1. Heuristic Search Planners

2. 2 USC INFORMATION SCIENCES INSTITUTE Planning as heuristic search Use standard search techniques, e.g. A*, best-first, hill-climbing etc. Attempt to extract heuristic state evaluator automatically from the Strips encoding of the domain Here, generate relaxed problem by assuming action preconditions are independent

3. 3 USC INFORMATION SCIENCES INSTITUTE Recap: A* search Best-first search using node evaluation f(n) = g(n) + h(n) where g(n) = accumulated cost h(n) = estimate of future cost For A*, h(.) should never overestimate the cost. In this case, the solution will be optimal. Then h is called an admissible heuristic.

4. 4 USC INFORMATION SCIENCES INSTITUTE Derive cost estimate from a relaxed planning problem Ignore the deletes on actions BUT – still NP-hard, so approximate: For individual propositions p: d(s, p) = 0 if p is true in s = 1 + min(d(s, pre(a))) otherwise [min over actions a that add p]

5. 5 USC INFORMATION SCIENCES INSTITUTE Cost of a conjunction How to compute d(s,pre(a)) or d(s,G) ? Different options:  Additive: d(s, P) = sum d(s, p) over p in P  Max: d(s, P) = max d(s, p) Then h(s) = d(s, G) Can compute d(.,.) in polynomial time

6. 6 USC INFORMATION SCIENCES INSTITUTE Admissibility and information Is h+ (additive version) admissible? How about h-max?

7. 7 USC INFORMATION SCIENCES INSTITUTE Admissibility and information II If h+ is not admissible, why would we use it rather than h-max?

8. 8 USC INFORMATION SCIENCES INSTITUTE HSP algorithm overview Hill-climbing search + restarts if plateau for too long Some ad hoc choices for the planning competition Hill-climbing search is not complete

9. 9 USC INFORMATION SCIENCES INSTITUTE HSP2 overview Best-first search, using h+ Based on WA* - weighted A*: f(n) = g(n) + W * h(n). If W = 1, it’s A* (with admissible h). If W > 1, it’s a little greedy – generally finds solutions faster, but not optimal. In HSP2, W = 5

10. 10 USC INFORMATION SCIENCES INSTITUTE Experiments Does ok compared with IPP (Graphplan derivative) and Blackbox.

11. 11 USC INFORMATION SCIENCES INSTITUTE Regression search Motivation for HSPr  HSP and HSP2 spend up to 80% of their time computing the evaluation function.  Slow to generate nodes compared to other heuristic search systems.  Search backwards from goal, then re-use cost estimates from s0 to the goal, since we always have a single start state s0.  Common wisdom: regression planning is good because the branching factor is much lower

12. 12 USC INFORMATION SCIENCES INSTITUTE HSPr problem space States are sets of atoms (correspond to sets of states in original space) initial state s0 is the goal G Goal states are those that are true in s0 Still use h+. h+(s) = sum g(s0, p)

13. 13 USC INFORMATION SCIENCES INSTITUTE Mutexes in HSPr Problem: many of the regressed goal states are ‘impossible’ – prune them with mutexes E.g in blocksworld (on(c,d), on(a,d),..) is probably unreachable.

14. 14 USC INFORMATION SCIENCES INSTITUTE Mutexes in HSPr First definition: A set M of pairs R = {p, q} is a mutex set if (1) R is not true in s0 (2) for every op o that adds p, o deletes q Sound, but too weak.

15. 15 USC INFORMATION SCIENCES INSTITUTE Mutexes in HSPr, take 2 Better definition: A set M of pairs R = {p, q} is a mutex set if (1) R is not true in s0 (2) for every op o that adds p, either o deletes q or o does not add q, and for some precond r of o, {r, q} is in M. Recursive definition allows for some interaction of the operators

16. 16 USC INFORMATION SCIENCES INSTITUTE Computing mutex sets Start with some set of potential mutex pairs Delete any that don’t satisfy (1) and (2) above Keep going until you don’t delete any more Initial set? – could be all pairs (usually too expensive)

17. 17 USC INFORMATION SCIENCES INSTITUTE Initial set of potential mutexes Ma = { {p, q} | some action adds p and deletes q} Mb = { {r, q} | {p, q} is in Ma, some action adds p, and has r in the precondition} Initial set = Ma u Mb Mutex set derived from Ma u Mb is M*

18. 18 USC INFORMATION SCIENCES INSTITUTE HSPr algorithm WA* search using h+(s0) and M* W = 5 as before Prune states that contain pairs in M*

19. 19 USC INFORMATION SCIENCES INSTITUTE Experiments comparing HSP2 and HSPr Sometimes HSPr does better, sometimes HSP2 does better. Why? Two reasons (per B & G):  Still have spurious states  Since HSP2 recomputes the estimate in each state, it actually has more information

20. 20 USC INFORMATION SCIENCES INSTITUTE Evidence for spurious states Re-run HSPr using mutex set derived from all possible pairs. No difference in most domains Improvement in tire-world domain (with complex interactions) Slows down in logistics domain

21. 21 USC INFORMATION SCIENCES INSTITUTE Branching factor Varies widely from instance to instance. (Always seems greater in forward chaining though) Performance of HSP2 vs HSPr doesn’t seem to correlate with branching factor  Other factors dominate, e.g. informedness of heuristic

22. 22 USC INFORMATION SCIENCES INSTITUTE Derivation of heuristics h+ has problems when there are positive or negative interactions Can efficient heuristics better capture the interactions? H^2 – use the cost of the most expensive pair of goals  Still admissible, more informative than hmax, still cheap  Room for domain-dependent options?

23. 23 USC INFORMATION SCIENCES INSTITUTE Comparing HSPr and Graphplan Both search forwards in relaxed space, then backwards Planning graph encodes an admissible heuristic: hg(s) = j if j is the first level where s appears without a mutex Graphplan encodes IDA* efficiently as solution extraction – but this makes it hard to use other search algorithms.

24. 24 USC INFORMATION SCIENCES INSTITUTE Overall Planning as heuristic search: HSP family are elegant, quite efficient for domain-independent, and use clear principles of search Simple algorithms and relatively thorough analysis – make it easy to consider lots of extensions

25. 25 USC INFORMATION SCIENCES INSTITUTE Ways to extend Improving automatically generated heuristics More flexible action representations  Probably easier to encode in forwards than backwards search Principles and format for encoding domain- dependent heuristics  Both the estimate function and other control