Projection Global Projection Global Consistency:Consistency:Application in AI PlanningApplication in AI Planning

Pavel SurynekPavel Surynek

Charles University, PragueCharles University, PragueCzech RepublicCzech Republic

Outline of the Outline of the presentationpresentation

Problem: Problem: select a set of non-mutex select a set of non-mutex actions supporting a goalactions supporting a goal

Obstacle: Obstacle: NP-completeNP-complete (Partial) solution: global consistency - (Partial) solution: global consistency -

projection consistencyprojection consistency Application:Application: AI Planning AI Planning using using

planning graphsplanning graphs Experiments: Experiments: several planning several planning

domainsdomains

CSCLP 2006 Pavel Surynek

Problem - support Problem - support problemproblem

GoalGoal = finite = finite set of atomsset of atoms Action Action = triple (preconditions, = triple (preconditions,

positive effects, negative effects)positive effects, negative effects)

CSCLP 2006 Pavel Surynek

Goal:

A1

A2

A3

A4

A5

A6

A7

A8

A9

A10 A11

A12

x

atom 1 atom 2 atom 3 atom 4

supports foratom 1

supports foratom 2

supports foratom 3

supports foratom 4

xx

x x

x

x

x

A1

A2

A3

A4

A6

A9

A8

A10 A11

A12

A5

A7 solution - mutex free

Mutex actionsMutex actions

CSCLP 2006 Pavel Surynek

3 1

2

1

(two) actionsload(small_truck,box1)load(big_truck,box2)are independent

(two) actionsload(small_truck,box1)load(big_truck,box1)are dependent

example of actionexample of action– load(small_truck, box1) load(small_truck, box1) = ({empty(truck), = ({empty(truck),

on(bottom, box1)}; {loaded(box1, truck)}; on(bottom, box1)}; {loaded(box1, truck)}; {{¬¬empty(truck), empty(truck), ¬¬on(bottom, box1)})on(bottom, box1)})

generalized dependency = generalized dependency = mutexmutex

Obstacle - NP Obstacle - NP completenesscompleteness

Instance:Instance:– goal goal augmented with augmented with finite sets of finite sets of

supporting actionssupporting actions– finite set of finite set of mutexes between actionsmutexes between actions

Answer:Answer:– set of set of non-mutex actions supporting the non-mutex actions supporting the

goalgoal NP-completeNP-complete

– SAT instanceSAT instance in in CNF CNF ►►► support ►►► support problemproblem

CSCLP 2006 Pavel Surynek

Projection consistencyProjection consistency

Interpret support problem as aInterpret support problem as a graph graph Greedily find (vertex disjoint)Greedily find (vertex disjoint) cliques cliques ►►►►►►

►►►►►► clique decompositionclique decomposition At most one action At most one action from each cliquefrom each clique can can

be selectedbe selected

CSCLP 2006 Pavel Surynek

a real support problem

Trucks, Cranes, Locations

Counting argumentCounting argument

CliqueClique decompositiondecomposition C C11, C, C22,..., C,..., Ckk

ContributionContribution of an action of an action a ac(a) = c(a) = number of supported atomsnumber of supported atoms

ContributionContribution of a clique of a clique CCc(C) = maxc(C) = maxaaC C c(a)c(a)

Counting argument Counting argument (simplest form)(simplest form)ifif ∑ ∑i=1...k i=1...k c(Cc(Cii) < size of the goal ►►►) < size of the goal ►►►►►► the goal is ►►► the goal is unsatisfiableunsatisfiable

Generalized form = projection consistencyGeneralized form = projection consistency– w.r.t. sub-goals w.r.t. sub-goals andand singleton approach singleton approach

CSCLP 2006 Pavel Surynek

Application: AI PlanningApplication: AI Planning

Planning problemPlanning problem– Initial state: Initial state: set of atomsset of atoms– Set of allowed actionsSet of allowed actions– Goal: Goal: set of atoms (literals)set of atoms (literals)

TaskTask– determine a determine a sequencesequence

of actions of actions transformingtransforminginitial stateinitial state to the to the goalgoal

Solution: Solution: planning graphsplanning graphs and and GraphPlanGraphPlan algorithm (Blum & Furst, 1997) - support algorithm (Blum & Furst, 1997) - support problem problem arise as a frequent sub-problemarise as a frequent sub-problem

CSCLP 2006 Pavel Surynek

location B location Clocation A

5 6

location D

location E

location F

31

42

location B location Clocation A

5

location D

location E

location F

31

2

64

Experiments:Experiments: towers of towers of HanoiHanoi

CSCLP 2006 Pavel Surynek

Original puzzle (3 pegs, 4 discs, and 1 hand)

Our generalization (more pegs, discs, and hands)

Experiments:Experiments: DWR DWR

CSCLP 2006 Pavel Surynek

5 631

42

987

Locations with several places for stacks of boxes and with several cranes

Each crane can reach some stacks within location (not all) Trucks of various capacities (small - 1 box, big - 2 boxes)

Experiments:Experiments: Refueling Refueling planesplanes

CSCLP 2006 Pavel Surynek

distance X

distance Y

distance Z

Several planes dislocated at several airports Transport a fleet of planes at destination airport Airport - unlimited source of fuel, planes can refuel in-flight

Experiments:Experiments: Results Results

CSCLP 2006 Pavel Surynek

Total Problem Solv ing Time

0.10

1.00

10.00

100.00

1000.00

10000.00

han0

1

dwr0

3

dwr0

4

han0

2

pln04

dwr0

2

dwr0

1

han0

4

pln01

han0

3

pln10

han0

7

pln05

dwr0

5

pln06

pln11

dwr0

7

han0

8

dwr1

6

pln13

dwr1

7

Tim

e (

se

co

nd

s)

Standard

Arc-consistency

Projection-consistency

Significant improvements on problems with high action parallelism (Dock Worker Robots, Refueling Planes, Hanoi Towers with more hands)

ConclusionsConclusions

Improvement of the GraphPlan algorithm– Better method for finding mutex free

set of actions supporting a goal– We use projection consistency

Experimental evaluation– Projection consistency especially

successful on problems with high action parallelism

CSCLP 2006 Pavel Surynek