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Planning as Satisfiability

Planning as Satisfiability Henry Kautz University of Rochester in collaboration with Bart Selman and J ö erg Hoffmann AI Planning Two traditions of research in planning: Planning as general inference (McCarthy 1969) Important task is modeling

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Planning as Satisfiability

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  1. Planning as Satisfiability Henry KautzUniversity of Rochester in collaboration with Bart Selman and Jöerg Hoffmann

  2. AI Planning • Two traditions of research in planning: • Planning as general inference (McCarthy 1969) • Important task is modeling • Planning as human behavior (Newell & Simon 1972) • Important task is to develop search strategies

  3. Satplan • Model planning as Boolean satisfiability • (Kautz & Selman 1992): Hard structured benchmarks for SAT solvers • Pushing the envelope: planning, propositional logic, and stochastic search (1996) • Can outperform best current planning systems

  4. Translating STRIPS • Ground action = a STRIPS operator with constants assigned to all of its parameters • Ground fluent = a precondition or effect of a ground action operator: Fly(a,b) precondition: At(a), Fueled effect: At(b), ~At(a), ~Fueled constants: NY, Boston, Seattle Ground actions: Fly(NY,Boston), Fly(NY,Seattle), Fly(Boston,NY), Fly(Boston,Seattle), Fly(Seattle,NY), Fly(Seattle,Boston) Ground fluents: Fueled, At(NY), At(Boston), At(Seattle)

  5. Clause Schemas • A large set of clauses can be represented by a schema

  6. Satplan in 15 Seconds • Time = bounded sequence of integers • Translate planning operators to propositional schemas that assert:

  7. Example • If an action occurs at time i, then its preconditions must hold at time i • If an action occurs at time i, then its effects must hold at time i+1

  8. SAT Encoding • If a fluent changes its truth value from time i to time i+1, one of the actions with the new value as an effect must have occurred at time i Like “for”, but connects propositions with OR

  9. Plan Graph Based Instantiation initial state: p action a: precondition: p effect: p action b: precondition:  p effect: p  q m0 m1 = = p0 p1 p2 a0 a1 b1 q2

  10. International Planning Competition • IPC-1998: Satplan (blackbox) is competitive

  11. International Planning Competition • IPC-2000: Satplan did poorly Satplan

  12. International Planning Competition • IPC-2002: we stayed home. Jeb Bush

  13. International Planning Competition • IPC-2004: 1st place, Optimal Planning • Best on 5 of 7 domains • 2nd best on remaining 2 domains PROLEMA / philosophers

  14. The IPC-4 Domains • Airport: control the ground traffic [Hoffmann & Trüg] • Pipesworld: control oil product flow in a pipeline network [Liporace & Hoffmann] • Promela: find deadlocks in communication protocols [Edelkamp] • PSR: resupply lines in a faulty electricity network [Thiebaux & Hoffmann] • Satellite & Settlers [Fox & Long], additional Satellite versions with time windows for sending data [Hoffmann] • UMTS: set up applications for mobile terminals [Edelkamp & Englert]

  15. International Planning Competition • IPC-2006: Tied for 1st place, Optimal Planning • Other winner, MAXPLAN, is a variant of Satplan!

  16. What Changed? • Small change in modeling • Modest improvement from 2004 to 2006 • Significant change in SAT solvers!

  17. What Changed? • In 2004, competition introduced the optimal planning track • Optimal planning is a very different beast from non-optimal planning! • In many domains, it is almost trivial to find poor-quality solutions by backtrack-free search! • E.g.: solutions to multi-airplane logistics planning problems found by heuristic state-space planners typically used only a single airplane! • See: Local Search Topology in Planning Benchmarks: A Theoretical Analysis (Hoffmann 2002)

  18. Why Care About Optimal Planning? • Real users want (near)-optimal plans! • Industrial applications: assembly planning, resource planning, logistics planning… • Difference between (near)-optimal and merely feasible solutions can be worth millions of dollars • Alternative: fast domain-specific optimizing algorithms • Approximation algorithms for job shop scheduling • Blocks World Tamed: Ten Thousand Blocks in Under a Second (Slaney & Thiébaux 1995)

  19. Domain-Independent Feasible Planning Considered Harmful

  20. Objections • Real-world planning cares about optimizing resources, not just make-span, and Satplan cannot handle numeric resources • We can extend Satplan to handle numeric constraints • One approach: use hybrid SAT/LP solver (Wolfman & Weld 1999) • Modeling as ordinary Boolean SAT is often surprisingly efficient! (Hoffmann, Kautz, Gomes, & Selman, under review)

  21. Projecting Variable Domains initial state: r=5 action a: precondition: r>0 effect: r := r-1 • Resource use represented as conditional effects a0 a1 r=5 r=5 r=5 r=4 r=4 r=4

  22. 2002 ICAPS Benchmarks

  23. Large Numeric Domains Directly encode binary arithmetic action: a precondition: r  k effect: r := r-k a1 -k r11 r12 r21 r22 + r31 r32 r41 r42

  24. Objections • If speed is crucial, you still must use feasible planners • For highly constrained planning problems, optimal planners can be faster than feasible planners!

  25. Constrainedness: Run Time

  26. Constrainedness: Percent Solved

  27. Further Extensions to Satplan • Probabilistic planning • Translation to stochastic satisfiability (Majercik & Littman 1998) • Alternative untested idea: • Encode action “failure” as conditional resource consumption • Can find solutions with specified probability of failure-free execution • (Much) less general than full probabilistic planning (no fortuitous accidents), but useful in practice

  28. plan failure free probability  0.90 action: a failure probability: 0.01 preconditions: p effects: q action: a precondition: p  s  log(0.89) effect: q  s := s + log(0.99) Encoding Bounded Failure Free Probabilistic Planning

  29. One More Objection! • Satplan-like approaches cannot handle domains that are too large to fully instantiate • Solution: SAT solvers with lazy instantiation • Lazy Walksat (Singla & Domingos 2006) • Nearly all instantiated propositions are false • Nearly all instantiated clauses are true • Modify Walksat to only keep false clauses and a list of true propositions in memory

  30. Summary • Satisfiability testing is a vital line of research in AI planning • Dramatic progress in SAT solvers • Recognition of distinct and important nature of optimizing planning versus feasible planning • SATPLAN not restricted to STRIPS any more! • Numeric constraints • Probabilistic planning • Large domains

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