Richard Howey and Derek Long

1. VAL's Progress:The Automatic Validation Tool for PDDL2.1 used in the International Planning Competition Richard Howey and Derek Long University of Durham VAL’s Progress

2. Introduction • VAL was the validation tool used in the 3rd IPC. • VAL is evolving along side PDDL to validate plans written in PDDL. • The latest development is to extend VAL to validate plans with continuous effects. • Validating plans with continuous effects is an important first step to developing planners capable of handling continuous effects.

3. Validation Example: Generator and Tanks Initial State and Problem: • A generator must run continuously for 100 time units. • The generator has a capacity of 60 fuel units, using 1 fuel unit for every 1 time unit to generate. • Two fuel tanks with 25 fuel units each can refuel the generator while it is generating. • The generator starts with a full tank of fuel.

4. Durative Actions Generate: • Duration: must be 100 time units. • Invariant condition: fuel-level of generator must be greater than zero. • Continuous Effect: dg/dt = – 1 t = time since durative action began g = fuel-level of generator f = fuel-level of a tank k(f) = flow constant of tank f Refuel: • Duration: less or equal to time to empty tank. • Invariant condition: fuel-level of generator must be less than the capacity of the generator. • Continuous Effects: dr/dt = 1 df/dt = 2 k(f) ( k(f) r – f(0) ) dg/dt = 2 k(f) ( f(0) – k(f) r )

5. Possible Plan 1: (generate generator) [100] 20: (refuel generator tank1) [25] 80: (refuel generator tank2) [12.5] generate refuel 1 refuel 2

6. Plan to Validate 1: (generate generator) - start 20: Invariant for (generate generator) 20: Update of Continuously Changing Primitive Numerical Expressions 20: (refuel generator tank1) - start 45: Invariant for (generate generator) Invariant for (refuel generator tank1) 45: Update of Continuously Changing Primitive Numerical Expressions 45: (refuel generator tank1) - end 101: Invariant for (generate generator) 101: Update of Continuously Changing Primitive Numerical Expressions 101: (generate generator) - end

7. Time 1 upto 20 1: (generate generator) - start 20: Invariant for (generate generator) 20: Update of Continuously Changing Primitive Numerical Expressions t = time since last simple action g = fuel-level of generator dg/dt = – 1  g(t) = – t + 60 for t in [0,19) Invariant g(t) > 0 for t in (0,19)  – t + 60 > 0 for t in (0,19) Update Generator fuel-level: g(19) = –19 + 60 = 41

8. Time 20 upto 45 t = time since last simple action g = fuel-level of generator a = fuel-level of tank 1 20: (refuel generator tank1) - start 45: Invariant for (generate generator) Invariant for (refuel generator tank1) 45: Update of Continuously Changing Primitive Numerical Expressions 45: (refuel generator tank1) - end dr/dt = 1 da/dt = 0.08r – 2 dg/dt = – 0.08r + 2  r(t) = t for t in [0,25)  a(t) = 0.04t2 – 2t + 25 for t in [0,25)  dg/dt = – 0.08t + 1  g(t) = – 0.04t2 + t + 41 for t in [0,25) dg/dt = – 1

9. Time 20upto 45 t = time since last simple action g = fuel-level of generator Invariants g(t) > 0 for t in [0,25)  – 0.04t2 + t + 41 > 0 for t in [0,25) g(t)  60 for t in (0,25)  – 0.04t2 + t + 41  60 for t in (0,25)  0.04t2 – t + 19  0 for t in (0,25) Update g(25) = – 0.04252 + 25 + 41 = 41, a(25) = 0

10. Time 45 upto 80 80: Invariant for (generate generator) 80: Update of Continuously Changing Primitive Numerical Expressions dg/dt = – 1  g(t) = – t + 41 for t in [0,35) t = time since last simple action g = fuel-level of generator Invariant g(t) > 0 for t in [0,35)  – t + 41 > 0 for t in [0,35) Update Generator fuel-level: g(35) = – 35 + 41= 6

11. Time 80 upto 92.5 t = time since last simple action g = fuel-level of generator b = fuel-level of tank 2 80: (refuel generator tank2) - start 92.5: Invariant for (generate generator) Invariant for (refuel generator tank2) 92.5: Update of Continuously Changing Primitive Numerical Expressions 92.5: (refuel generator tank2) - end dr/dt = 1 db/dt = 0.32r – 4 dg/dt = – 0.32r + 4  r(t) = t for t in [0,12.5)  b(t) = 0.16t2 – 4t + 25 for t in [0,12.5)  dg/dt = – 0.32t + 3  g(t) = – 0.16t2 + 3t + 6 for t in [0,12.5) dg/dt = – 1

12. Time 80upto 92.5 t = time since last simple action g = fuel-level of generator Invariants g(t) > 0 for t in [0,12.5)  – 0.16t2 + 3t + 6 > 0 for t in [0,12.5) g(t)  60 for t in (0,12.5)  – 0.16t2 + 3t + 6  60 for t in (0,12.5)  0.16t2 – 3t –54  0 for t in (0,12.5) Update Generator and Tank fuel-levels: g(12.5) = 18.5, b(12.5) = 0

13. Time 92.5 upto 101 101: Invariant for (generate generator) 101: Update of Continuously Changing Primitive Numerical Expressions 101: (generate generator) - end dg/dt = – 1  g(t) = – t + 18.5 for t in [0,8.5) Invariant g(t) > 0 for t in [0,8.5)  – t + 18.5 > 0 for t in [0,8.5) Update Generator fuel-level: g(8.5) = – 8.5 + 18.5 = 10

14. Goal

15. LaTeX Report • Plan • Plan to validate • Plan validation • Diagrams • See handouts for generator validation report

16. Gantt Chart Actions of different executives can be displayed on different rows

17. Graphs Graphs of the primitive numerical expressions are plotted for the duration of the plan.

18. Conclusions • Restrictions on types of continuous effects are required to guarantee validation of plans: all continuous effects must be polynomial • Validating plans with continuous effects is an important first step to developing planners capable of handling continuous effects.