AAAI-2006 1 of 20

Deconstructing Planning as SatisfiabilityA Revised Report

Henry KautzUW University of Rochester

in collaboration with Bart Selman and Jöerg Hoffmann

AAAI-2006 2 of 20

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

AAAI-2006 3 of 20

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

AAAI-2006 4 of 20

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

AAAI-2006 5 of 20

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

= =

AAAI-2006 6 of 20

International Planning Competition

• IPC-1998: Satplan (blackbox) is competitive

AAAI-2006 7 of 20

International Planning Competition• IPC-2000: Satplan did poorly

Satplan

AAAI-2006 8 of 20

International Planning Competition

• IPC-2002: we stayed home.

Jeb Bush

AAAI-2006 9 of 20

International Planning Competition

• IPC-2004: 1st place, Optimal Planning– Best on 5 of 7 domains– 2nd best on remaining 2 domains

PROLEMA /

philosophers

AAAI-2006 10 of 20

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

AAAI-2006 11 of 20

What Changed?

• Small change in modeling– Modest improvement from 2004 to 2006

• Significant change in SAT solvers!

AAAI-2006 12 of 20

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)

AAAI-2006 13 of 20

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)

AAAI-2006 14 of 20

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

? ?

AAAI-2006 15 of 20

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)

AAAI-2006 16 of 20

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

AAAI-2006 17 of 20

2002 ICAPS Benchmarks

AAAI-2006 18 of 20

Large Numeric Domains

Directly encode binary arithmetic

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

a1

r11

+

-k

r21

r31

r41

r12

r22

r32

r42

AAAI-2006 19 of 20

Objections

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

optimal planners can be faster than feasible planners!

AAAI-2006 20 of 20

Constrainedness: Run Time

AAAI-2006 21 of 20

Constrainedness: Percent Solved

AAAI-2006 22 of 20

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

AAAI-2006 23 of 20

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)

AAAI-2006 24 of 20

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

AAAI-2006 25 of 20

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