IJCAI Introduction

1
Temporal Plan Execution Dynamic Scheduling and
Simple Temporal Networks
Brian C. Williams 16.412J/6.834J December 2nd,
2002
1
2
Goals and Environment Constraints
Projective Task Expansion
Temporal NetworkSolver
Temporal Planner
Temporal Plan
Task Dispatch
Dynamic Scheduling and Task Dispatch
Goals
Modes
Reactive Task Expansion
Commands
Observations
3
Goals and Environment Constraints
Projective Task Expansion
Temporal NetworkSolver
Temporal Planner
Temporal Plan
Task Dispatch
Dynamic Scheduling and Task Dispatch
Goals
Modes
Reactive Task Expansion
Commands
Observations
4
Outline

• Temporal Representation
• Qualitative temporal relations
• Metric constraints
• Temporal Constraint Networks
• Temporal Plans
• Temporal Reasoning forPlanning and Scheduling
• Flexible Execution throughDynamic Scheduling

5
Qualitative Temporal Constraints(Allen 83)

• x before y
• x meets y
• x overlaps y
• x during y
• x starts y
• x finishes y
• x equals y

• y after x
• y met-by x
• y overlapped-by x
• y contains x
• y started-by x
• y finished-by x
• y equals x

X
Y
X
Y
X
Y
Y
X
Y
X
Y
X
Y
X
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Example Deep Space One Remote Agent Experiment
Timer
Max_Thrust
Idle
Idle
SEP_Segment
Accum
SEP Action
Attitude
Poke
7
Qualitative Temporal ConstraintsMaybe Expressed
as Inequalities (Vilain, Kautz 86)

• x before y X lt Y-
• x meets y X Y-
• x overlaps y (Y- lt X) (X- lt Y)
• x during y (Y- lt X-) (X lt Y)
• x starts y (X- Y-) (X lt Y)
• x finishes y (X- lt Y-) (X Y)
• x equals y (X- Y-) (X Y)

Inequalities may be expressed as binary interval
relations X - Y- lt -inf, 0
8
Metric Constraints

• Going to the store takes at least 10 minutes and
  at most 30 minutes.
• 10 lt T(store) T-(store) lt 30
• Bread should be eaten within a day of baking.
• 0 lt T(baking) T-(eating) lt 1 day
• Inequalities, X lt Y- , may be expressed as
  binary interval relations
• - inf lt X - Y- lt 0

9
Metric Time Quantitative Temporal Constraint
Networks(Dechter, Meiri, Pearl 91)

• A set of time points Xi at which events occur.
• Unary constraints (a0 lt Xi lt b0 ) or (a1 lt Xi lt
  b1 ) or . . .
• Binary constraints (a0 lt Xj - Xi lt b0 ) or (a1
  lt Xj - Xi lt b1 ) or . . .

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Temporal Constraint Satisfaction Problem (TCSP)

• lt Xi, Ti , Tij gt
• Xi continuous variables
• I1, . . . ,In interval constraints
• where Ii ai,bi interval
• Ti (ai Xi bi) or . . . or (ai Xi bi)
• Tij (a1 Xi - Xj b1) or ... or (an Xi - Xj
  bn)

Dechter, Meiri, Pearl, aij89
11
TCSP Are Visualized UsingDirected Constraint
Graphs
12
Simple Temporal Networks(Dechter, Meiri, Pearl
91)

• Simple Temporal Networks
• A set of time points Xi at which events occur.
• Unary constraints (a0 lt Xi lt b0 ) or (a1 lt Xi lt
  b1 ) or . . .
• Binary constraints (a0 lt Xj - Xi lt b0 ) or (a1
  lt Xj - Xi lt b1 ) or . . .

• Sufficient to represent
• most Allen relations
• simple metric constraints

• Cant represent
• Disjoint tokens

13
Simple Temporal Network

• Tij (aij Xi - Xj bij)

14
A Completed Plan Forms an STN
Thrust Goals
Power
Attitude
Thrust (b, 200)
Engine
Off
Warm Up
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A Completed Plan Forms an STN
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16
Outline

• Temporal Representation
• Temporal Reasoning forPlanning and Scheduling
• TCSP Queries
• Induced Constraints
• Plan Consistency
• Static Scheduling
• Flexible Execution throughDynamic Scheduling

17
TCSP Queries(Dechter, Meiri, Pearl, AIJ91)

• Is the TCSP consistent?
• What are the feasible times for each Xi?
• What are the feasible durations between each Xi
  and Xj?
• What is a consistent set of times?
• What are the earliest possible times?
• What are the latest possible times?

18
Goals
Histories(?)
Projective Task Expansion
Scheduler
Flexible Sequence (Plans)
Task Dispatch
Plan Runner
Goals
Modes
Reactive Task Expansion
Commands
Observations
19
TCSP Queries(Dechter, Meiri, Pearl, AIJ91)

• Is the TCSP consistent? Planning
• What are the feasible times for each Xi?
• What are the feasible durations between each Xi
  and Xj?
• What is a consistent set of times? Scheduling
• What are the earliest possible times? Scheduling
• What are the latest possible times?

20
To Query an STN Map to aDistance Graph Gd lt
V,Ed gt
Edge encodes an upper bound on distance to target
from source.
Xj - Xi bij Xi - Xj - aij
Tij (aij Xj - Xi bij)
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Gd Induces Constraints

• Path constraint i0 i, i1 . . ., ik j
• Conjoined path constraints result in the
  shortest path as bound
• where dij is the shortest path from i to j

22
Shortest Paths of Gd
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STN Minimum Network
d-graph
STN minimum network
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Conjoined Paths are Computed using All Pairs
Shortest Path(e.g., Floyd-Warshalls algorithm )

• 1. for i 1 to n do dii 0
• 2. for i, j 1 to n do dij aij
• 3. for k 1 to n do
• 4. for i, j 1 to n do
• 5. dij mindij, dik dkj

k
i
j
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Testing Plan Consistency
No negative cycles -5 gt TA TA 0
d-graph
26
Latest Solution
Node 0 is the reference.
20
40
0
1
2
-10
-30
-10
20
50
4
3
-40
-60
70
d-graph
27
Solution Latest Times
S1 (d01, . . . , d0n)
20
40
0
3
1
-10
-30
-10
20
50
4
2
-40
-60
70
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Earliest Solution
Node 0 is the reference.
20
40
0
1
2
-10
-30
-10
20
50
4
3
-40
-60
70
d-graph
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Solution Earliest Times
S1 (-d10, . . . , -dn0)
20
40
0
1
3
-10
-30
-10
20
50
4
2
-40
-60
70
30
SchedulingFeasible Values
Latest Times

• X1 in 10, 20
• X2 in 40, 50
• X3 in 20, 30
• X4 in 60, 70

d-graph
Earliest Times
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Scheduling without Search Solution by
Decomposition

• Select value for 1
• 15 10,20

d-graph
32
Scheduling without Search Solution by
Decomposition

• Select value for 1
• 15 10,20

d-graph
33
Solution by Decomposition

• Select value for 1
• 15

• Select value for 2, consistent with 1
• 45 40,50, 1530,40

d-graph
34
Solution by Decomposition

• Select value for 1
• 15

• Select value for 2, consistent with 1
• 45 45,50

d-graph
35
Solution by Decomposition

• Select value for 1
• 15

• Select value for 2, consistent with 1
• 45 45,50

d-graph
36
Solution by Decomposition

• Select value for 1
• 15

• Select value for 2, consistent with 1
• 45

• Select value for 3, consistent with 1 2
• 30 20,30, 1510,20,45-20,-10

d-graph
37
Solution by Decomposition

• Select value for 1
• 15

• Select value for 2, consistent with 1
• 45

• Select value for 3, consistent with 1 2
• 30 25,30

d-graph
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Solution by Decomposition

• Select value for 1
• 15

• Select value for 2, consistent with 1
• 45

• Select value for 3, consistent with 1 2
• 30 25,30

d-graph
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Solution by Decomposition

• Select value for 1
• 15

• Select value for 2, consistent with 1
• 45

• Select value for 3, consistent with 1 2
• 30

d-graph

• Select value for 4, consistent with 1,2 3

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In General Solving TCSPs is NP Hard

• Forward(I1, . . ., Ii)
• 1. if i m then
• 2. M M union Solve-STP(I1, . . ., Im), and
• 3. Go-Back(I1, . . ., Im)
• 4. Ci1 empty
• 5. for every Ij in Di1 do
• 6. if Consistent-STP (I1, . . . , Ii , Ij), then
• 7. for i, j 1 to n do
• 8. dij mindij, dik dkj

41
Outline

• Temporal Representation
• Temporal Reasoning forPlanning and Scheduling
• Flexible Execution throughDynamic Scheduling

42
Goals and Environment Constraints
Projective Task Expansion
Temporal NetworkSolver
Temporal Planner
Temporal Plan
Task Dispatch
Dynamic Scheduling and Task Dispatch
Goals
Modes
Reactive Task Expansion
Commands
Observations
43
Executing Flexible Temporal Plans Muscettola,
Morris, Pell et al.

• Handling delays and fluctuations in task
  duration
• Least commitment temporal plans leave room to
  adapt

• flexible execution adapts through dynamic
  scheduling
• Assigns time to event when executed.

44
Issues in Flexible Execution

• How do we minimize execution latency?
• How do we schedule at execution time?

45
Time Propagation Can Be Costly
EXECUTIVE
CONTROLLED SYSTEM
46
Time Propagation Can Be Costly
EXECUTIVE
CONTROLLED SYSTEM
47
Time Propagation Can Be Costly
EXECUTIVE
CONTROLLED SYSTEM
48
Time Propagation Can Be Costly
EXECUTIVE
CONTROLLED SYSTEM
49
Time Propagation Can Be Costly
EXECUTIVE
CONTROLLED SYSTEM
50
Time Propagation Can Be Costly
EXECUTIVE
CONTROLLED SYSTEM
51
Issues in Flexible Execution

• How do we minimize execution latency?
• Propagate through a small set of neighboring
  constraints.
• How do we schedule at execution time?

52
Compile to Efficient Network
EXECUTIVE
CONTROLLED SYSTEM
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Compile to Efficient Network
EXECUTIVE
CONTROLLED SYSTEM
54
Compile to Efficient Network
EXECUTIVE
CONTROLLED SYSTEM
55
Issues in Flexible Execution

• How do we minimize execution latency?
• Propagate through a small set of neighboring
  constraints.
• How do we schedule at execution time?

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Issues in Flexible Execution

• How do we minimize execution latency?
• Propagate through a small set of neighboring
  constraints.
• How do we schedule at execution time?
• Through decomposition?

57
Dynamic Scheduling by Decomposition
Simple Example
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Dynamic Scheduling by Decomposition

• Compute APSP graph
• Decomposition enables assignment without search

Equivalent Distance Graph Representation
59
Assignment by Decomposition
11
lt0,10gt
10
-1
0
1
lt2,11gt
t 0
1
-1
0
-2
2
lt0,10gt
10
-2
60
Assignment by Decomposition
11
t 3
10
-1
0
1
lt0,0gt
lt2,11gt
1
-1
0
-2
2
lt0,10gt
10
-2
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Assignment by Decomposition
But C now has to be executed at t 2, which is
already in the past!

• Solution
• Assignments must monotonically increase in
  value.
• Execute first, all APSP neighbors with negative
  delays.

11
t 3
10
-1
0
1
lt0,0gt
lt4,4gt
1
-1
0
-2
2
lt2,2gt
10
-2
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Dispatching Execution Controller

• Execute an event when enabled and active
• Enabled - APSP Predecessors are completed
• Predecessor a destination of a negative edge
  that starts at event.
• Active - Current time within bound of task.

63
Dispatching Execution Controller

• Initially
• E Time points w/o predecessors
• S
• Repeat
• Wait until current_time has advanced st
• Some TP in E is active
• All time points in E are still enabled.
• Set TPs execution time to current_time.
• Add TP to S.
• Propagate time of execution to TPs APSP
  immediate neighbors.
• Add to A, all immediate neighbors that became
  enabled.
• TPx enabled if all negative edges starting at TPx
  have their destination in S.

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Propagation is Focused

• Propagate forward along positive edges to tighten
  upper bounds.
• forward prop along negative edges is useless.
• Propagate backward along negative edges to
  tighten lower bounds.
• Backward prop along positive edges useless.

65
Propagation Example
S A
11
10
-1
lt0,0gt
-1
1
-1
1
0
-2
9
2
-2
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Propagation Example
S A
11
lt1,10gt
10
-1
lt0,0gt
-1
1
-1
1
0
-2
9
2
-2
67
Propagation Example
S A
11
lt1,10gt
10
-1
lt0,0gt
-1
1
-1
1
0
-2
9
2
lt0,9gt
-2
68
Propagation Example
S A
11
lt1,10gt
10
-1
lt2,11gt
lt0,0gt
-1
1
-1
1
0
-2
9
2
lt0,9gt
-2
69
Propagation Example
S A
E
C
11
lt1,10gt
10
-1
lt2,11gt
lt0,0gt
-1
1
-1
1
0
-2
9
2
lt0,9gt
-2
70
Reducing Execution Latency

• Execution time is O(n)
• worst case
• best case

• Filtering
• some edges are redundant
• remove redundant edges

11
t 3
10
-1
0
1
lt0,0gt
lt2,11gt
1
-1
0
-2
2
lt0,10gt
10
-2
71

• Edge Dominance
• Eliminate edge that is redundant due to the
  triangle inequality AB BC AC

72

• Edge Dominance
• Eliminate edge that is redundant due to the
  triangle inequality AB BC AC

150
-100
100
80
-50
-20
73

• Edge Dominance
• Eliminate edge that is redundant due to the
  triangle inequality AB BC AC

150
-100
80
-50
-20
74

• Edge Dominance
• Eliminate edge that is redundant due to the
  triangle inequality AB BC AC

150
-100
80
-50
75
An Example of Edge Filtering

• Start off with the APSP network

76
An Example of Edge Filtering

• Start at A-B-C triangle

11
10
-1
-1
1
-1
1
0
-2
9
2
-2
77
An Example of Edge Filtering

• Look at B-D-C triangle

11
-1
-1
1
-1
1
0
-2
9
2
-2
78
An Example of Edge Filtering

• Look at B-D-C triangle

11
-1
-1
1
-1
1
0
-2
9
2
79
An Example of Edge Filtering

• Look at D-A-B triangle

11
-1
-1
1
-1
1
0
9
2
80
An Example of Edge Filtering

• Look at D-A-C triangle

11
-1
1
-1
1
0
9
2
81
An Example of Edge Filtering

• Look at B-C-D triangle

-1
1
-1
1
0
9
2
82
An Example of Edge Filtering

• Look at B-C-D triangle

-1
1
-1
1
0
9
83
An Example of Edge Filtering

• Resulting network has less edges than the original

9
1
1
-1
-1
0
84
Avoiding Intermediate Graph Explosion

• Problem
• APSP consumes O(n2) space.
• Solution
• Interleave process of APSP construction with edge
  elimination
• Never have to build whole APSP graph

85
Additional Filtering

• Node Contraction
• Collapse two events with fixed time between them

86
Additional Filtering

• Node Contraction
• Collapse two events with fixed time between them

87
Additional Filtering

• Node Contraction
• Collapse two events with fixed time between them

88
Goals and Environment Constraints
Projective Task Expansion
Temporal NetworkSolver
Temporal Planner
Temporal Plan
Task Dispatch
Dynamic Scheduling and Task Dispatch
Goals
Modes
Reactive Task Expansion
Commands
Observations