planning as satisfiability cs672. 2 outline 0. overview of planning 1. modeling and solving planning...
TRANSCRIPT
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Planning as SatisfiabilityPlanning as Satisfiability
CS672
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OutlineOutline
0. Overview of Planning
1. Modeling and Solving Planning Problems as SAT - SATPLAN
2. Improved Encodings using Graph Analysis - BLACKBOX
3. Improved Encodings using Compiled Control Knowledge
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Overview of PlanningOverview of 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)
Action = a partial function State State
• specified by three sets of variables:precondition, add list, delete
list
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Abdundance of Negative Complexity Results
Abdundance of Negative Complexity Results
I. Domain-independent planning: PSPACE-complete
(Chapman 1987; Bylander 1991; Backstrom 1993)
II. Domain-dependent planning: NP-complete(Chenoweth 1991; Gupta and Nau 1992)
III. Approximate planning: NP-complete(Selman 1994)
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Planning as InferencePlanning 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
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Part 1: Modeling and Solving Planning Problems as SAT
Part 1: Modeling and Solving Planning Problems as SAT
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SAT EncodingsSAT Encodings
Planning Problem -> Propositional CNF by axiom schemas
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
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Encoding ConventionsEncoding Conventions
• Actions imply preconditions and effects
fly(x,y,i) at(x,i) & route(x,y) & at(y,i+1)
• Conflicting actions cannot occur at same time (A deletes a precondition of B)
fly(x,y,i) & yz fly(x,z,i)
• If something changes, an action must have caused it (Explanatory Frame Axioms)
at(x,i) & at(x,i+1) y . route(x,y) & fly(x,y,i)
• Initial and final states hold
at(NY,0) & ... & at(LA,9) & ...
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Modeling TricksModeling Tricks
Can often dramatically reduce size of problem by modeling techniques
move(x,y,z,i) requires n4 vars
pickup(x,y,i), putdown(x,z,i) requires 2n3 vars
State-based encodings: eliminate all action variables (“compile away”)
at(x,i) at(x,i+1) y . route(x,y) & at(y,i+1)
at(x,i) & xy at(y,i)
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Solution to a Planning ProblemSolution 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
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SATPLANSATPLAN
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• Inspired by Mins-Conflict algorithm
(Adorf, Johnson, Minton, Philips, & Laird)
• GSAT (Selman), Walksat (Selman, Kautz & Cohen)greedy local search + noise to escape minima
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Planning Benchmark Test SetPlanning Benchmark Test Set
Extension of Graphplan test set
blocks world - up to 18 blocks, 1019 states
logistics - complex, highly-parallel transportation domain.
Logistics.d:
• 2,165 possible actions per time slot
• 1016 legal configurations (22000 states)
• optimal solution contains 74 distinct actions over 14 time slots
Problems of this size never previously handled by state-space planning systems
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Scaling Up Logistics PlanningScaling Up Logistics Planning
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DP
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Randomized RestartsRandomized Restarts
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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What SATPLAN ShowsWhat SATPLAN Shows
General propositional theorem provers can compete with state of the art specialized planning systems
• New, highly tuned variations of DP surprising powerful
– result of sharing ideas and code in large SAT/CSP research community
– specialized engines can catch up, but by then new general techniques
• Radically new stochastic approaches to SAT can provide very low exponential scaling
– 2+ orders magnitude speedup on hard benchmark problems
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Why SATPLAN WorksWhy SATPLAN Works
More flexible than forward or backward chaining
• Systematic: most unit propagation at most highly constrained states
• Stochastic: iterative repair
Randomized algorithms less likely to get trapped along bad paths
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Part 2: Improved Encodings by Graph Analysis: The BLACKBOX Planner
Part 2: Improved Encodings by Graph Analysis: The BLACKBOX Planner
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GraphplanGraphplan
Planning as graph search (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
Graphplan - better instantiation
SATPLAN - better search
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Graph PruningGraph Pruning
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!
In logical terms: limited application of negative binary propagation
• given: P V Q, P V R V S V ...
• infer: Q V R V S V ...
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The Plan GraphThe Plan Graph
Facts FactsActions
... ...
Facts FactsActions
... ...
preconditions add effects
mutually exclusive
delete effects
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Translation of Plan GraphTranslation of Plan Graph
Fact Act1 Act2
Act1 Pre1 Pre2
¬Act1 ¬Act2
Act1
Act2
Fact
Pre1
Pre2
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General Limited InferenceGeneral Limited Inference
Generated wff can be further simplified by consistency propagation techniques
Compact (Crawford & Auton 1996)
• 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)
Complements domain specific limited inference
Discovers 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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BlackboxBlackbox
STRIPSPlan Graph
Mutex computation
CNF
GeneralStochastic / Systematic SAT engines
Solution
SimplifierTranslator
CNF
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Blackbox ResultsBlackbox Results
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Graphplan
BB-walksat
BB-rand-sys
Handcoded-walksat
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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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Part 3: Improved Encodings: Compiling Control KnowledgePart 3: Improved Encodings: Compiling Control Knowledge
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Kinds of Control KnowledgeKinds of Control Knowledge
About domain itself• a truck is only in one location
About good plans• do not remove a package from its destination location
About how to search• plan air routes before land routes
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Expressing KnowledgeExpressing Knowledge
Such information is traditionally incorporated in the planning algorithm itself
– or in a special programming language
Instead: use additional declarative axioms– (Bacchus 1995; Kautz 1998; Chen, Kautz, & Selman 1999)
• Problem instance: operator axioms + initial and goal axioms + control axioms
• Control knowledge constraints on search and solution spaces
• Independent of any search engine strategy
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Axiomatic Control KnowledgeAxiomatic Control Knowledge
State 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 Assumption: 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)
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Adding Control Kx to SATPLANAdding Control Kx to SATPLAN
ProblemSpecification
Axioms
Control Knowledge
Axioms
Instantiated Clauses
SAT Simplifier
SAT Engine
SAT “Core”
As control knowledge increases, Core shrinks!
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Tradeoffs of Control KnowledgeTradeoffs of Control Knowledge
If the planning domain is inherently intractable, how can any amount of control knowledge make planning tractable?
• by reducing solution quality
• optimal planning - NP-Hard
• non-optimal - (maybe) Polynomial
Issue: speed / quality tradeoff
Case study: Control Knowledge in TLPLAN and BlackBox
• TLPLAN (Bacchus 1996): simple forward-chaining search with strong control rules
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TLPlanTLPlan
Temporal Logic Control Formula
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I. Rules involves only static information
II. Rules depends on the current state
III. Rules depends on the current state and
requires dynamic user-defined predicates
Temporal Logic for ControlTemporal Logic for Control
( at(obj1, loc1) => at(obj1, loc1) )
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a
Category I Control RulesCategory I Control Rules
a
Do NOT unload an object from an airplane unless the object is at its goal destination
GoalInitial
a
SFO ORLNYC
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Pruning the Planning GraphCategory I Rules
Pruning the Planning GraphCategory I Rules
Facts FactsActions
... ...
Facts FactsActions
... ...
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Effect of Graph PruningEffect of Graph Pruning
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Category II Control RulesCategory II Control Rules
a
ORL NYC
Do NOT move an airplane if there is an object in the airplane that needs to be unloaded at that location.
SFO
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Control by Adding ConstraintsControl by Adding Constraints
Control Rules
))ORLORL )next(at(p,)at(p,(in(pkg,p)
Planning Formula Constraints Clauses
)( 1 iii yyx
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Blackbox with Control Knowledge(Logistics domain)
Blackbox with Control Knowledge(Logistics domain)
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tim
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ec)
blackbox blackbox(I) blackbox(I&II)
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Comparison Comparison between Blackbox and TLPlan Blackbox and TLPlan((Parallel Plan Length) Plan Length)
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Comparison between Blackbox and TLPlan(Running Time)
Comparison between Blackbox and TLPlan(Running Time)
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ComparisonComparison
TLPlan (without Control): Intractable.TLPlan (with Control): fastest, but limited parallelism
Blackbox (without Control): slower, high parallelismBlackbox (with Control): faster, high parallelism
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SummarySummary
Easy to encode domain-specific knowledge in the planning as satisfiablity frame
• Key to order-of-magnitude scaling
• Propositional logic, temporal logic, ...
• Can be applied before/after SAT encoding
Can control time / quality tradeoff• Power of underlying SAT engines gives option of
finding higher quality solutions
Heuristics are independent from the SAT engine• Can use same axioms for radically different
problem solvers
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How to Generate Control KxHow to Generate Control Kx
Introspection• Try to capture “obvious” inferences that are hard to
deduce
EBL (Minton, Kambhampati)
• Generalize trace of previous problem solving
Static analysis (Smith, Etzioni, Knoblock, Peot)
• Analyze operators
Inductive Logic Programming (Huang, Selman, Kautz)
• Find rules that hold for a set of previous high-quality solution plans
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ConclusionsConclusions
• Propositional approaches to Open-Loop planning using general SAT engines are highly competitive with specialized planning algorithms
• Synergy with Plan Graph approaches
• Can effectively employ purely declarative control knowledge
• Biggest limitation: domains where number of objects is too large to instantiate