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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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Translating STRIPS• Ground action = a STRIPS operator with

constants assigned to all of its parameters• Ground fluent = a precondition or effect of a

ground actionoperator: Fly(a,b)

precondition: At(a), Fueledeffect: At(b), ~At(a), ~Fueled

constants: NY, Boston, SeattleGround actions: Fly(NY,Boston), Fly(NY,Seattle),

Fly(Boston,NY), Fly(Boston,Seattle), Fly(Seattle,NY), Fly(Seattle,Boston)

Ground fluents: Fueled, At(NY), At(Boston), At(Seattle)

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Clause Schemas

• A large set of clauses can be represented by a schema

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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

1 2

0

negates a precondition

action( ) pre( ) effect( 1)

( ) ( ) if interfering

fact( ) fact( 1) ( )

initial_state ,

o

goal_stat

f

frame

e

axioms

n

i i i

action i action i

i i action i acti

action action

on

⊃ ∧ +¬ ∨¬

¬ ∧ + ⊃ ∨ ∨L

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Example• If an action occurs at time i, then its preconditions must

hold at time i• If an action occurs at time i, then its effects must hold at

time i+1

(fly(a,b,i) at(a,i

for (1 i<K)

for (a {NY,B

))

(fly(a,b,i)

oston,Seattle})

for (b {NY,Boston,Seattl

fuel(i))

(fly(a,b,i) at(b,i+1))

(fly(a,b,i) f

e

u

} & a b

l

)

e (

⊃ ∧⊃ ∧⊃ ∧⊃¬

≤∈

∈ ≠

i+1))

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SAT Encoding

• If a fluent changes its truth value from time i to time i+1, one of the actions with the new value as an effect must have occurred at time i

( at(b,i) at(b,i+1))

for (1 i<K)

for (b {NY,Bo

exists (a {NY,Boston,Sea

st

tt

on,Seattle}

le} & a b)

)

fly(a,b,i)

)

¬ ∧∈

≤∈

≠⊃

Like “for”, but connects propositions

with OR

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Plan Graph Based Instantiation

initial state: p

action a:precondition: p

effect: p

action b:precondition: p

effect: p q

a0 a1

p0 p1 p2

b1

m0 m1

q2

= =

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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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The IPC-4 Domains

• Airport: control the ground traffic [Hoffmann & Trüg] • Pipesworld: control oil product flow in a pipeline network [Liporace &

Hoffmann] • Promela: find deadlocks in communication protocols [Edelkamp]• PSR: resupply lines in a faulty electricity network [Thiebaux &

Hoffmann]• Satellite & Settlers [Fox & Long], additional Satellite versions with

time windows for sending data [Hoffmann]• UMTS: set up applications for mobile terminals [Edelkamp &

Englert]

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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 (near)-optimal and merely

feasible solutions can be worth millions of dollars

• Alternative: fast domain-specific optimizing algorithms – 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 Feasible Planning Considered Harmful

Solution Quality?

Speed?

General optimizing planning algorithms

Best Moderate

Domain-specific optimizing planning algorithms

High Fast

Domain-independent feasible planning

? ?

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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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Projecting Variable Domains

initial state: r=5

action a:precondition: r>0

effect: r := r-1

• Resource use represented as conditional effects

a1

r=5 r=5 r=5

r=4 r=4

a0

r=4

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2002 ICAPS Benchmarks

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Large Numeric Domains

Directly encode binary arithmetic

action: aprecondition: r keffect: r := r-k

a1

r11

+

-k

r21

r31

r41

r12

r22

r32

r42

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Objections

• If speed is crucial, you still must use feasible planners– For highly constrained planning problems,

optimal planners can be faster than feasible planners!

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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)– Alternative untested idea:

• Encode action “failure” as conditional resource consumption

• Can find solutions with specified probability of failure-free execution

• (Much) less general than full probabilistic planning (no fortuitous accidents), but useful in practice

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Encoding Bounded Failure Free Probabilistic Planning

plan failure free probability 0.90

action: afailure probability: 0.01

preconditions: p

effects: q

action: aprecondition: p

s log(0.89)

effect: q s := s + log(0.99)

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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

optimizing planning versus feasible planning

• SATPLAN not restricted to STRIPS any more!– Numeric constraints– Probabilistic planning– Large domains