Deconstructing Planning as
Satisfiability
Henry Kautz
University of Rochester
in collaboration with Bart Selman and
Jöerg Hoffmann
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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
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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
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Satplan (satz)
Graphplan (IPP)
log.a
5 sec
31 min
log.b
7 sec
13 min
log.c
9 sec
> 4 hours
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Satplan in 15 Seconds
• Time = bounded sequence of integers
• Translate planning operators to
propositional schemas that assert:
action(i )  pre(i )  effect(i  1)
action1(i )  action2 (i ) if interfering
fact(i )  fact(i  1)  action1(i )  action2 
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International Planning Competition
• IPC-1998: Satplan (blackbox) is
competitive
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International Planning Competition
• IPC-2000: Satplan did poorly
Satplan
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International Planning Competition
• IPC-2002: we stayed home.
Jeb Bush
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International Planning Competition
• IPC-2004: 1st place, Optimal Planning
– Best on 5 of 7 domains
– 2nd best on remaining 2 domains
PROLEMA /
philosophers
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International Planning Competition
• IPC-2006: Tied for 1st place, Optimal Planning
– Other winner, MAXPLAN, is a variant of Satplan!
CPT2
MIPS-BDD SATPLAN
Maxplan
FDP
Propositional
Domains
(1st / 2nd Places)
0/1
1/1
3/2
0/3
Temporal
Domains
(1st / 2nd Places)
2/0
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What Changed?
• Small change in modeling
– Modest improvement from 2004 to 2006
• Significant change in SAT solvers!
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What Changed?
• In 2004, competition introduced the optimal
planning track
– Optimal planning is a very different beast from nonoptimal planning!
– In many domains, it is almost trivial to find poorquality 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)
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Why Care About Optimal Planning?
• Real users want (near)-optimal plans!
– Industrial applications: assembly planning, resource
planning, logistics planning…
– Difference between optimal and merely feasible
solutions can be worth millions of dollars
• Alternative: fast domain-specific approximation
algorithms that provide near-optimal solutions
– Approximation algorithms for job shop scheduling
– Blocks World Tamed: Ten Thousand Blocks in Under
a Second (Slaney & Thiébaux 1995)
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Domain-Independent Heuristic
Planning Considered Harmful
Solution
Quality?
Speed?
Optimal planning
algorithms
Best
Moderate
Domain-specific
algorithms
High
Fast
Domain-independent Poor
heuristic planning
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Hard to predict
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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)
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Objections
• If speed is crucial, you still must use
heuristic planners
– For highly constrained planning problems,
optimal planning is often faster than heuristic
planning!
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Constrainedness: Run Time
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Constrainedness: Percent Solved
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Further Extensions to Satplan
• Probabilistic planning
– Translation to stochastic satisfiability (Majercik
& Littman 1998)
– Translation to weighted model-counting
(Hoffmann 2006)
• Solved by modified DPLL solver, Cachet (Sang,
Beame, & Kautz 2005)
• Competitive with best probabilistic planners
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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
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Summary
• Satisfiability testing is a vital line of
research in AI planning
– Dramatic progress in SAT solvers
– Recognition of distinct and important nature of
optimal planning
• Not restricted to STRIPS any more!
– Numeric constraints
– Probabilistic planning
– Large domains
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