1. 1 The LPSAT Engine and its Application to Metric Planning Steve Wolfman University of Washington CS&E Advisor: Dan Weld

2. 2 Background: Planning Problem Given –Domain: a set of actions –Problem: an initial state and a set of goal states Produce a list of actions which leads from the initial state to the goal Metric planners reason about real-valued variables (mappings from preconditions to effects) (usually a partially specified state)

3. 3 Metric Planning Metric planning –Incorporate metric quantities; e.g. y  2 + x –Important for modeling expendable resources Fast back end metric solver will help –Shown effective with SAT solvers for planning [Kautz&Selman] –Compile from planning to simple intermediate language –Solve problems in simple language Steve Wolfman: Note MAJSAT success Steve Wolfman: Note MAJSAT success

4. 4 Sample Metric Planning Problem Ship RedVines from one location to another –Fuel used by moving –Maximum fuel level in truck –Maximum load for truck –Limited amount of available RedVines

5. 5 Contributions Architecture –Intermediate language with metric values –Efficient propositional logic + metric constraints solver –Proof of concept in metric planning domain System –LCNF intermediate language –LPSAT solver –Metric planning compiler/decoder extended SAT solver to work with metric constraints and Cassowary. Extended Cassowary to discover minimal conflict sets. Designed LCNF language. Designed minimal conflict set discovery technique. Wrote LiPSyNC’s translator from planning to LCNF and the decoder from LCNF solution to plan. extended SAT solver to work with metric constraints and Cassowary. Extended Cassowary to discover minimal conflict sets. Designed LCNF language. Designed minimal conflict set discovery technique. Wrote LiPSyNC’s translator from planning to LCNF and the decoder from LCNF solution to plan.

6. 6 Outline Motivation Planning System LCNF LPSAT Experimental Results Conclusions planning compiler LPSAT solver planning decoder input planning language LCNF intermediate language value assignment solution plan

7. 7 Metric Planning Compiler/Decoder Compiler –inputs planning problems in LPDDL –accepts constraints over operators +, -, *, and  –outputs LCNF problem and mapping tables Decoder –inputs solution to LCNF problem and mapping tables –outputs plan with instantiated metric values So non-linear OK, so far Extension of McDermott’s PDDL (truth assignment and constraint variable values)

8. 8 Sample Metric Planning Domain Action: MOVE-TRUCK(load) preconditions: load  45 ; max. avail. RedVines fuel  7 + load / 2 ; min. required fuel fuel  15 ; fuel capacity load  30 ; load capacity effects: (deliver) if (load = 45) then ; good-trip moves (good-trip) ; all the RedVines Steve Wolfman: I will use this as a running example Steve Wolfman: I will use this as a running example

9. 9 Sample Metric Planning Problem Problem: Too-Big initial conditions: not (deliver) not (good-trip) goal conditions: (deliver) (good-trip) Mention unsolvable for goals deliver _and_ good-trip

10. 10 LCNF Language CNF formula with triggered constraints –Boolean variable triggers one (or no) constraint –Constraint triggered iff its trigger variable’s truth assignment is true –Truth/real-value assignment is a solution iff CNF formula is satisfied All triggered constraints are satisfied

11. 11 Sample Problem in LCNF good-trip 1 deliver 1 move-truck 0  max-load move-truck 0  max-fuel move-truck 0  min-fuel move-truck 0  deliver 1 (move-truck 0  all-loaded)  good-trip 1... max-load: load  30 max-fuel: fuel  15 min-fuel: fuel  7 + load/2 all-loaded: load = 45... CNF formulaConstraint triggers Variable names and, in particular, trigger names added for clarity That’s the front end of the system; the back end simply reads off which actions are assigned true and what the constraint vars values are Set aside for later use!

12. 12 LPSAT Inputs LCNF subset –Supports operators +, -, *, and  –Restricts constraints to linear (in)equalities Outputs truth/real-value assignment or reports failure –Sound –Complete –Satisficing As opposed to optimizing

13. 13 LPSAT Architecture Satisfiability Solver Linear Programming System new/revoked constraints consistency info, real variable values input problem solution

14. 14 LPSAT Design SAT controller calls LP black box SAT solver based on RelSAT [Bayardo&Schrag] –Systematic solver –Sound and complete –Gradual changes to constraint set –Learning/backjumping based on conflict sets LP system is Cassowary [Badros&Borning] –supports linear constraints over real variables –fast updates to constraint set Explores all possible truth assignments without repeating any

15. 15 LPSAT Algorithm: Redefinitions Important changes to RelSAT concepts satisfied = statement is empty and active constraint set is consistent inconsistent = clause in statement is empty or active constraint set is inconsistent pure literal = any literal whose negation never appears in statement except positive trigger variables

16. 16 LPSAT Algorithm Procedure LPSAT(φ: LCNF problem) If φ is satisfied, return YES Else if φ is inconsistent, return NO Else if there is a unit clause {  } or pure literal  in φ, return LPSAT(φ|  ) Else choose a variable in φ. If LPSAT(φ| ), return YES Else, return LPSAT(φ|  ) Put on other projector

17. 17 Modifications to RelSAT Altered to support trigger variables –Constraints tied to variables –Pure literal rule modified Incorporated constraints in solution –Constraint consistency checked after each add –Real values reported in final solution Added trigger-aware heuristic function

18. 18 Modifications to Cassowary Support RelSAT learning/backjumping Global conflict set discovery –Reports entire set of active constraints –Correct but conservative Minimal conflict set discovery –Set of constraints reported is inconsistent –Every proper subset of the set is consistent –Correct and almost optimal Implemented in Cassowary framework Linear in # of constraints + # of variables Determines most constraining conflict set Implemented in Cassowary framework Linear in # of constraints + # of variables Determines most constraining conflict set Minimal, not minimum Explain why smaller is better!

19. 19 Sample Problem Conflict Sets fuel load load  30 fuel  15 load = 45 fuel  7+load/2 Solution region without load = 45 constraint

20. 20 Experimental Results Tested against Zeno planner –Many times faster than Zeno –Solved problems Zeno could not solve Tested conflict set discovery techniques –Without learning/backjumping (slow) –With global conflict sets (faster) –With minimal conflict sets (fastest) Under resource bounds (memory)

21. 21 Conflict Set Discovery Results The domain is a metric version of Kautz and Selman’s logistics domain. Log-c is difficult in the original domain even for modern planners

22. 22 Contributions Created LCNF intermediate language Implemented LPSAT, an LCNF solver –Solves LCNF with linear constraints –Incorporates new technique for finding conflict sets in incremental Simplex Implemented a full metric planning system –Compiler translates metric planning to LCNF –Decoder translates LCNF solutions to plans

23. 23 Conclusions Enhanced satisfiability solvers are effective for solving metric planning problems Effective heuristics for combined solution processes must use information from both processes The translate/solve/decode architecture of SAT-based planning can be profitably extended to more complex solvers

24. 24 Future Work: Compiler/Decoder Optimize current compiler –Speed –Encoding quality Use metric IPP [Koehler] to compile metric planning problems to LCNF Compiler/decoders for new domains –Scheduling –Temporal planning –Analog circuit verification

25. 25 Future Work: LPSAT/LCNF Implement LPSAT with a stochastic SAT solver Decompose nonlinearities in LCNF Construct an LCNF solver with native support for nonlinear constraints Investigate other combinations of solution processes