10/18: Temporal Planning (Contd)

1
10/18 Temporal Planning (Contd)

• 10/25 Rao out of town midterm
• Today
• ? Temporal Planning with progression/regression/
  Plan-space
• ? Heuristics for temporal planning (contd. Next
  class)

2
State-Space SearchSearch is through
time-stamped states
Review
Search states should have information about --
what conditions hold at the current time slice
(P,M below) -- what actions have we already
committed to put into the plan (?,Q below)
S(P,M,?,Q,t)
In the initial state, P,M, non-empty
Q non-empty if we have exogenous
events
3
Review
Light-match
Let current state S be Phave_light_at_0
at_steps_at_0
Qhave_light_at_15
t 0 (presumably after doing the light-candle
action) Applying
cross_cellar to this state gives S
Phave_light_at_0 crossing_at_0
?have_light,lt0,10gt Qat_fuse-box_at_10hav
e_light_at_15 t 0
Time-stamp
Light-match
Cross-cellar
15
10
4
Advancing the clock as a device for concurrency
control
Review
In the cellar plan above, the clock, If advanced,
will be advanced to 15, Where an event
(have-light will occur) This means cross-cellar
can either be done At 0 or 15 (and the latter
makes no sense)

• To support concurrency, we need to consider
  advancing the clock
• How far to advance the clock?
• One shortcut is to advance the clock to the time
  of the next earliest event event in the event
  queue since this is the least advance needed to
  make changes to P and M of S.
• At this point, all the events happening at that
  time point are transferred from Q to P and M (to
  signify that they have happened)
• This
• This strategy will find a plan for every
  problembut will have the effect of enforcing
  concurrency by putting the concurrent actions to
  align on the left end
• In the candle/cellar example, we will find plans
  where the crossing cellar action starts right
  when the light-match action starts
• If we need slack in the start times, we will have
  to post-process the plan
• If we want plans with arbitrary slacks on
  start-times to appears in the search space, we
  will have to consider advancing the clock by
  arbitrary amounts (even if it changes nothing in
  the state other than the clock time itself).

have-light
Light-match
Cross-cellar
Cross-cellar
15
10
5
Search Algorithm (cont.)

• Goal Satisfaction
• S(P,M,?,Q,t) ? G if ?ltpi,tigt? G either
• ? ltpi,tjgt ? P, tj lt ti and no event in Q deletes
  pi.
• ? e ? Q that adds pi at time te lt ti.
• Action Application
• Action A is applicable in S if
• All instantaneous preconditions of A are
  satisfied by P and M.
• As effects do not interfere with ? and Q.
• No event in Q interferes with persistent
  preconditions of A.
• A does not lead to concurrent resource change
• When A is applied to S
• P is updated according to As instantaneous
  effects.
• Persistent preconditions of A are put in ?
• Delayed effects of A are put in Q.

S(P,M,?,Q,t)
TLplan Sapa 2001
6
Interference
? Clearly an overkill
7
(No Transcript)
8
Regression Search is similar
We can either work On R at tinf or R and Q At
tinf-D(A3)
R W X y

• In the case of regression over durative actions
  too, the main generalization we need is
  differentiating the advancement of clock and
  application of a relevant action
• Can use same state representation S(P,M,?,Q,t)
  with the semantics that
• P and M are binary and resource subgoals needed
  at current time point
• Q are the subgoals needed at earlier time points
• ? are subgoals to be protected over specific
  intervals
• We can either add an action to support something
  in P or Q, or push the clock backward before
  considering subgoals
• If we push the clock backward, we push it to the
  time of the latest subgoal in Q
• TP4 uses a slightly different representation
  (with State and Action information)

Q
A3W
A2X
A1Y
To work on have_light_at_ltt1,t2gt, we can either
--support the whole interval directly
with one action --or first split ltt1,t2gt to two
subintervals ltt1,tgt ltt,t2gt and work on
supporting have-light on both intervals
TP4 1999
9
Let current state S be Pat_fuse_box_at_0
t 0
Regressing cross_cellar over this
state gives S P ?have_light,lt 0
, -10gt Qhave_light_at_ -10at_stairs_at_-10
t 0
Cross_cellar
Have_light
Notice that in contrast to progression, Regression
will align the end points of Concurrent
actions(e.g. when we put in Light-match to
support have-light)
10
Notice that in contrast to progression, Regression
will align the end points of Concurrent
actions(e.g. when we put in Light-match to
support have-light)
Cross_cellar
S P ?have_light,lt 0 , -10gt
Qhave_light_at_-10at_stairs_at_-10 t
0 If we now decide to support the subgoal in
Q Using light-match SP
Qhave-match_at_-15at_stairs_at_-10
?have_light,lt0 , -10gt t 0
Have_light
Cross_cellar
Have_light
Light-match
11
PO (Partial Order) Search
Involves LPsolving over Linear
constraints (temporal constraints Are linear
too) Waits for nonlinear constraints To become
linear.
Involves Posting temporal Constraints,
and Durative goals
Split the Interval into Multiple
overlapping intervals
Zeno 1994
12
More on Temporal planningby plan-space planners
(Zeno)

• The accommodation to complexity that Zeno makes
  by refusing to handle nonlinear constraints
  (waiting instead until they become linear) is
  sort of hilarious given it doesnt care much
  about heuristic control otherwise
• Basically Zeno is trying to keep the per-node
  cost of the search down (and if you do nonlinear
  constraint consistency check, even that is quite
  hard)
• Of course, we know now that there is no obvious
  reason to believe that reducing the per-node cost
  will, ipso facto, also lead to reduction in
  overall search.
• The idea of goal reduction by splitting a
  temporal subgoal to multiple sub-intervals is
  used only in Zeno, and helps it support a
  temporal goal over a long duration with multiple
  actions. Neat idea.
• Zeno doesnt have much of a problem handling
  arbitrary concurrencysince we are only posting
  constraints on temporal variables denoting the
  start points of the various actions. In
  particular, Zeno does not force either right or
  left alignment of actions.
• In addition to Zeno, IxTeT is another influential
  metric temporal planner that uses plan-space
  planning idea.

13
At_fusebox
Have_light_at_t1
t1
Cross_cellar
G
I
at_fuse_box_at_G
t2
Have_light_at_ltt1,t2gt
t2-t1 10 t1 lt tG tI lt t1
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The have_light effect at t4 can violate the
lthave_light, t3,t1gt causal link! Resolve by
Adding T4ltt3 V t1ltt4
have-light
t3
t4
Burn_match
At_fusebox
Have_light_at_t1
t1
Cross_cellar
G
I
at_fuse_box_at_G
t2
Have_light_at_ltt1,t2gt
t2-t1 10 t1 lt tG tI lt t1 T4lttG T4-t315 T3ltt1 T4lt
t3 V t1ltt4
15
Notice that zeno allows arbitrary slack
between the two actions
have-light
t3
t4
Burn_match
At_fusebox
Have_light_at_t1
t1
Cross_cellar
G
I
at_fuse_box_at_G
t2
Have_light_at_ltt1,t2gt
t2-t1 10 t1 lt tG tI lt t1 T4lttG T4-t315 T3ltt1 T4lt
t3 V t1ltt4 T3ltt2 T4ltt3 V t2ltt4
To work on have_light_at_ltt1,t2gt, we can either
--support the whole interval directly by
adding a causal link lthave-light, t3,ltt1,t2gtgt
--or first split ltt1,t2gt to two subintervals
ltt1,tgt ltt,t2gt and work on supporting
have-light on both intervals
16
Tradeoffs Progression/Regression/PO Planning for
metric/temporal planning

• Compared to PO, both progression and regression
  do a less than fully flexible job of handling
  concurrency (e.g. slacks may have to be handled
  through post-processing).
• Progression planners have the advantage that the
  exact amount of a resource is known at any given
  state. So, complex resource constraints are
  easier to verify. PO (and to some extent
  regression), will have to verify this by posting
  and then verifying resource constraints.
• Currently, SAPA (a progression planner) does
  better than TP4 (a regression planner). Both do
  oodles better than Zeno/IxTET. However
• TP4 could be possibly improved significantly by
  giving up the insistence on admissible heuristics
• Zeno (and IxTET) could benefit by adapting ideas
  from RePOP.

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10/30 (Dont print hidden slides)
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Multi-objective search

• Multi-dimensional nature of plan quality in
  metric temporal planning
• Temporal quality (e.g. makespan, slackthe time
  when a goal is needed time when it is
  achieved.)
• Plan cost (e.g. cumulative action cost, resource
  consumption)
• Necessitates multi-objective optimization
• Modeling objective functions
• Tracking different quality metrics and heuristic
  estimation
• ? Challenge There may be inter-dependent
  relations between different quality metric

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Example

• Option 1 Tempe ?Phoenix (Bus) ? Los Angeles
  (Airplane)
• Less time 3 hours More expensive 200
• Option 2 Tempe ?Los Angeles (Car)
• More time 12 hours Less expensive 50
• Given a deadline constraint (6 hours) ? Only
  option 1 is viable
• Given a money constraint (100) ? Only option 2
  is viable

20
Solution Quality in the presence of multiple
objectives

• When we have multiple objectives, it is not clear
  how to define global optimum
• E.g. How does ltcost5,Makespan7gt plan compare
  to ltcost4,Makespan9gt?
• Problem We dont know what the users utility
  metric is as a function of cost and makespan.

21
Solution 1 Pareto Sets

• Present pareto sets/curves to the user
• A pareto set is a set of non-dominated solutions
• A solution S1 is dominated by another S2, if S1
  is worse than S2 in at least one objective and
  equal in all or worse in all other objectives.
  E.g. ltC4,M9gt dominated by ltC5M9gt
• A travel agent shouldnt bother asking whether I
  would like a flight that starts at 6pm and
  reaches at 9pm, and cost 100 or another ones
  which also leaves at 6 and reaches at 9, but
  costs 200.
• A pareto set is exhaustive if it contains all
  non-dominated solutions
• Presenting the pareto set allows the users to
  state their preferences implicitly by choosing
  what they like rather than by stating them
  explicitly.
• Problem Exhaustive Pareto sets can be large
  (exponentially large in many cases).
• In practice, travel agents give you
  non-exhaustive pareto sets, just so you have the
  illusion of choice ?
• Optimizing with pareto sets changes the nature of
  the problemyou are looking for multiple rather
  than a single solution.

22
Solution 2 Aggregate Utility Metrics

• Combine the various objectives into a single
  utility measure
• Eg w1costw2make-span
• Could model grad students preferences with
  w1infinity, w20
• Log(cost) 5(Make-span)25
• Could model Bill Gates preferences.
• How do we assess the form of the utility measure
  (linear? Nonlinear?)
• and how will we get the weights?
• Utility elicitation process
• Learning problem Ask tons of questions to the
  users and learn their utility function to fit
  their preferences
• Can be cast as a sort of learning task (e.g.
  learn a neual net that is consistent with the
  examples)
• Of course, if you want to learn a true nonlinear
  preference function, you will need many many more
  examples, and the training takes much longer.
• With aggregate utility metrics, the multi-obj
  optimization is, in theory, reduces to a single
  objective optimization problem
• However if you are trying to good heuristics to
  direct the search, then since estimators are
  likely to be available for naturally occurring
  factors of the solution quality, rather than
  random combinations there-of, we still have to
  follow a two step process
• Find estimators for each of the factors
• Combine the estimates using the utility measure
• THIS IS WHAT IS DONE IN SAPA

23
Sketch of how to get cost and time estimates

• Planning graph provides level estimates
• Generalizing planning graph to temporal planning
  graph will allow us to get time estimates
• For relaxed PG, the generalization is quite
  simplejust use bi-level representation of the
  PG, and index each action and literal by the
  first time point (not level) at which they can be
  first introduced into the PG
• Generalizing planning graph to cost planning
  graph (i.e. propagate cost information over PG)
  will get us cost estimates
• We discussed how to do cost propagation over
  classical PGs. Costs of literals can be
  represented as monotonically reducing step
  functions w.r.t. levels.
• To estimate cost and time together we need to
  generalize classical PG into Temporal and
  Cost-sensitive PG
• Now, the costs of literals will be monotonically
  reducing step functions w.r.t. time points
  (rather than level indices)
• This is what SAPA does

24
SAPA approach

• Using the Temporal Planning Graph (Smith Weld)
  structure to track the time-sensitive cost
  function
• Estimation of the earliest time (makespan) to
  achieve all goals.
• Estimation of the lowest cost to achieve goals
• Estimation of the cost to achieve goals given the
  specific makespan value.
• Using this information to calculate the heuristic
  value for the objective function involving both
  time and cost
• Involves propagating cost over planning graphs..

25
Heuristic Control
Temporal planners have to deal with more
branching possibilities ? More critical to have
good heuristic guidance
Design of heuristics depends on the objective
function
? In temporal Planning heuristics focus on richer
obj. functions that guide both planning and
scheduling
26
Objectives in Temporal Planning

• Number of actions Total number of actions in the
  plan.
• Makespan The shortest duration in which we can
  possibly execute all actions in the solution.
• Resource Consumption Total amount of resource
  consumed by actions in the solution.
• Slack The duration between the time a goal is
  achieved and its deadline.
• Optimize max, min or average slack values
• Combinations there-of

27
Deriving heuristics for SAPA
We use phased relaxation approach to derive
different heuristics
Relax the negative logical and resource
effects to build the Relaxed Temporal Planning
Graph
AltAlt,AIJ2001
28
Heuristics in Sapa are derived from the
Graphplan-style bi-level relaxed temporal
planning graph (RTPG)
Progression so constructed anew for each
state..
29
Relaxed Temporal Planning Graph
Note Bi-level rep we dont actually stack
actions multiple times in PGwe just
keep track the first time the action entered
RTPG is modeled as a time-stamped plan! (but Q
only has ve events)

• Relaxed Action
• No delete effects
• May be okay given progression planning
• No resource consumption
• Will adjust later

while(true) forall A?advance-time
applicable in S S
Apply(A,S) Involves changing P,?,Q,t Update Q
only with positive effects and only when there
is no other earlier event giving that effect
if S?G then Terminatesolution
S Apply(advance-time,S) if ?(pi,ti) ?G
such that ti lt Time(S) and pi?S
then
Terminatenon-solution else S S end
while
Deadline goals
30
Details on RTPG Construction

• ?All our heuristics are based on the relaxed
  temporal planning graph structure (RTPG). This is
  a Graphplanstyle
• 2 bi-level planning graph generalized to
  temporal domains.
• Given a state S (PM ? Q t), the RTPG is
  built from S using the set of relaxed actions,
  which are generated from original actions by
  eliminating all effects which (1) delete some
  fact (predicate) or (2) reduce the level of some
  resource. Since delete effects are ignored, RTPG
  will not contain any mutex relations, which
  considerably reduces the cost of constructing
  RTPG. The algorithm to build the RTPG structure
  is summarized in Figure 4.
• ?To build RTPG, we need three main
  datastructures a fact level, an action level,
  and an unexecuted event queue
• ?Each fact f or action A is marked in, and
  appears in the RTPGs fact/action level at time
  instant tf /tA if it can be
• achieved/executed at tf /tA.
• ?In the beginning, only facts which appear in P
  are marked in at t, the action level is empty,
  and the event queue holds all the unexecuted
  events in Q that add new predicates.
• ?Action A will be marked in if (1) A is not
  already marked in and (2) all of As
  preconditions are marked in.
• When action A is in, then all of As unmarked
  instant add effects will also be marked in at t.
• ?Any delayed effect e of A that adds fact f
  is put into the event queue Q if (1) f is not
  marked in and (2) there is no event e0 in Q that
  is scheduled to happen before e and which also
  adds f. Moreover, when an event e is added to Q,
  we will take out from Q any event e0 which is
  scheduled to occur after e and also adds f.
• ?When there are no more unmarked applicable
  actions in S, we will stop and return no-solution
  if either
• (1) Q is empty or (2) there exists some unmarked
  goal with a deadline that is smaller than the
  time of the
• earliest event in Q.
• ?If none of the situations above occurs, then we
  will apply advance-time action to S and
• activate all events at time point te0 of the
  earliest event e in Q.
• ?The process above will be repeated until all the
  goals are marked in or one of the conditions
  indicating non-solution occurs.

From Do Kambhampati ECP 01
31
Heuristics directly from RTPG
A D M I S S I B L E

• For Makespan Distance from a state S to the
  goals is equal to the duration between time(S)
  and the time the last goal appears in the RTPG.
• For Min/Max/Sum Slack Distance from a state to
  the goals is equal to the minimum, maximum, or
  summation of slack estimates for all individual
  goals using the RTPG.
• Slack estimate is the difference between the
  deadline of the goal, and the expected time of
  achievement of that goal.

Proof All goals appear in the RTPG at times
smaller or equal to their achievable times.
32
Heuristics from Relaxed Plan Extracted from RTPG
RTPG can be used to find a relaxed solution which
is then used to estimate distance from a given
state to the goals
Sum actions Distance from a state S to the goals
equals the number of actions in the relaxed plan.
Sum durations Distance from a state S to the
goals equals the summation of action durations in
the relaxed plan.
33
Resource-based Adjustments to Heuristics
Resource related information, ignored originally,
can be used to improve the heuristic values
Adjusted Sum-Action h h ?R ?
(Con(R) (Init(R)Pro(R)))/?R?
Adjusted Sum-Duration h h ?R
(Con(R) (Init(R)Pro(R)))/?R.Dur(AR)
? Will not preserve admissibility
34
Aims of Empirical Study

• Evaluate the effectiveness of the different
  heuristics.
• Ablation studies
• Test if the resource adjustment technique helps
  different heuristics.
• Compare with other temporal planning systems.

35
Empirical Results

• Sum-action finds solutions faster than sum-dur
• Admissible heuristics do not scale up to bigger
  problems
• Sum-dur finds shorter duration solutions in most
  of the cases
• Resource-based adjustment helps sum-action, but
  not sum-dur
• Very few irrelevant actions. Better quality than
  TemporalTLPlan.
• So, (transitively) better than LPSAT

36
Empirical Results (cont.)
Logistics domain with driving restricted to
intra-city (traditional logistics domain)
Sapa is the only planner that can solve all 80
problems
37
Empirical Results (cont.)
Logistics domain with inter-city driving actions
The sum-action heuristic used as the default
in Sapa can be mislead by the long duration
actions...
?
Future work on fixed point time/level propagation
38
The (Relaxed) Temporal PG
39
Time-sensitive Cost Function
cost
?
300
220
100
0
time
1.5
2
10
Drive-car(Tempe,LA)
Airplane(P,LA)
Heli(T,P)
Shuttle(Tempe,Phx) Cost 20 Time 1.0
hour Helicopter(Tempe,Phx) Cost 100 Time 0.5
hour Car(Tempe,LA) Cost 100 Time 10
hour Airplane(Phx,LA) Cost 200 Time 1.0 hour
Shuttle(T,P)
t 10
t 0
t 0.5
t 1
t 1.5

• Standard (Temporal) planning graph (TPG) shows
  the time-related estimates e.g. earliest time to
  achieve fact, or to execute action
• TPG does not show the cost estimates to achieve
  facts or execute actions

40
Estimating the Cost Function
?
Shuttle(Tempe,Phx) Cost 20 Time 1.0
hour Helicopter(Tempe,Phx) Cost 100 Time 0.5
hour Car(Tempe,LA) Cost 100 Time 10
hour Airplane(Phx,LA) Cost 200 Time 1.0 hour
300
220
100
20
time
0
1.5
2
10
1
Cost(At(LA))
Cost(At(Phx)) Cost(Flight(Phx,LA))
41
Observations about cost functions
ADDED

• Because cost-functions decrease monotonically, we
  know that the cheapest cost is always at
  t_infinity (dont need to look at other times)
• Cost functions will be monotonically decreasing
  as long as there are no exogenous events
• Actions with time-sensitive preconditions are in
  essence dependent on exogenous events (which is
  why PDDL 2.1 doesnt allow you to say that the
  precondition must be true at an absolute time
  pointonly a time point relative to the beginning
  of the action
• If you have to model an action such as Take
  Flight such that it can only be done with valid
  flights that are pre-scheduled (e.g. 940AM,
  1130AM, 315PM etc), we can model it by having a
  precondition Have-flight which is asserted at
  940AM, 1130AM and 315PM using timed initial
  literals)
• Becase cost-functions are step funtions, we need
  to evaluate the utility function U(makespan,cost)
  only at a finite number of time points (no matter
  how complex the U(.) function is.
• Cost functions will be step functions as long as
  the actions do not model continuous change (which
  will come in at PDDL 2.1 Level 4). If you have
  continuous change, then the cost functions may
  change continuously too

42
Cost Propagation

• Issues
• At a given time point, each fact is supported by
  multiple actions
• Each action has more than one precondition
• Propagation rules
• Cost(f,t) min Cost(A,t) f ?Effect(A)
• Cost(A,t) Aggregate(Cost(f,t) f ?Pre(A))
• Sum-propagation ? Cost(f,t)
• The plans for individual preconds may be
  interacting
• Max-propagation Max Cost(f,t)
• Combination 0.5 ? Cost(f,t) 0.5 Max Cost(f,t)

Cant use something like set-level idea here
because That will entail tracking the costs of
subsets of literals
Probably other better ideas could be tried
43
Termination Criteria
cost

• Deadline Termination Terminate at time point t
  if
• ? goal G Dealine(G) ? t
• ? goal G (Dealine(G) lt t) ? (Cost(G,t) ?
• Fix-point Termination Terminate at time point t
  where we can not improve the cost of any
  proposition.
• K-lookahead approximation At t where Cost(g,t) lt
  ?, repeat the process of applying (set) of
  actions that can improve the cost functions k
  times.

?
300
220
100
0
time
1.5
2
10
Earliest time point
Cheapest cost
Drive-car(Tempe,LA)
Plane(P,LA)
H(T,P)
Shuttle(T,P)
t 0
0.5
1.5
1
t 10
44
Heuristic estimation using the cost functions
The cost functions have information to track both
temporal and cost metric of the plan, and their
inter-dependent relations !!!

• If the objective function is to minimize time h
  t0
• If the objective function is to minimize cost h
  CostAggregate(G, t?)
• If the objective function is the function of both
  time and cost
• O f(time,cost) then
• h min f(t,Cost(G,t)) s.t. t0 ? t ? t?
• Eg f(time,cost) 100.makespan Cost then
• h 100x2 220 at t0 ? t 2 ? t?

cost
?
300
220
100
0
t01.5
2
t? 10
time
Cost(At(LA))
Earliest achieve time t0 1.5 Lowest cost time
t? 10
45
Heuristic estimation by extracting the relaxed
plan

• Relaxed plan satisfies all the goals ignoring the
  negative interaction
• Take into account positive interaction
• Base set of actions for possible adjustment
  according to neglected (relaxed) information
  (e.g. negative interaction, resource usage etc.)
• ? Need to find a good relaxed plan (among
  multiple ones) according to the objective function

46
Heuristic estimation by extracting the relaxed
plan
cost

• Initially supported facts SF Init state
• Initial goals G Init goals \ SF
• Traverse backward searching for actions
  supporting all the goals. When A is added to the
  relaxed plan RP, then
• SF SF ? Effects(A)
• G (G ? Precond(A)) \ Effects
• If the objective function is f(time,cost), then A
  is selected such that
• f(t(RPA),C(RPA)) f(t(Gnew),C(Gnew))
• is minimal (Gnew (G ? Precond(A)) \ Effects)
• When A is added, using mutex to set orders
  between A and actions in RP so that less number
  of causal constraints are violated

?
300
220
100
0
t01.5
2
t? 10
time
Tempe
L.A
Phoenix
f(t,c) 100.makespan Cost
47
Heuristic estimation by extracting the relaxed
plan
cost

• General Alg. Traverse backward searching for
  actions supporting all the goals. When A is added
  to the relaxed plan RP, then
• Supported Fact SF ? Effects(A)
• Goals SF \ (G ? Precond(A))
• Temporal Planning with Cost If the objective
  function is f(time,cost), then A is selected such
  that
• f(t(RPA),C(RPA)) f(t(Gnew),C(Gnew))
• is minimal (Gnew (G ? Precond(A)) \ Effects)
• Finally, using mutex to set orders between A and
  actions in RP so that less number of causal
  constraints are violated

?
300
220
100
0
t01.5
2
t? 10
time
Tempe
L.A
Phoenix
f(t,c) 100.makespan Cost
48
End of 10/30 lecture
49
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
50
Partialization Example
A position-constrained plan with makespan 22
A1(10) gives g1 but deletes p 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
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) .
51
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
52
Validity Requirements for a partialization

• An o.c plan Poc is a valid partialization of a
  valid p.c plan Ppc, if
• Poc contains the same actions as Ppc
• Poc is executable
• Poc satisfies all the top level goals
• (Optional) Ppc is a legal dispatch (execution) of
  Poc
• (Optional) Contains no redundant ordering
  relations

redundant
X
P
P
Q
Q
53
Greedy Approximations

• Solving the optimization problem for makespan and
  number of orderings is NP-hard (Backstrom,1998)
• Greedy approaches have been considered in
  classical planning (e.g. Kambhampati Kedar,
  1993, Veloso et. al.,1990)
• Find a causal explanation of correctness for the
  p.c plan
• Introduce just the orderings needed for the
  explanation to hold

54
Partialization A simple example
Pickup(A)
Stack(A,B)
Pickup(C)
Stack(C,D)
On(A,B)
Stack(A,B)
Holding(C)
Pickup(A)
Stack(C,D)
On(C,D)
Hand-empty
Pickup(C)
Holding(B)
Hand-empty
55
Modeling greedy approaches as value ordering
strategies
Key insight We can capture many of the greedy
approaches as specific value ordering strategies
on the CSOP encoding

• Variation of Kambhampati Kedar,1993 greedy
  algorithm for temporal planning as value
  ordering
• Supporting variables SpA A such that
• etpA lt stpA in the p.c plan Ppc
• ? B s.t. etpA lt et?pB lt stpA
• ? C s.t. etpC lt etpA and satisfy two above
  conditions
• Ordering and interference variables
• ?pAB lt if et?pB lt stpA ?pAB gt if st?pB gt
  stpA
• ?rAA lt if etrA lt strA in Ppc ?rAA gt if strA
  gt etrA in Ppc ?rAA ? other wise.

56
CSOP Variables and values

• Continuous variables
• Temporal stA D(stA) 0, ?, D(stinit) 0,
  D(stGoals) Dl(G).
• Resource level VrA
• Discrete variables
• Resource ordering ?rAA Dom(?rAA) lt,gt or
  Dom(?rAA) lt,gt,?
• Causal effect SpA Dom(SpA) B1, B2,Bn, p?
  E(Bi)
• Mutex ?pAA Dom(?pAA) lt,gt p ?
  E(A),?p?E(A) U P(A)

A2
A3
Exp Dom(SQA2) Aibit, A1 Dom(SRA3) A2,
Dom(SGAg) A3 ?RA1A2, ?RA1A3
Q
R
R
G
?R
Q
A1
G
Q
57
Constraints

• Causal-link protection
• SpA B ? ?A, ?p?E(A) (?pAB lt) ? (?pAA gt)
• Ordering and temporal variables
• SpA B ? etpB lt stpA
• ?pAB lt ? et?pA lt stpA ?pAB gt ? et?pA gt
  stpA
• ?rAA lt ? etrA lt strA ?rAA gt ? strA gt etrA
• Optional temporal constraints
• Goal deadline stAg ? tg
• Time constraints on individual actions L ? stA
  ? U
• Resource precondition constraints
• For each precondition VrA ? K, ? gt,lt,?,?,
  set up one constraint involving all ?rAA such
  as
• Exp Initr ?AltAUrA ?A?A,Ult0 UrA gt K if ? gt

58
Modeling Different Objective Functions

• Temporal quality
• Minimum Makespan Minimize MaxA (stA durA)
• Maximize summation of slacks
• Maximize ?(stgAg - etgA) SgAg A
• Maximize average flexibility
• Maximize Avg(Dom(stA))
• Fewest orderings
• Minimize (stA lt stA)

59
Empirical evaluation

• Objective
• Demonstrate that metric temporal planner armed
  with our approach is able to produce plans that
  satisfy a variety of cost/makespan tradeoff.
• Testing problems
• Randomly generated logistics problems from TP4
  (HasslumGeffner)

Load/unload(package,location) Cost 1 Duration
1 Drive-inter-city(location1,location2) Cost
4.0 Duration 12.0 Flight(airport1,airport2)
Cost 15.0 Duration 3.0 Drive-intra-city(loc
ation1,location2,city) Cost 2.0 Duration
2.0