Exam post-mortem

1
10/27

• Exam post-mortem
• Phased-relaxation approach
• Order generalization (Partialization)
• Temporal Networks

2
Phased Relaxation in SAPA
3
Adjusting the Heuristic Values
Ignored resource related information can be used
to improve the heuristic values (such like ve
and ve interactions in classical planning)
Adjusted Cost C C ?R ?
(Con(R) (Init(R)Pro(R)))/?R? C(AR)
? Cannot be applied to admissible heuristics
Similar phased relaxation is also applied to
negative interactions
4
SAPA converts its position constrained plans to
order constrained plans
A position-constrained plan with makespan 22
A1(10) gives g1 but deletes p at the start A3(8)
gives g2 but requires p at start A2(4) gives p at
end We want g1,g2
A1
A2
A3
p
Order Constrained plan
The best makespan dispatch of the
order-constrained plan
A2
g2
A3
G
p
A2
A3
14e
A1
A1
g1
There could be multiple O.C. plans because of
multiple possible causal sources. Optimization
will involve Going through them all.
et(A1) lt et(A2) or st(A1) gt st(A3) et(A2)
lt st(A3) .
5
Order generalization as Explanation-based learning

• The order generalization (partialization) in the
  previous slide is an example of Explanation-based
  learning
• The idea is to explain (prove) why an example
  (in our case, a specific plan) is an instance of
  a concept (in our case, a correct solution
  plan), and realize that only the aspects of the
  example that took part in the proof/explanation
  are relevant (needed) for the proof to proceed.
• The explanation is done with respect to some
  background domain theory (in our case, the theory
  is the theory of what makes a plan
  correctwhich is given by the causal proof).
• If the background theory is incorrect, then
  explanation will be incorrect too (and you will
  learn superstitions -)
• Sometimes, background theory may be partial
  (correct but incomplete). For example, you may
  not know how to explain why the example is an
  instance of the conceptbut may know that certain
  attributes qualitatively influence (determine)
  certain class labels
• E.g. you get down at Sao Paolo, Brazil and the
  first three people you see
• Are speaking portugese
• Are wearing red shirts
• You induce that Brazilians speak Portugese, but
  you dont induce that Brazilians all wear red
  clothes. This is because you think nationality
  does determine language, but does not determine
  color of clothes.
• EBL can be used in conjunction with inductive
  learning. EBL can pick the relevant attributes
  over which you then do your induction
• Recall that one major headache in many
  classification learning strategies is the issue
  of irrelevant attributes. The domain theory and
  EBL analysis can help you identify relevant
  attributes over which to do learning

6
Plan representation
Is this representation general?
An executable plan must provide -- the actions
that need to be executed -- the start times
for each of the actions ? Or a set of
simple temporal constraints on the
set of actions (S.T.C. are generalization of
partial orders) E.g.
A14,5?A2 (means 4 lt
ST(A2) ST(A1) lt 5 )
Plan views Pert and Gantt charts GANTT Chart
is what is shown on the right PERT shows the
Causal links
7
Problem Definitions

• Position constrained (p.c) plan The execution
  time of each action is fixed to a specific time
  point
• Can be generated more efficiently by state-space
  planners
• Order constrained (o.c) plan Only the relative
  orderings between actions are specified
• More flexible solutions, causal relations between
  actions
• Partialization Constructing a o.c plan from a
  p.c plan

t1
t2
t3
Q
R
Q
R
R
R
G
G
?R
?R
Q
Q
Q
G
Q
G
p.c plan
o.c plan
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9
Goals and Environment Constraints
Projective Task Expansion
Temporal NetworkSolver
Temporal Planner
Temporal Plan
Task Dispatch
Dynamic Scheduling and Task Dispatch
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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
12
Intervals can be handled directly

• The 13 in the previous page are primitive
  relations. The relation between a pair of
  intervals may well be a disjunction of these
  primitive ones
• A meets B OR A starts B
• There are transitive axioms for computing the
  relations between A and C, given the relations
  between A and B B and C
• A meets B B starts C gt A starts C
• A starts B B during C gt C before A
• Using these axioms, we can do constraint
  propagation directly on interval relations to
  check for tight relations among any given pair of
  relations (as well as consistency of a set of
  relations)
• Allens Interval Algebra
• Intervals can also be handled in terms of their
  start and end points. This latter is what we will
  see next.

13
Example Deep Space One Remote Agent Experiment
Timer
Max_Thrust
Idle
Idle
SEP_Segment
Accum
SEP Action
Attitude
Poke
14
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
15
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

16
Jerseyvotes
Incumbent rule
11/1 Temporal Networks Scheduling
Median Outcome
Turnout
The Cell-phone correction
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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
  . . .

Not n-ary constraints
19
Digression the less-than-fully-rational bias for
binary CSP problems in CSP community

• Much work in CSP community (including temporal
  networks) is directed at binary CSPsi.e. csps
  where all the constraints are between exactly 2
  variables.
• E.g. Arc-consistency, Conflit-directed-backjumping
  etc are only clearly articulated for binary CSPs
  first. Temporal networks studied in Dechter et al
  are all binary.
• Binary CSPs are a canonical subset of CSPany
  n-ary CSP can be compiled into a binary CSP by
  introducing additional (hidden) variables. The
  conversion is not always good
• Bacchus and Vanbeek, 98 provides a tradeoff
  analysis
• The ostensible reason for the interest in binary
  CSPs is ostensibly that most naturally occuring
  constraints are between 2-entities.
• A less charitable characterization is that the
  constraint graphs in binary CSPs are normal
  graphs so they can be analyzed better
• The constraint graphs in n-ary CSPs will be
  hyper graphs (edges are between sets of
  vertices)
• In the case of temporal networks that will arise
  in planning,even for simple constraints caused by
  causal threats, the disjunctive constraint that
  is posted is a 3-ary constraint (between threat,
  producer and consumer)not a binary one
• If you split the disjunction into the search
  space however, we will get two Simple temporal
  networks that are both binary.

20
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)
• Simple Temporal Network if each constraint has
  only one interval

Dechter, Meiri, Pearl, aij89
21
TCSPs vs CSPs

• TCSP is a subclass of CSPs with some important
  properties
• The domains of the variables are totally ordered
• The domains of the variables are continuous
• Most queries on TCSPs would involve reasoning
  over all solutions of a TCSP (e.g.
  earliest/latest feasible time of a temporal
  variable)
• Since there are potentially an infinite number of
  solutions to a TCSP, we need to find a way of
  representing the set of all solutions compactly
• Minimal TCSP network is such a representation

22
TCSP Are Visualized UsingDirected Constraint
Graphs
23
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?

All of these can be done if we compute the
minimal equivalent network
24
Minimal Networks

• A TCSP N1 is considered minimal network if there
  is no other network N2 that has the same
  solutions as N1, and has at least one tighter
  constraint than N1
• Tightness means there are fewer valid composite
  labels for the variables. This has nothing to do
  with the syntactic complexity of the constraint
• A Constraint a 1 3b is tighter than a
  constraint a0 10b
• A constraint a1 1.51.6 1.91.9 2.3 2.3 4.8
  5 6b is tighter than a constraint a0 10b
• Computation of minimal networks, in general,
  involves doing two operations
• Intersection over constraints
• Composition over constraints
• For each path p in the network, connecting a pair
  of nodes a and b, find the path constraint
  between a and b (using composition)
• Intersect all the constraints between a pair of
  nodes a and b to find the tightest constraint
  between a and b
• Can lead to fragmentation of constraints in the
  case of disjunctive TCSPs

25
Operations on Constraints Intersection And Co
mposition
Compose 10,20 with 30,4060,inf to get
constraint between 0 and 3
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An example where minimal network is different
from the original one.
40,60
10,20
30,40
10,20
30,40
0
1
3
0
1
3
0,100
0,100
To compute the constraint between 0 and 3, we
first compose 10,20 and 30,40 to get
40,60 we then intersect 40,60 and
0,100 to get 40,60
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Computing Minimal Network for a STP

• Minimal networks for STPs can be computed by
  ensuring path consistency
• For each triple of vertices i,j,k
• C(i,k) C(i,k) .intersection. C(i,j) .compose.
  C(j,k)
• For STPs we are guaranteed to reach fixpoint by
  the time we visit each constraint once.
• An alternative is to convert STP to a distance
  graph and do All pairs shortest path algorithm

28
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

30
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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Shortest Paths of Gd
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STN Minimum Network
d-graph
STN minimum network
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Disjunctive TCSPs

• Suppose we have a TCSP, where just one of the
  constraints is dijunctive a 1 25 6 b
• We have two STPs one in which the constraint
  a1 2b is there and the other contains a5 6b
• Disjunctive TCSPs can be solved by solving the
  exponential number of STPs
• Minimal network for DTP is the union of minimal
  networks for the STPs
• This is a brute-force method Exponential number
  of STPsmany of which have significant
  overlapping constraints.
• There are better approaches that work directly on
  DTPs Decther, Schwalb, 97
• Scheduling can be seen as solving a DTP (the
  disjunction is induced because of the resource
  contention constraints)

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Testing Plan Consistency
No negative cycles -5 gt TA TA 0
d-graph
35
Latest Solution
Node 0 is the reference.
20
40
0
1
2
-10
-30
-10
20
50
4
3
-40
-60
70
d-graph
36
Earliest Solution
Node 0 is the reference.
20
40
0
1
2
-10
-30
-10
20
50
4
3
-40
-60
70
d-graph
37
Solution Earliest Times
S1 (-d10, . . . , -dn0)
20
40
0
1
3
-10
-30
-10
20
50
4
2
-40
-60
70
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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
40
Scheduling without Search Solution by
Decomposition

• Select value for 1
• 15 10,20

d-graph
41
Solution by Decomposition

• Select value for 1
• 15

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

d-graph
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Solution by Decomposition

• Select value for 1
• 15

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

d-graph
43
Solution by Decomposition

• Select value for 1
• 15

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

d-graph
44
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
45
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
46
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
47
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