2000.9.20 CS 541 1

Compilation Approaches to AI Planning 1

José Luis Ambite*

Some slides are taken from presentations by Kautz and Selman. Please visit their websites:

http://www.cs.washington.edu/homes/kautz/ http://www.cs.cornell.edu/home/selman/

2000.9.20 CS 541 2

Complexity of Planning

• Domain-independent planning: PSPACE-complete or worse– (Chapman 1987; Bylander 1991; Backstrom 1993)

• Bounded-length planning: NP-complete– (Chenoweth 1991; Gupta and Nau 1992)

• Approximate planning: NP-complete or worse– (Selman 1994)

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Compilation Idea

• Use any computational substrate that is (at least) NP-hard.

• Planning as:– SAT: Propositional Satisfiability

• SATPLAN, Blackbox (Kautz&Selman, 1992, 1996, 1999)• OBDD: Ordered Binary Decision Diagrams (Cimatti et al, 98)

– CSP: Constraint Satisfaction • GP-CSP (Do & Kambhampati 2000)

– ILP: Integer Linear Programming• Kautz & Walser 1999, Vossen et al 2000

– …

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Planning as SAT

• Bounded-length planning can be formalized as propositional satisfiability (SAT)

• Plan = model (truth assignment) that satisfies logical constraints representing:

– Initial state– Goal state– Domain axioms: actions, frame axioms, …

for a fixed plan length• Logical spec such that any model is a valid plan

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Architecture of a SAT-based planner

Compiler(encoding)

satisfyingmodel

Plan

mappingIncrement plan length

If unsatisfiable

Problem Description• Init State• Goal State• Actions

CNF Simplifier(polynomial inference)

Solver(SAT engine/s)

Decoder

CNF

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Parameters of SAT-based planner

• Encoding of Planning Problem into SAT

• General Limited Inference: Simplification

• SAT Solver(s)

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Encodings of Planning to SAT

• Discrete Time– Each proposition and action have a time parameter:

– drive(truck1 a b) ~> drive(truck1 a b 3)

– at(p a) ~> at(p a 0)

• Common Axiom schemas:– INIT: Initial state completely specified at time 0

– GOAL: Goal state specified at time N

– A => P,E: Action implies preconditions and effects

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Encodings of Planning to SATCommon Schemas Example

• INIT: on(a b 0) ^ clear(a 0) ^ …

• GOAL: on(a c 2)

• A => P,EMove(x y z) pre: clear(x) ^ clear(z) ^ on(x y) eff: on(x z) ^ not clear(z) ^ not on(x y)

Move(a b c 1) => clear(a 0) ^ clear(b 0) ^ on(a b 0)Move(a b c 1) => on(a c 2) ^not clear(a 2) ^ not clear(b 2)

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Encodings of Planning to SATFrame Axioms

• Classical: (McCarthy & Hayes 1969)– state what fluents are left unchanged by an action– clear(d i-1) ^ move(a b c i) => clear(d i+1)– Problem: if no action occurs at step i nothing can be

inferred about propositions at level i+1 – Sol: at-least-one axiom: at least one action occurs

• Explanatory: (Haas 1987)– State the causes for a fluent change– clear(d i-1) ^ not clear(d i+1) => (move(a b d i) v move(a c d i) v … move(c Table d i))

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Encodings of Planning to SATOperator Splitting

• To reduce size of instantiated formula (#vars)• Normal = plan-length #actions objectsmax-act-arity

• Split = plan-length #actions objects max-act-arity– Replaces: drive(truck1 LA SF 5)

– With: (drive-arg1(truck1 5) ^ drive-arg2(LA 5) ^

drive-arg3(SF 5))

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KS96 Encodings: Linear (sequential)

• Same as KS92• Initial and Goal States• Action implies both preconditions and its effects• Only one action at a time• Some action occurs at each time

(allowing for do-nothing actions)

• Classical frame axioms• Operator Splitting

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KS96 Encodings: Graphplan-based

• Goal holds at last layer (time step)• Initial state holds at layer 1• Fact at level i implies disjuntion of all

operators at level i–1 that have it as an add-efffect

• Operators imply their preconditions• Conflicting Actions (only action mutex

explicit, fact mutex implicit)

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Graphplan Encoding

Fact => Act1 Act2

Act1 => Pre1 Pre2

¬Act1 ¬Act2

Act1

Act2

Fact

Pre1

Pre2

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KS96 Encodings: State-based

• Assert conditions for valid states • Combines graphplan and linear• Action implies both preconditions and its effects• Conflicting Actions (only action mutex explicit,

fact mutex implicit)• Explanatory frame axioms• Operator splitting• Eliminate actions ( state transition axioms)

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Algorithms for SAT

• Systematic (complete: prove sat and unsat)– Davis-Putnam (1960)– Satz (Li & Anbulagan 1997)– Rel-Sat (Bayardo & Schrag 1997)

• Stochastic (incomplete: cannot prove unsat)– GSAT (Selman et al 1992)– Walksat (Selman et al 1994)

• Randomized Restarts (Gomes et al 1998)

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Davis-Putnam algorithm

function Satisfiable ( clause set S ) return boolean

repeat /* unit propagation */

for each unit clause L in S do

delete from S every clause containing L /* unit subsumption */

delete not L from every clause of S in which it occurs /*unit resolution*/

if S is empty then return true

else if a clause becomes null in S then return false

until no further changes result

choose a literal L occurring in S /* splitting */

if Satisfiable ( S U L ) then return true

else if Satisfiable ( S U {not L}) then return true

else return false

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Walksat

For i=1 to max-tries

A:= random truth assigment

For j=1 to max-flips

If solution?(A) then return A else

C:= random unsatisfied clause

With probability p flip a random variable in C

With probability (1- p) flip the variable in C that

minimizes the number of unsatisfied clauses

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General Limited InferenceFormula Simplification

• Generated wff can be further simplified by consistency propagation techniques

• Compact (Crawford & Auton 1996)– unit propagation (unit clauses) O(n)– failed literal rule O(n2)

• if Wff + { P } unsat by unit propagation, then set p to false

– binary failed literal rule: O(n3)• if Wff + { P, Q } unsat by unit propagation, then add (not p V not q)

• Experimentally reduces number of variables and clauses by 30% (Kautz&Selman 1999)

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Randomized Sytematic Solvers

• Stochastic local search solvers (walksat)– when they work, scale well

– cannot show unsat

– fail on some domains

• Systematic solvers (Davis Putnam)– complete

– seem to scale badly

• Can we combine best features of each approach?

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Heavy Tails

• Bad scaling of systematic solvers can be caused by heavy tailed distributions

• Deterministic algorithms get stuck on particular instances– but that same instance might be easy for a

different deterministic algorithm!

• Expected (mean) solution time increases without limit over large distributions

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Heavy Tailed Cost Distribution

0.1

1

1 10 100 1000 10000 100000

log( Backtracks )

log

( 1

- F

(x)

)

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Randomized Restarts

• Solution: randomize the systematic solver– Add noise to the heuristic branching (variable

choice) function– Cutoff and restart search after a fixed number

of backtracks

• Eliminates heavy tails

• In practice: rapid restarts with low cutoff can dramatically improve performance

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Rapid Restart Speedup

1000

10000

100000

1000000

1 10 100 1000 10000 100000 1000000

log( cutoff )

log

( b

ackt

rack

s )

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Blackbox Results

0.01

0.1

1

10

100

1000

10000

rocket.a rocket.b log.a log.b log.c log.d

Graphplan

BB-walksat

BB-rand-sys

Handcoded-walksat

1016 states6,000 variables125,000 clauses

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AI Planning Systems CompetitionCMU, 1998

Team Number of Average Fastest Shortestproblems solution on solutionssolved time (msec) for

Blackbox 10 3171 3 6(AT&T Labs)HSP 9 25875 1 5(Venezuela)IPP 8 (11) 11036 1(3) 6(8)(Germany)STAN 7 20947 5 4(UK)