1. Cost-Optimal Planning with Constraints and Preferences in Large State Spaces Stefan Edelkamp, Shahid Jabbar, Mohammed Nazih University of Dortmund

2. PDDL3

3. Outcome IPC-6 Optimal Planning STRIPS: SATPLAN, MAXPLAN SIMPLE TIME: CPT PDDL3: MIPS-XXL Suboptimal Planning PDDL3, 1st: SGPLAN, PDDL3, Special: YoChan (Simple), MIPS-XXL (Simple/Complex), ??? (Qualitative)

4. Motivation The Metric-FF Planner computes good plans for PDDL2.1 Level 2 Planning Problems How can it be extended to feature full expressiveness for the 2006 International Planning Competition?

5. Overview 3-Phase Compiler LTL Translation Automata translation Merging Constraint Hierarchy Hard- and Soft Constraints Large-Scale Search Conclusion, Future Work

6. Automata-based Model Checking Model M L(M)  L(S) Specification S L(M)  L(S) = {} L(M)@L(S) = {}

7. Images T(x,x‘) encodes Transition Relation for Image(x‘) =  x States(x)  T(x,x‘) Pre-Image(x) =  x States(x‘)  T(x,x‘)  Forward and Backward Search very much the same Partition Computation: 1 Transition Relation for each Planning Operator (  and v commute)  Disjunctive Partition

8. PDDL3 Expressions and LTL PDDL3 State Trajectory Constraints model temporally extended goals and control rules They are „close“ to LTL-expressions, e.g sometimes  <>, always  [] LTL Standard in Automata-based Model Checking (e.g. SPIN) Gerevini and Long give semantics in LTL Core Difference STC / LTL: Expression evaluated on finite / infinite trajectories

9. Three Phase-Compiler: PDDL3  Grounded PDDL 2 - 1- For each constraint: parse specificatoin, flatten quantifiers and flush corresponding LTL formula to disk -2- Derive Büchi Automaton for each LTL formula and generate corresponding PDDL code -3- Merge results of first and second phase  Poster on Main Conference

10. Compilation problem.pddldomain.pddl grounded-problemgrounded-domain.pddl LTL1 LTL2 LTL3 LTLn preprocess 1.pddl 2.pddl 3.pddl n.pddl merge domain.pddlproblem.pddl State Trajectory Constraints PDDL3 Grounded PDDL 2 Grounded PDDL 2

11. Constraint Hierarchy State Trajectory Constraints Hard Constraints Normal within TimedNormalTimed Soft Constraints withinat end

12. „Hard“ Automata Translation (sometime (at person4 city3))  <> (at person4 city3) (:predicates (accepting-a) (sync-automaton-a) (at-a-init) (at-a-accept)) (:init (at-a-init)) (:goal (accepting-a)) (at person4 city3) true

13. „Hard“ Automata Translation (:action sync-trans-a-init-a-accept :precondition (and (at-a-init) (sync-automaton-a) (at person4 city3)) :effect (and (accepting-a) (not (at-a-init)) (at-a-accept (not (sync-automaton-a)) (sync-next))) (at person4 city3) true

14. „Soft“ Automata Translation (preference z (sometime (at person4 city3)))  <> (at person4 city3) (:predicates (alive-p-z) (sync-automaton-p-z) (at-p-z-init) (at-p-z-accept)) (:init (alive-p-z) (at-p-z-init) (= (IS_VIOLATED p-z) 1)) (at person4 city3) true

15. „Soft“Automata Translation (:action sync-trans-p-z-init-p-z-accept :precondition (and (alive-p-z) (at-p-z-init) (sync-automaton-p-z) (at person4 city3)) :effect (and (assign (IS_VIOLATED p-z) 0) (not (at-p-z-init)) (at-p-z-accept) (not (sync-automaton-p-z)) (sync-next))) (at person4 city3) true

16. „Hard“ Compilation of Hold (thanks to Derek) (hold-during t1 t2 P)  Timed Initial Literal (at t1 dummy) (at t2 (not (dummy)) (hold-after t)  (at t dummy) (durative-action supplement :duration (= ?duration 0) :condition (and (over all (P)) (over all (dummy))) :effect (and (over all (done-dummy))))

17. „Soft“ Compilation of Hold (thanks to Derek) (hold-during t1 t2 P)  Timed Initial Literal (at t1 dummy) (at t2 (not (dummy)) (hold-after t)  (at t dummy) (durative-action supplement :duration (= ?duration 0) :condition (and (over all (P)) (over all (dummy))) :effect (and (over all (assign (is-violated p) 0))))

18. Compilation of „Within“ Timed Initial Literal (TI) (at 0 wfact) (at t (not (wfact)) LTL/BA (within t p)  (<> p) (always-within t p q)  ([ ] (p  (<> q))) Constraint automata acceptance with TIL

19. Compilation of „Within“ (:durative-action sync-trans-a-accept-a-accept :duration (= ?duration 0 ) :condition (and (at start (at-a-accept)) (over all (sync-automaton-a)) (over all (wfact))) :effect (and (at start (not (sync-automaton-a))) (at end (sync-next))))

20. Planner Architecture Metric-FF + PERT Scheduling (Durative FF)  PDDL2.1 Level 3 Multiple Time Windows for Action Execution (Timed Initial Literals)  PDDL2.2 PDDL3  PDDL2 Compiler Branch-And-Bound Wrapper Externalization

21. (Internal) Branch-And-Bound Finding Plans: f = |PrefixPlan| + k |RelaxedPlan| (standard) f = PERT(PrefixPlan o RelaxedPlan) (temporal)  Plan P, quality Q Optimizing Plans wrt. Metric F Wrapper includes F(P) < Q as new goal for next iteration terminates (full exploration, time-out) with best Q If Plan-Finding / Optimization fails  Externalize

22. External Branch-And-Bound Planning External Algorithms utilize Secondary Memory Primitives: External Scanning, External Scanning - sort(|E|), scan(|V|) External BFBnB External Enforced Hill-Climbing Solution Reconstruction: Double State Vector: (State, Predecessor)

23. External BF Branch-And-Bound Constraints Upper Bound O(sort(|E|)+scan(|V|))

24. External Enforced Hill Climbing  AAAI-06 h Every BFS-Layer is a file … actually executed in (g-h Matix)  External A* O(h(s) *(sort(|E|)+scan(|V|)))

25. Conclusion 1st PDDL3-to-PDDL2 Compiler Full PDDL Expressiveness (ass. linear expr.) Advantages for Heuristic Search Planning: Accepting states Wrapper on Metric Eager Preference Evaluation  Early Application of Pruning Rules in BnB 1st External Planner - Useful even in Competition Context (cf. Results)

26. Future Work Proving Correctness of the Compilation Integration of Acceleration Techniques: Symmetry Pruning, Partial Order Reduction, Goal Ordering … Application to Model Checking Extension to PDDL4 (Processes and Events) Evaluate Automata Compilation vs. Formula Progression

27. BDD-based Planning Symbolic Representation of Planning State Sets 1 Bit per Proposition Inefficient  Use Multivariate (SAS+) Encoding Variable Ordering is important Breadth-First Symbolic Search: Si(x) represents all states reachable in i steps

28. Images T(x,x‘) encodes Transition Relation for Image(x‘) =  x States(x)  T(x,x‘) Pre-Image(x) =  x States(x‘)  T(x,x‘)  Forward and Backward Search very much the same Partition Computation: 1 Transition Relation for each Planning Operator (  and v commute)  Disjunctive Partition

29. Weaknesses Automata Translation in LTL2BA not optimal Wrapper Approach with Relaxed Planning Heuristic not optimal (Duplicate Anomaly)