = max(h1(s), h2(s)) is admissible • h3 dominates h1 and h2

10. Exercise: Rubik’s Cube • State: position and orientation of each cubie • Corner cubie: 8 positions, 3 orientations • Edge cubie: 8 positions, 2 orientations • Center cubie: 1 position – fixed • Move: Turn one face • Center cubits never move! • Devise an admissible heuristic

11. Heuristics • # cubies out of place / 8 • Must divide up 8 so is admissible • Too optimistic / weak •  Better:  •  Korf 1997 – Solves Rubik’s cube optimally

12. Automatically DerivingHeuristics for STRIPS planning • Goal: quickly compute upper bound on distance from S to a state containing Goals • Consider: |Goals – S|/m where m is the maximum number of goals added by any one action • Admissible? • Accurate? Yes! No, ignores preconditions!

13. Another Attempt • Create relaxed planning problem: Solve • Initial = S • Same Goals • Eliminate negative effects from all operators • Use length of shortest solution to relaxed problem as h • Admissible? • Accurate? • Easy to compute? Yes! Pretty good No, hard as set covering!

14. Plan Graph Heuristic • Idea: Quickly estimate the minimum length of solution of relaxed problem • Method: • For each literal, create a dummy “persistence” action with that literal as precondition and effect • Compute a Plan Graph: • S0: initial literals • A0: actions whose preconditions are in S0 • S1: add effects of actions in A0 • A1: actions whose preconditions are in S1 …

15. Plan Graph Heuristic • Method (continued): 3. Propagate mutual exclusions (mutexes): • Two non-persistence actions at the same level are mutex • A persistence action is mutex with an action that whose effect negates it • Two literals A and B are mutex at level i if all the actions that add A are mutex with those that add B 4. Estimated length = lowest level containing all goals with no mutexes between them

16. Example Plan Graph

17. Is Planning Solved? • A* with plan graph heuristic works well for domains with relatively few bad interactions between actions • For “harder” domains: • Solve non-optimally using non-admissible heuristics • Solve optimally using other approaches – such as satisfiability testing • More later in the course!