Unifying SAT-based and Graph-based Planning

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Unifying SAT-based and Graph-based Planning

• Henry Kautz
• ATT Labs
• Bart Selman
• Cornell University

IJCAI-99
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SATPLAN (Kautz Selman 1996)
instantiated propositional clauses
instantiate
axiom schemas
problem description
length
mapping
SAT engine(s)
interpret
satisfying model
plan
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SAT 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
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Tradeoffs 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

• 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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Blackbox
Plan Graph
Reachability Analysis
STRIPS
Translator
CNF
Simplifier
General Stochastic / Systematic SAT engines
Solution
CNF
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Staged Inference
Polytime domain specific inference
Domain specific model
Abstract problem specification
General language encoding
Polytime general inference
Encoding scheme
Full general inference (NP complete)
Combinatorial CORE
Solution
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Intuition

• 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 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 Graph
Facts
Facts
Actions
Facts
Facts
Actions
...
...
...
...
mutually exclusive
preconditions
add effects
delete effects
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Component 2 Translation
Act1
Pre1
Fact
Pre2
Act2
Fact ? Act1 ? Act2 Act1 ? Pre1 ? Pre2 Act1 ?
Act2
Backward-chaining axioms force groundedness Preven
ts underconstrained variables from taking on
arbitrary values
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Mutex 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 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 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 Inference
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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
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Randomized 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 Predictability
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Summary of Results
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Observations

• 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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Applicability

• 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 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...

• 1. Domain-specific Control Knowledge
• Encode state invariants heuristics
  axiomatically
• Trucks always in one location
• Dont 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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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 BB
  solvers
• (Weld et al 1999) - new SATLP engine

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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 Picture
Polytime domain specific inference
Domain specific model
Abstract problem specification
General language encoding
Polytime general inference
Encoding scheme
Full general inference (NP complete)
Combinatorial CORE
Solution