1. The Role of Domain-Specific Knowledge in the Planning as Satisfiability Framework Henry Kautz AT&T Labs Bart Selman Cornell University

2. 2 A Core Computation Problem Focus: Classical state-space planning, extended with parallel actions Core problem - similar computational issues arise in many other models Reactive plans Planning with uncertainty and utilities Continuous processes Metric time Many results directly extend to richer formalisms

3. 3 State-space Planning Find a sequence of operators that transform an initial state to a goal state State = complete truth assignment to a set of variables (fluents) Goal = partial truth assignment (set of states) Operator = a partial function State  State specified by three sets of variables: precondition, add list, delete list

4. 4 Parallelism Operators may be applied in parallel when all orderings are well defined and equivalent (Op1 || Op2)(s) = Op2(Op1(s)) = Op1(Op2(s)) A special form of non-linear plans Only allows parallel actions, not parallel action sequences Easy to serialize

5. 5 Abdundance of Negative Complexity Results I. Domain-independent planning: PSPACE- complete or worse (Chapman 1987; Bylander 1991; Backstrom 1993) II. Domain-dependent planning: NP-complete or worse (Chenoweth 1991; Gupta and Nau 1992) III. Approximate planning: NP-complete or worse (Selman 1994)

6. 6 Practice Domain-independent planning systems can only generate relatively short plans. Practical systems minimize or eliminate search by employing Search control rules, hand-tailored to the search engine and the particular search space (Sacerdoti 1975, Slaney 1996, Bacchus 1996) Pre-compiling entire state-space to a reactive finite-state machine (Agre & Chapman 1997, Williams & Nayak 1997) Scaling remains problematic when state space is large or not well understood!

7. 7 Planning as Inference Planning as first-order theorem proving (Green 1969) computationally infeasible STRIPS (Fikes & Nilsson 1971) very hard Partial-order planning (modal truth criteria) (Tate 1977, Chapman 1985, McAllester 1991, Smith & Peot 1993) can be more efficient, but still hard (Minton, Bresina, & Drummond 1994) SATPLAN: planning as propositional reasoning

8. 8 Approach SAT encodings are designed so that plans correspond to satisfying assignments Use recent efficient satisfiability procedures (systematic and stochastic) to solve Evaluation performance on benchmark instances

9. 9 SATPLAN axiom schemas instantiated propositional clauses satisfying model plan mapping length problem description SAT engine(s) instantiate interpret (New implementation blackbox takes STRIPS input - these experiments based on older hand-created “state-based” encodings)

10. 10 SAT Encodings Propositional CNF: no variables or quantifiers Sets of clauses specified by axiom schemas fully instantiated before problem-solving Discrete time, modeled by integers state predicates: indexed by time at which they hold action predicates: indexed by time at which action begins –each action takes 1 time step –many actions may occur at the same step fly(Plane, City1, City2, i)  at(Plane, City2, i +1)

11. 11 Solution to a Planning Problem A solution is specified by any model (satisfying truth assignment) of the conjunction of the axioms describing the initial state, goal state, and operators Easy to convert back to a STRIPS-style plan

12. 12 Satisfiability Testing Procedures Systematic, complete procedures Depth-first backtrack search (Davis, Putnam, & Loveland 1961) –unit propagation, shortest clause heuristic One of the best implementations: ntab (Crawford & Auton 1997) –see also: satz (Li), CSAT (Dubois), MODOC (van Gelder) Stochastic, incomplete procedures Inspired by Mins-Conflict algorithm (Adorf & Johnson 1990; Minton, Johnson, Philips, & Laird 1992) GSAT (Selman et. al 1993) Current fastest: Walksat (Selman & Kautz 1993) –greedy local search + noise to escape local minima

13. 13 Planning Benchmark Test Set Extension of Graphplan test set blocks world - up to 18 blocks, 10 19 states logistics - complex, highly-parallel transportation domain. Logistics.d: 2,165 possible actions per time slot 10 16 legal configurations (2 2000 states) optimal solution contains 74 distinct actions over 14 time slots Problems of this size never previously handled by state-space planning systems

14. 14 Solution of Logistics Problems

15. 15 Why SATPLAN Works More flexible than forward or backward chaining Systematic: most unit propagation at most highly constrained states Stochastic: iterative repair Space for time tradeoff Less overhead since does not have to instantiate variable during search Randomized algorithms less likely to get trapped along bad paths

16. 16 Domain Dependent Planning Efficient representations and (randomized) SAT engines extend the range of domain- independent planning Ways to further improvement: Better general search algorithms Incorporate (more) domain dependent knowledge

17. 17 Kinds of Knowledge * About domain itself a truck is only in one location airplanes are always at some airport * About good plans do not remove a package from its destination location do not unload a package and immediate load it again X About how to search plan air routes before land routes work on hardest goals first

18. 18 Expressing Knowledge Such information is traditionally incorporated in the planning algorithm itself –or in a special programming language Instead: we propose expressing as additional declarative axioms Problem instance: operator axioms + initial and goal axioms + heuristic axioms Domain knowledge  constraints on search and solution spaces Independent of any search engine strategy NOT logic programming!

19. 19 Logical Status of Heuristics 1. Entailed by operator axioms: conflicts and derived effects fly(plane,d1,i) and fly(plane,d2,i) conflict 2. Entailed by operators + initial state axioms: state invariants a truck is at only one location 3. Entailed by operators + initial + goal + length: optimality conditions do not return a package to a location 4. New constraints on problem instance: simplifying assumptions Once a truck is loaded, it should immediately move

20. 20 Axiomatic Form Invariant: A truck is at only one location at(truck,loc1,i) & loc1  loc2   at(truck,loc2,i) Optimality: Do not return a package to a location at(pkg,loc,i) &  at(pkg,loc,i+1) & i<j   at(pkg,loc,j) Simplifying: Once a truck is loaded, it should immediately move  in(pkg,truck,i) & in(pkg,truck,i+1) & at(truck,loc,i+1)   at(truck,loc,i+2)

21. 21 Questions Does it work? Additional axioms might just blow up instance with redundant information Is effect independent of search engine? Simplifying assumptions might hurt stochastic methods by reducing solution density Can we predict the most useful level of heuristic axioms? What is relation of difficulty to problem size?

22. 22 Experiment 1: Blocks World Observation: Original BW examples for SATPLAN included some state invariant axioms After simplification by unit propagation, invariant axioms make instance smaller and easier to solve (Ernst, Millstein, Weld 1997) Experiment: 3 versions of each instance minimal set of axioms + state invariants + optimality (“don’t move a block twice in a row”)

23. 23 Experiment 2: Logistics h1: Optimality conditions Once a package leaves a location, it never returns h2, h3: Simplifying assumptions A package is never in any city other than its origin or destination cities –rules out solutions where packages are transferred between airplanes in an intermediate city Once a vehicle is loaded, it should immediately move –rules out solutions where vehicles are loaded incrementally h4: More optimality conditions A package never leaves its destination city

24. 24 logistics size in variables

25. 25 logistics size in clauses

26. 26 walksat solution of logistics

27. 27 ntab solution of logistics

28. 28 Answers Does it work? YES, 10 to 100+ times speedup Is effect independent of search engine? YES, same heuristics best for systematic and stochastic engines - but needs more investigation Can we predict the most useful level of heuristic axioms? USUALLY point at which problem size is minimized after simplification by unit propagation (40% - 70% reduction) BUT larger simplified instances still easier than “no heuristic” case

29. 29 How to Generate Heuristics Introspection Try to capture “obvious” inferences that are hard to deduce (McAllester) Compilation Fix operators and initial or goal state, generate tractable equivalent theory (Kautz & Selman) In worse case, tractable theory is exponentially large but can be efficient in practice EBL (Minton, Kambhampati) Static analysis (Smith, Etzioni, Knoblock, Peot) How to adapt to generate constraints on plans instead of explicit control rules?

30. 30 Summary Easy to encode domain-specific knowledge in the planning as satisfiablity frame Key to next order-of-magnitude scaling Purely declarative assertions Not logic programming! Makes clear logical status of different heuristics invariants, optimality, simplifying entailed  admissible Heuristics are independent from the SAT engine Can use same axioms for radically different problem solvers

31. 31 Summary (continued) Dramatic speedup possible Stochastic: speedup 6X - 10X Systematic: speedup 100X + For largest, hardest problems, additional domain knowledge allows systematic solvers to become competitive but in some domains stochastic solvers still show better scaling

32. 32 Logistics Best Solution Times

33. 33 Current & Future Work Automatic generation of axioms from STRIPS input efficient interleaving instantiation and pruning (Kambhampati 1997) Randomized systematic search broad coverage across domains robust against instances that are hard for deterministic strategies (“heavy tails”) (Gomes, Selman, Kautz 1997) blackbox adapts graphplan for generating wffs randomized version of satz (Li 1997) now solves log.d in 1 minute directly from STRIPS input, without hand-encodings or domain knowledge!

34. 34 blackbox version 9B command line: blackbox -o logistics.pddl -f logistics_prob_d_len.pddl -solver compact -l -then satz -cutoff 25 -restart 10 ---------------------------------------------------- Converting graph to wff 6151 variables 243652 clauses Invoking simplifier compact Variables undetermined: 4633 Non-unary clauses output: 139866 ---------------------------------------------------- Invoking solver satz version satz-rand-2.1 Wff loaded [1] begin restart [1] reached cutoff 25 --- back to root [2] begin restart [2] reached cutoff 25 --- back to root [3] begin restart [3] reached cutoff 25 --- back to root [4] begin restart [4] reached cutoff 25 --- back to root [5] begin restart **** the instance is satisfiable ***** **** verification of solution is OK **** total elapsed seconds = 25.930000 ---------------------------------------------------- Begin plan 1 drive-truck_ny-truck_ny-central_ny-po_ny

35. 35

36. 36 Current & Future Work (continued) Future extensions to blackbox: Specification of heuristic axioms in the STRIPS-style input –state invariants - straightforward –optimality conditions - temporal logic? Automatic generation of heuristic axioms DOWNLOAD latest version from www.research.att.com/~kautz/blackbox

37. 37 What is Planning? Informally: find a (partially) ordered set of actions that transform a given initial state to a specified goal state. in most general case, can cover most forms of problem solving special case of program synthesis scheduling: fixes set of actions, need to find optimal total ordering - scheduling problems often (near) linear, may be solvable by linear relaxation techniques - planning problems typically highly non-linear, require combinatorial search

38. 38 Some Applications of Planning Autonomous systems Many NASA applications: Deep Space One Remote Agent Long-range mission planning Communication planning & scheduling Softbots - software robots Internet agents, program assistants AI “characters” in games, entertainment User modeling / discourse Synthesis, diagnosis, experiment design, bug- finding (goal = undesirable state),...

39. 39 Walksat Procedure Start with random initial assignment. Pick a random unsatisfied clause. Select and flip a variable from that clause: With probability p, pick a random variable. With probability 1-p, pick greedily –a variable that minimizes the number of unsatisfied clauses Repeat to predefined maximum number flips; if no solution found, restart.