Date post: | 31-Dec-2015 |
Category: |
Documents |
Upload: | morwenna-awena |
View: | 18 times |
Download: | 0 times |
Unifying SAT-based and Graph-based Planning
Unifying SAT-based and Graph-based Planning
Henry KautzAT&T Labs
Bart SelmanCornell University
IJCAI-99
2
SATPLAN (Kautz & Selman 1996)SATPLAN (Kautz & Selman 1996)
axiomschemas instantiated
propositionalclauses
satisfyingmodelplan
mapping
length
problemdescription
SATengine(s)
instantiate
interpret
3
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
4
Critically-constrained Logistics Planning Problems
Critically-constrained Logistics Planning Problems
0.01
0.1
1
10
100
1000
10000
rocket.a
rocket.b
log.b
log.a
log.c
log.d
log
so
luti
on
tim
e
Graphplan
DP
DP/Satz
Walksat
5
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
6
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)
7
BlackboxBlackbox
STRIPSPlan Graph
Reachability Analysis
CNF
GeneralStochastic / Systematic SAT engines
Solution
SimplifierTranslator
CNF
8
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
9
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
10
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)
11
The Plan GraphThe Plan Graph
Facts FactsActions
... ...
Facts FactsActions
... ...
preconditions add effects
mutually exclusive
delete effects
12
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
13
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
14
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!
15
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
16
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%
17
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!
18
Unpredictability of Systematic Search
Unpredictability of Systematic Search
0.01
0.1
1
10
100
1000
10000
rocket.a
rocket.b
log.a
log.b
log.c
log.d
log
so
luti
on
tim
e
Satz
19
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)
20
Increased PredictabilityIncreased Predictability
0.01
0.1
1
10
100
1000
10000
rocket.a
rocket.b
log.a
log.b
log.c
log.d
log
so
luti
on
tim
e
Satz
Satz/Rand
21
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
22
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)
23
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
24
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
25
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
26
Comparison between Blackbox and TLPlan(Parallel Plan Length)
0
5
10
15
20
25
30
35
log-c log-d log-e log-1 log-2
Par
alle
l P
lan
Len
gth
TLPlan Blackbox
27
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
28
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
29
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