Planning as Satisfiability

1. Planning as Satisfiability CSE 574 April 8, 2003 Dan Weld

2. Schedule • BASICS • Intro • Graphplan • SATplan • State-space • Refinement • SPEEDUP TECHNIQUES • SIMULTANEOUS + DURATIVE ACTIONS • INCOMPLETE INFORMATION • INTERLEAVED PLANNING + EXECUTION • DECISION-THEORETIC PLANNING

3. The Idea • Suppose a plan of length n exists • Encode this hypothesis in SAT • Init state true at t0 • Goal true at Tn • Actions imply effects, etc • Look for satisfying assignment • Decode into plan RISC: The Revolutionary Excitement

4. History • Green IJCAI-69 • STRIPS AIJ-71 • Decades of work on “specialized theorem provers” • Kautz+Selman ECAI-92 • Rapid progress on SAT solving • Kautz+Selman AAAI-96 • Electrifying results (on hand coded formulae) • Kautz, McAllester & Selman KR-96 • Variety of encodings (but no compiler) • CSE 573 => Ernst et al. IJCAI-97

5. Blackbox • Blackbox solves planning problems by converting them into SAT. • Very fast • Tried different solvers • Local search (GSAT) • Systematic search with EBL (RelSAT) • In 2000, GP-CSP could beat Blackbox • But in 2001, a newer “SUPER-DUPER” SAT solver called CHAFF was developed, that significantly speeds up Graphplan • By folks in CAD community (EE folks—see getting other people to do our work) • Currently, CSP people are trying to copy over the ideas from CHAFF to CSP.

6. Action Representation more vars more clses Paint-A-Red = 5

7. Small # Variables, Literals • Variables for actions at t • Regular = n • Bitwise = log(n)

8. Main Ideas • Clear taxonomy • Utility of • Explanatory frame axioms (most things don’t change) • Parallelism & conflict exclusion • Type inference • Domain axioms • Surprising • Effectiveness of regular action encodings

9. Comparison Among Encodings • Explanatory Frames are superior to classical • - few actions affect each fluent • - explanatory frames aid simplifications • Parallelism is a major factor • - fewer mutual exclusion clauses • - fewer time steps • Regular actions representation is smallest! • - exploits full parallelism • - aids simplification • Overloaded, bitwise reps. are infeasible • - prohibitively many clauses • - sharing hinders simplification

10. Optimization 1: Factored Splitting - use partially-instantiated actions HasColor-A-Blue-(t-1) ^ Paint-Arg1-B-t ^ Paint-Arg2-Red-t HasColor-A-Blue-(t+1) Explanatory Frames factored unfactored

11. Optimization 2: Types • A type is a fluent which no actions affects. • type interference • prune impossible operator instantiations • type elimination Type opts No type opts

12. Domain-Specific Axioms Adding domain-specific axioms decreases solve time dramatically. domain info no domain info

13. Weaknesses • Too much info, poor organization • Graphs – lots of data, little insight • Characterize best/worst domain / encoding • No experiments on effects of type opt

14. Future Work • Negation, disjunctive preconds,  • Domain axioms • t clear(x, t)   y on(y, x, t)

15. Future Work • Automatically choose best encoding • Might do this for frame axioms • Automating domain axioms (invariants) • Use bounds on plan-length • Rather than linear or binary search • Resource Planning • Compilation to …?

16. Domain Axioms • Domain knowledge • Synchronicvs.Diachronic constraints • Speedup knowledge • Action conflicts (=> by action schemata alone) • Domain invariants (=> by initial state+schemata) • Optimality heuristics • Simplifying assumptions

17. 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

18. Sample Metric Planning Domain Steve Wolfman: I will use this as a running example 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

19. LPSAT Architecture Systematic SAT Solver + Incremental LP Solver input planning language planning compiler LCNF intermediate language LPSAT solver value assignment planning decoder solution plan

20. 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

21. Sample Problem in LCNF 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 good-trip1 deliver1 move-truck0 max-load move-truck0 max-fuel move-truck0 min-fuel move-truck0 deliver1 (move-truck0 all-loaded)  good-trip1 ... CNF formula Constraint triggers max-load: load 30 max-fuel: fuel  15 min-fuel: fuel  7 + load/2 all-loaded: load = 45 ... Set aside for later use!

22. 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

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

24. 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

25. LPSAT Algorithm Put on other projector 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|)

26. 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)

27. 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

28. Alternative encodings.. • The problem of finding a valid plan from the planning graph can be encoded on any combinatorial substrate • Alternatives: • CSP [GP-CSP] • SAT [Blackbox; SATPLAN] • IP [Vossen et al.] • BDDs [Cimatti ….] SAT is CSP with Boolean Variables

29. Compilation to CSP Goals: In(A),In(B) [Do & Kambhampati, 2000] CSP: Given a set of discrete variables, the domains of the variables, and constraints on the specific values a set of variables can take in combination, FIND an assignment of values to all the variables which respects all constraints Variables: Propositions (In-A-1, In-B-1, ..At-R-E-0 …) Domains: Actions supporting that proposition in the plan In-A-1 : { Load-A-1, #} At-R-E-1: {P-At-R-E-1, #} Constraints: Mutual exclusion ~[ ( In-A-1 = Load-A-1) & (At-R-M-1 = Fly-R-1)] ; etc.. Activation In-A-1 != # & In-B-1 != # (Goals must have action assignments) In-A-1 = Load-A-1 => At-R-E-0 != # , At-A-E-0 != # (subgoal activation constraints) [Corresponds to a regression-based proof]

30. CSP Encodings can be more compact: GP-CSP Do & Kambhampati, 2000

31. Compilation to LP Integer Linear Programming: Given a set of real valued variables, a linear objective function on the variables, a set of linear inequalities on the variables, and a set of integrality restrictions on the variables, Find the values of the feasible variables for which the objective function attains the maximum value -- 0/1 integer programming corresponds closely to SAT problem Motivations Ability to handle numeric quantities, and optimize Heuristic value of the LP relaxation of ILP problems Conversion Convert a SAT/CSP encoding to ILP inequalities E.g. X v ~Y v Z => x + (1 - y) + z >= 1 Explicitly set tighter ILP inequalities (Cutting constraints) If X,Y,Z are pairwise mutex, we can write x+y+z <= 1 (instead of x+y <=1 ; y+z <=1 ; z +x <= 1) [ Walser & Kautz; Vossen et. al; Bockmayr & Dimopolous]

32. Vossen/Ball/Lotem/Nau work • Vossen et. Al. [IJCAI 99] tried two different IP encodings • Simple conversion of PG SAT encoding to IP (last slide) • Two new ideas • Compile away proposition variables. • The change of truth value of a proposition is equated to the occurrence of one of the actions that can give that proposition. • This is basically a known transformation—see the McAllester/Kautz/Selman paper. (see next slide) • Reformulate the constraints to make better use of expressiveness provided by IP • E.g. IP is particularly good at compactly encoding n-ary mutex constraints—even for n>2 • To say only one of X1..Xn can be true, we just say X1+X2+…Xn <= 1 • In SAT, we have to write n constraints of the form • Xi => ~X1,~X2…~Xi-1,~Xi+1,….~Xn • (IP, on the other hand, will have harder time representing “All-different” constraints)

34. Performance of IP Encodings • As seen in the previous slide, IP encodings are not competitive with SAT/CSP encodings for classical planning • Aren’t exercizing the “optimizing” properties of IP, • Aren’t exercizing its ability to encode numerical variables • IP may well do better if we are interested in finding “cost-optimal” plans of a given makespan (length) • No one has confirmed this as yet… • IP encodings may well be more useful when we have numerical quantities such as actions consuming metric resources (preconditions and effects involve numbers) • See the work by Kautz and Walser(AAAI-99) or • LPSat (Wolfman/Weld—IJCAI 99).

35. Tradeoffs as f(compilation substrates) • CSP encodings support implicit representations • More compact encodings [Do & Kambhampati, 2000] • Easier integration with Scheduling techniques • ILP encodings support numeric quantities • Seamless integration of numeric resource constraints [Walser & Kautz, 1999] • Not competitive with CSP/SAT for problems without numeric constraints • SAT encodings support axioms in propositional logic form • May be more natural to add (for whom ;-) • Very active research on finding faster solvers.. • Davis Putnam ( Satz; RelSat; Chaff) • Gsat

36. Direct vs. compiled solution extraction Need to adapt CSP/SAT techniques Can exploit approaches for compacting the plan Can make the search incremental across iterations Compilation stage can be time consuming, leads to memory blow-up Makes it harder to exploit search from previous iterations Can exploit the latest advances in SAT/CSP solvers Makes it easier to add declarative control knowledge Compiled DIRECT

37. Resource Planning

38. Challenges in SAT