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Coping with Time & Continuous Quantities
David E. Smith
Collaborators:Jeremy Frank, Ari Jónsson
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The Ambitious Spacecraft
Observation-1priority
time windowtargetinstrumentsduration
Observation-2
Observation-3
Observation-4
…Objective:
maximize science return
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Complications
Observation-1priority
time windowtargetinstrumentsduration
Observation-2
Observation-3
Observation-4
…
calibrationtarget1target2…
consumables:
fuel
power
data storage
cryogen
angle between targets
turn duration
Objective:
maximize science return
lin
ked
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The Spacecraft’s Choices
Observations
Instrument
Calibration target
Ordering
Communication
Instrument Status
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The Spacecraft’s Choices
Observations
Instrument
Calibration target
Ordering
Communication
Instrument statuscontinuous time
time constraints
concurrent actions
metric resources
optimization
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The Choice Spectrum
planningjob-shopscheduling
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The Choice Spectrum
planningjob-shopscheduling
Job1task1 < task2 < task3 < …
Job2
Job3…
R3 R7 R1
Ordering choices only
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The Choice Spectrum
cascading levels of choice
planningjob-shopscheduling
…
…… … …
… …
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The Choice Spectrum
job-shopscheduling
planningresourcechoices(RCSP)
umfagogginclavitracle
fernambulator5 11
Task1
Task2
Task3
Task4
Task5
Task6
Task8
Task7
[8,17]
Ordering choicesResource choices
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The Choice Spectrum
job-shopscheduling
planning
resourcechoices(RCSP)
alternativeprocesses
process3 process7process8
Task1
Task2
Task3
Task4
Task5
Task6
Task8
Task7
Ordering choicesResource choicesProcess choices
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The Choice Spectrum
planningambitious spacecraftjob-shop
scheduling
resourcechoices(RCSP)
alternativeprocesses
Observation choices
Instrument choices
Calibration target choices
Ordering choices
Communication choices
Instrument status choices
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The Choice Spectrum
planning
Subset Selectionambitious spacecraftobservation schedulingprocess planning
job-shopscheduling
resourcechoices(RCSP)
alternativeprocesses
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Strips Assumptions
Instantaneous actions
No time constraints
No concurrent actions
No continuous quantities
Bleak?
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Some STRIPS Operators
TakeImage (?target, ?instr):Pre: Status(?instr, Calibrated), Pointing(?target)Eff: Image(?target)
Calibrate (?instrument):Pre: Status(?instr, On), Calibration-Target(?target), Pointing(?target)Eff: ¬Status(?inst, On), Status(?instr, Calibrated)
Turn (?target):Pre: Pointing(?direction), ?direction ≠ ?targetEff: ¬Pointing(?direction), Pointing(?target)
no time, no resources
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Outline
Resources
Reusable
Consumable
Time
Uncertainty?
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Resources
Discrete Continuoussingle/multiple (metric)
Reusable instruments bandwidth
Consumable solid rockets fuel
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Reusable Resources(single-capacity)
TakeImage (?target, ?instr):Res: ?instrPre: Status(?instr, Calibrated), Pointing(?target)Eff: Image(?target)
Prevent concurrent actions
Mutual exclusion between actions:Graphplan trivial
Satplan ¬A1 ¬A2
POCL A1<A2 A2<A1
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Reusable Resources(multiple-capacity)
TakeImage (?target, ?instr):Res: ?instrPre: Status(?instr, Calibrated), Pointing(?target)Eff: Image(?target)
Suppose: 2 camerasSimple approach: name the cameras
Mutual exclusion between actions:Graphplan trivial
Satplan ¬A1 ¬A2
POCL ?instr1≠?instr2 A1<A2 A2<A1
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Reusable Resources(multiple-capacity)
TakeImage (?target):Res: Camera(1)Pre: Pointing(?target)Eff: Image(?target)
2 Cameras:Not named
Explicit checking of concurrent actions to ensure:
usage ≤ capacity
Capacity(Camera)=2
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Reusable Resources(multiple-capacity)
usage > capacity
Mutual exclusion between sets of actions:Graphplan n-ary mutex
Satplan ¬(A1 A2 A3 …)
POCL A1<A2 A2<A1 A1<A3 A3<A1 …
Graph TechniquesIxTeT
Profiling heuristicsSadeh, Fox, BeckO-Plan
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Summary
ResourcesReusable
Single-capacity binary mutex easy
Multiple-capacity n-ary mutex tricky
specialized algorithms
search heuristics
Consumable
Time
Uncertainty?
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Consumable Resources
Turn (?target):Pre: Pointing(?direction), ?direction ≠ ?target
Fuel ≥ Angle(?direction,?target) *ConsumptionRateEff: ¬Pointing(?direction), Pointing(?target)
Fuel –= Angle(?direction,?target) *ConsumptionRate
Simple approach:wait until all variables are instantiated
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LP Approaches
Zeno:
open conditions: P1, P2, P3, P4, P5, E1, E2, E3
constraints: Fuel ≤ 10
Simplex
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LP Approaches: LPSAT
Turn (?target):Pre: Pointing(?direction), ?direction ≠ ?target
Fuel ≥ Angle(?direction,?target) *ConsumptionRateEff: ¬Pointing(?direction), Pointing(?target)
Fuel –= Angle(?direction,?target) *ConsumptionRate
¬Turn-T3-T71 v
{Pointing-T31 C11
¬Pointing-T32 Pointing-T72 C22 }
C11 Fuel1 ≥ Angle(T3,T7) *ConsumptionRate
C22 Fuel2 = Fuel1 – Angle(T3,T7) *ConsumptionRate
Simplex
SAT engine
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ILP planning
¬Turn-T3-T71 v
{Pointing-T31 C11
¬Pointing-T32 Pointing-T72 C22 }
C11 Fuel1 ≥ Angle(T3,T7) *ConsumptionRate
C22 Fuel2 = Fuel1 – Angle(T3,T7) *ConsumptionRate
¬Turn-T3-T71 v Pointing-T31
¬Turn-T3-T71 v C11
¬Turn-T3-T71 v ¬ Pointing-T32
¬Turn-T3-T71 v Pointing-T72
¬Turn-T3-T71 v C22
C11 Fuel1 ≥ Angle(T3,T7) *ConsumptionRate
C22 Fuel2 = Fuel1 – Angle(T3,T7) *ConsumptionRate
Kautz & WalserVossen et al.
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ILP planning
(1 - Turn-T3-T71) + Pointing-T31 ≥ 1
M(1 - Turn-T3-T71) + (1 + D11 ) ≥ 1
(1 - Turn-T3-T71) + (1 - Pointing-T32 ) ≥ 1
(1 - Turn-T3-T71) + Pointing-T72 ≥ 1
M(1 - Turn-T3-T71) + (1 ± D21 ) ≥ 1
D11 = Fuel1 – Angle(T3,T7) *ConsumptionRate
D22 = Fuel2 – (Fuel1 – Angle(T3,T7) *ConsumptionRate)
¬Turn-T3-T71 v Pointing-T31
¬Turn-T3-T71 v C11
¬Turn-T3-T71 v ¬ Pointing-T32
¬Turn-T3-T71 v Pointing-T72
¬Turn-T3-T71 v C22
C11 Fuel1 ≥ Angle(T3,T7) *ConsumptionRate
C22 Fuel2 = Fuel1 – Angle(T3,T7) *ConsumptionRate
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Summary
ResourcesReusable
Consumable
Equality and Inequality conditions
POCL + LP slow
SAT + LP fast
ILP medium
Graphplan ?
Concurrent continuous change ugly!
Time
Uncertainty?
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Concurrent Continuous Change
Turn (?target):Pre: Pointing(?direction), ?direction ≠ ?target
Fuel ≥ Angle(?direction,?target) *ConsumptionRateEff: ¬Pointing(?direction), Pointing(?target)
Fuel –= Angle(?direction,?target) *ConsumptionRate
Fuel = F(t) Fu
el
Time
Fu
el
Time
Fu
el
Time
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Outline
Resources
Time
Uncertainty?
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World Description
State-centric:for each time describe propositions that are true
Proposition-centric:for each proposion describe times it is true
Pointing(A7)
Status(Cam2, Calibrated)
Turn(A7)
Pointing(Earth)Status(Cam2, Calibrated)
¬ Image(A7)
Turn(A7) Pointing(A7)Status(Cam2, Calibrated)
¬ Image(A7)
Pointing(Earth)
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Allen’s Ordering Relationships
A BA before B
A BA meets B
A
BA overlaps B
A contains BA
B
A = BA
B
A
BA starts B
A
BA ends B
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TakeImage Schema
TakeImage(?target, ?instr)A
P {Status(?instr, Calibrated)P Contains(P, A)}
Q {Pointing(?target)Q Contains(Q, A)}
R {Image(?target)R Meets(A, R)}
TakeImage (?target, ?instr):Pre: Status(?instr, Calibrated), Pointing(?target)Eff: Image(?target)
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Shorthand
Pointing(?target)
Status(?instr, Calibrated)
TakeImage(?target, ?instr) Image(?target)meets
contains
contains
TakeImage (?target, ?instr)contained-by Status(?instr, Calibrated)contained-by Pointing(?target)meets Image(?target)
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Turn
Turn (?target)met-by Pointing(?direction)meets Pointing(?target)
Pointing(?target)Pointing(?direction) Turn(?target)meetsmeets
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Calibrate
Status(?instr, Calibrated)
Pointing(?target)
CalibrationTarget(?target)
Calibrate(?instr)meetsmeets
contains
contains
Status(?instr, On)
Calibrate (?instr)met-by Status(?instr, On)contained-by CalibrationTarget(?target)contained-by Pointing(?target)meets Status(?instr, Calibrated)
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Algorithm
Choose:
introduce an action & instantiate constraints
coalesce propositions
Propagate constraints
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Initial Plan
Past Image(?target)meets
Pointing(Earth)
Status(Cam1, Off)
Status(Cam2, On)
CalibrationTarget(T17)
Futuremeets
-∞ ∞
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Expansion 1
Image(A7) Futuremeets
Past meets
Pointing(Earth)
Status(Cam1, Off)
Status(Cam2, On)
CalibrationTarget(T17)
∞
Pointing(A7)
Status(?instr, Calibrated)
TakeImage(A7, ?instr)meets
contains
contains
before
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Expansion 2
Pointing(Earth)
Status(Cam1, Off)
Status(Cam2, On)
CalibrationTarget(T17)
Image(A7)
Pointing(A7)
Status(?instr, Calibrated)
TakeImage(A7, ?instr)meets
contains
contains
Pointing(?direction) Turn(A7)
Pointing(?caltarget)
CalibrationTarget(?caltarget)
Calibrate(?instr)
meetsmeets
meetsmeets
contains
contains
Status(?instr, On)
before
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Coalescing
Pointing(Earth)
Status(Cam1, Off)
Status(Cam2, On)
CalibrationTarget(T17)
Image(A7)
Pointing(A7)
Status(Cam2, Calibrated)
TakeImage(A7, Cam2)meets
contains
contains
Pointing(?direction) Turn(A7)
Pointing(T17)
Calibrate(Cam2)
meetsmeets
meetsmeets
contains
contains
before before
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Coalescing
Pointing(Earth)
Status(Cam1, Off)
Status(Cam2, On)
CalibrationTarget(T17)
Image(A7)
Pointing(A7)
Status(Cam2, Calibrated)
TakeImage(A7, Cam2)meets
contains
contains
Turn(A7)
Pointing(T17)
Calibrate(Cam2)
meets
meets
meetsmeets
contains
contains
before
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Expansion 3
Pointing(Earth)
Status(Cam1, Off)
Status(Cam2, On)
CalibrationTarget(T17)
Image(A7)
Pointing(A7)
Status(Cam2, Calibrated)
TakeImage(A7, Cam2)meets
contains
contains
Turn(A7)
Pointing(T17)
Calibrate(Cam2)
meets
meets
meetsmeets
contains
contains
Turn(T17)meets
meets
Pointing(?direction)
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Coalescing
Pointing(Earth)
Status(Cam1, Off)
Status(Cam2, On)
CalibrationTarget(T17)
Image(A7)
Pointing(A7)
Status(Cam2, Calibrated)
TakeImage(A7, Cam2)meets
contains
contains
Turn(A7)
Pointing(T17)
Calibrate(Cam2)
meets
meets
meetsmeets
contains
contains
Turn(T17)meets
meets
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POCL vs CBI
POCL CBI
search goal-directed goal-directedstrategy
temporal before/after complexreasoning
protection causal links meets
conflicts threats mutex
conflict proactive lazyresolution
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Causal Links & Threats
propositionaction
meets meetsactionaction action
proposition
action actionproposition
action
threatens
proposition
action
action
proposition
mutex
POCL CBI
Causal links:
Threats:
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CBI Planners
Zeno (Penberthy) intervals, no CSP
Trains (Allen)
Descartes (Joslin) extreme least commitment
IxTeT (Ghallab) functional rep.
HSTS (Muscettola) functional rep., activities
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Timelines
Pointing=Turn (?target)met-by Pointing=?directionmeets Pointing=?target
Earth A7Turn(A7)
T17
?direction
?
Pointing:
Turn (?target)met-by Pointing(?direction)meets Pointing(?target)
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Explanatory Frame Axioms
Pointing(?target)met-by Pastormet-by Turn (?target)
Turn (?target)met-by Pointing(?direction)meets Pointing(?target)
Pointing(?target)Pointing(?direction) Turn(?target)meetsmeets
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A Stupid Question
Why should CBI perform any better than POCL?
• less commitment
• better propagation
• timelines
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A Stupid Question
Why should CBI perform any better than POCL?
• less commitment
• better propagation
• timelines
• reachability analysis
• heuristic estimators
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Reachability Analysis
Graphplan:optimistic projection of achievability
…
…
…
level 0 level 1 level 2 level 3
level 0 level 1 level 2
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Reachability Analysis
Pointing(Earth) Pointing(A7)
Calibrate(Cam1,T5) Status(Cam1, Calibrated)meets
contains
Turn(A7)meets
[0, ∞] [0, ∞] [5, ∞]meets
[5, ∞]
Pointing(T5)Turn(T5)
[3, ∞]meets
[3, ∞]meets [0, ∞]
[3, ∞] [8, ∞]
[0, 13]
mutex rules more complex (Smith & Weld - IJCAI99)
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Benefits?
1. Pruning during plan construction
2. Basis for heuristic distance measuresMcDermott (UNPOP)
Geffner (HSP-R)
RAO
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Extracting Heuristic Estimates
Pointing(Earth) Pointing(A7)
Calibrate(Cam1,T5) Status(Cam1, Calibrated)meets
contains
Turn(A7)meets
[0, ∞] [0, ∞] [5, ∞]meets
[5, ∞]
Pointing(T5)Turn(T5)
[3, ∞]meets
[3, ∞]meets [0, ∞]
[3, ∞] [8, ∞]
[0, 13]
Distance metricstime requiredfuel required…
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Summary
Multi-capacity Metric Optimization Continuous Speed Resources Quantities Time
Graphplan hard? hard hard? hard good IPP TGP
SAT moderate? moderate moderate? very hard? goodplanning LPSAT
ILP easy? easy easy very hard? fair?planning ILP-plan ILP-plan
CBI moderate moderate moderate? easy fair?planning IxTeT Zeno HSTS
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Critical Issues
Multi-capacity Metric Optimization Continuous Speed Resources Quantities Time
Graphplan hard? hard hard? hard good IPP TGP
SAT moderate? moderate moderate? very hard? goodplanning LPSAT
ILP easy? easy easy very hard? fair?planning ILP-plan ILP-plan
CBI moderate moderate moderate? easy fair?planning IxTeT Zeno HSTS