unifying sat-based and graph-based planning
DESCRIPTION
Unifying SAT-based and Graph-based Planning. Henry Kautz AT&T Labs Bart Selman Cornell University. IJCAI-99. SATPLAN (Kautz & Selman 1996). instantiated propositional clauses. instantiate. axiom schemas. problem description. length. mapping. SAT engine(s). interpret. - PowerPoint PPT PresentationTRANSCRIPT
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Unifying SAT-based and Graph-based Planning
Unifying SAT-based and Graph-based Planning
Henry KautzAT&T Labs
Bart SelmanCornell University
IJCAI-99
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SATPLAN (Kautz & Selman 1996)SATPLAN (Kautz & Selman 1996)
axiomschemas instantiated
propositionalclauses
satisfyingmodelplan
mapping
length
problemdescription
SATengine(s)
instantiate
interpret
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SAT AlgorithmsSAT Algorithms
Systematic Search• DP (Davis Putnam Logemann Loveland)
backtrack search + unit propagation
• satz (Chu Min Li) - variable selection by forward checking: max unit props
• relsat (Bayardo) - dependency directed backtracking: add new clauses at dead-ends
Local Search• Walksat (Selman, Kautz & Cohen)
local search + noise to escape minima
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Critically-constrained Logistics Planning Problems
Critically-constrained Logistics Planning Problems
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rocket.a
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Graphplan
DP
DP/Satz
Walksat
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Tradeoffs of SAT ApproachTradeoffs of SAT Approach
Advantages
• Can trade space for time by avoiding variable binding during search
• Domain modeling can substitute for algorithm development
• New high powered SAT algorithms can take advantage of implicit structure of encoded problems
Disadvantages
• Instantiated formulas huge, much redundancy
• Good domain models can be hard to develop - automatic STRIPS translations disappointing
• No way to explicitly leverage structure
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SATPLAN & Graphplan: Disjunctive Planners
SATPLAN & Graphplan: Disjunctive Planners
Graphplan (Blum & Furst 1995)
Set new paradigm for planning
Like SATPLAN...
• Two phases: instantiation of propositional structure, followed by search
Unlike SATPLAN...
• Interleaves instantiation and pruning of plan graph
• Employs specialized search engine
Neither approach best for all domains or all instances
• Graphplan - better instantiation
• SATPLAN - better search
IJCAI Challenge in Bridging Plan Synthesis Paradigms (Kambhampati 1997)
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BlackboxBlackbox
STRIPSPlan Graph
Reachability Analysis
CNF
GeneralStochastic / Systematic SAT engines
Solution
SimplifierTranslator
CNF
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Staged InferenceStaged Inference
Domain specific model
Polytime domain specific inference
General language encoding
Full general inference(NP complete)
Solution
Polytime general inference
Abstract problem specification
Encoding scheme
Combinatorial CORE
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IntuitionIntuition
Many real-world problems not tractable, but are nearly so
• polytime inference takes advance of special kinds of structure
• structure may be visible at the level of a domain specific representation, or only after the problem is encoded
• small number of practical methods for combinatorial core
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Component 1: Reachability AnalysisComponent 1: Reachability Analysis
Graphplan instantiates in a forward direction, pruning unreachable nodes • conflicting actions are mutex
• if all actions that add two facts are mutex, the facts are mutex
• if the preconditions for an action are mutex, the action is unreachable
Reachability analysis in unfolded Petri Nets(K. McMillian 1992)
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The Plan GraphThe Plan Graph
Facts FactsActions
... ...
Facts FactsActions
... ...
preconditions add effects
mutually exclusive
delete effects
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Component 2: TranslationComponent 2: Translation
Fact Act1 Act2
Act1 Pre1 Pre2
¬Act1 ¬Act2
Act1
Act2
Fact
Pre1
Pre2
Backward-chaining axioms force groundedness
Prevents underconstrained variables from taking on arbitrary values
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Mutex Algorithm as ResolutionMutex Algorithm as Resolution
Each mutex computation equivalent to a series of resolutions
• one resolvant always negative binary clause
K actions add P (1 clause)
K actions add Q (1 clause)
all P adders mutex Q adders (K2 clauses)
Inferring (~P v ~Q) requires 4K2 resolutions
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Improved EncodingsImproved Encodings
Translations of Logistics.a:
STRIPS Axiom Schemas SAT(Medic system, Weld et. al 1997)
• 3,510 variables, 16,168 clauses
• 24 hours to solve
STRIPS Plan Graph SAT
• 2,709 variables, 27,522 clauses
• 5 seconds to solve!
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Component 3: SimplificationComponent 3: Simplification
Generated wff can be further simplified by consistency propagation techniques
• unit propagation: is Wff inconsistant by resolution against unit clauses?
O(n)
• failed literal rule: is Wff + { P } inconsistant by unit propagation?
O(n2)
• binary failed literal rule: is Wff + { P V Q } inconsistant by unit propagation?
O(n3)
General limited inference complements domain specific limited inference (mutex)
Reveals hidden local structure
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General Limited InferenceGeneral Limited Inference
Percent vars set byProblem Varsunitprop
failedlit
binaryfailed
bw.a 2452 10% 100% 100%bw.b 6358 5% 43% 99%bw.c 19158 2% 33% 99%log.a 2709 2% 36% 45%log.b 3287 2% 24% 30%log.c 4197 2% 23% 27%log.d 6151 1% 25% 33%
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Component 4: Improved Systematic SAT Solvers
Component 4: Improved Systematic SAT Solvers
Systematic search generally best for wffs derived from STRIPS operators
• Wffs not as “flat” - long chains of unit propagations
Problem:
Solution time for backtrack search highly variable as problem instance varied
• “easier” problems may take orders of magnitude longer to solve than “harder” ones!
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Unpredictability of Systematic Search
Unpredictability of Systematic Search
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Satz
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Randomized RestartsRandomized Restarts
Heavy tailed distribution of running times
Solution: randomize the systematic solver
• Add noise to the heuristic branching (variable choice) function
• Cutoff and restart search after a fixed number of backtracks
In practice: rapid restarts with low cutoff can dramatically improve performance
(Gomes 1996, Gomes, Kautz, and Selman 1997, 1998)
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Increased PredictabilityIncreased Predictability
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Satz/Rand
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Summary of ResultsSummary of Results
Blackbox /satz-rand
Graphplan /IPP
SATPLANmake (walk) satz-rand
rocket.b 5 sec 55 sec 41 (1) 1 sec
log.a 5 sec 31 min 72 (2) 4 sec
log.b 7 sec 13 min 78 (3) 7 sec
log.c 9 sec * 102 (2) 1 sec
log.d 28 sec * 210 (7) 96 sec
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ObservationsObservations
SAT engines can outperform direct search of plan graph
• when problems critically constrained
• bottleneck is extraction (not reachability)
• when graphplan/IPP heuristics for non-optimal planning (e.g. RIFO) not applicable
Solution time using best randomized systematic SAT algorithm virtually identical for BlackBox and SATPLAN wffs
• although SATPLAN wffs included much extra explicit domain knowledge - invariants, etc.
Scaling of BlackBox/satz-rand closely matches scaling of SATPLAN/walksat (~ 4x)
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ApplicabilityApplicability
When is the BlackBox approach not a good idea?
• when domain too large for propositional planning approaches
• when long sequential plans are needed
• when solution time dominated by reachability analysis (plan-graph generation), not extraction
• when optimal or near optimal planning not necessary
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Efficiency of Translation ApproachEfficiency of Translation Approach
Translation usually not a bottleneck• wff grows linearly in size of plan graph
• modified translation reduces explicit mutex clauses by 75%
• new compact representations of plan graph will challenge this approach!
(Koehler, Fox & Long, Smith & Weld...)
Loss of cached information acceptable on hardest problems
• Graphplan caches info when searching “too short” graphs, use to speed up search of expanded graph
• For critically constrained problems, nearly all effort goes into searching last (or next to last) size problem
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Next Steps...Next Steps...
1. Domain-specific Control Knowledge• Encode state invariants & heuristics axiomatically
– Trucks always in one location
– Don’t move a package from a destination location
Dramatic speedup possible (Kautz & Selman 1998)
• For non-admissible control knowledge, tradeoff between speed / solution quality (Huang, Selman, Kautz AAAI-99)
– Temporal logic specification used to generate axioms and/or prune plan graph
– Using control knowledge from TLPlan (Bacchus 1996), can find better parallel plans
• Current work: inductive learning of control knowledge
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Comparison between Blackbox and TLPlan(Parallel Plan Length)
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log-c log-d log-e log-1 log-2
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TLPlan Blackbox
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Next Steps...Next Steps...
2. Beyond SAT: Planning with Resources and Optimization Criteria
• SAT special case of 0/1 integer linear programming
• ILPPlan (Kautz & Walser AAAI-99)Model extended STRIPS in AMPL, solve with
– Branch and bound
– Local search WSAT(OIP)
• Current work: IP translator for BlackBox(Nau et al 1999) - better encodings for B&B solvers
(Weld et al 1999) - new SAT+LP engine
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Next Steps...Next Steps...
3. Planning with Incomplete & Uncertain Information
• The “Holy Grail”
• SAT-encoding approaches
– Contingent planning via QBF (Rintanen 1999)
– C-MAXPLAN, ZANDER (Littman & Majercik 1999)Probabilistic planning via stochastic SAT
state of the art performance on (small, hard) POMDP problems
• Extensions to Graphplan
– contingent plans (Weld, Anderson, Smith 1998)
– probabilistic plans (Blum & Langford 1998)
• GOAL: a universal BlackBox
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Big PictureBig Picture
Domain specific model
Polytime domain specific inference
General language encoding
Full general inference(NP complete)
Solution
Polytime general inference
Abstract problem specification
Encoding scheme
Combinatorial CORE