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Deconstructing Planning as Satisfiability

Henry KautzUniversity 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

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:

1 2

1 2

action( ) pre( ) effect( 1)( ) ( ) if interfering

fact( ) fact( 1) ( )

i i iaction i action i

i i action i action

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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 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 non-

optimal planning!– In many domains, it is almost trivial to find poor-

quality 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 heuristic planning

Poor 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