10/18: temporal planning (contd) 10/25: rao out of town; midterm today: temporal planning with...

10/18: Temporal Planning (Contd)

10/25: Rao out of town; midtermToday: Temporal Planning with progression/regression/Plan-space Heuristics for temporal planning (contd. Next class)

State-Space Search:Search is through time-stamped states

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)

Set <pi,ti> of predicates pi and thetime of their last achievement ti < t.

Set of functions represent resource values.

Set of protectedpersistent conditions(could be binary or resource conds).

Event queue (contains resource as wellAs binary fluent events).

Time stamp of S.

In the initial state, P,M, non-empty Q non-empty if we have exogenous events

Review

(:durative-action cross_cellar:parameters ():duration (= ?duration 10):condition (and (at start have_light)

(over all have_light)(at start at_steps))

:effect (and (at start (not at_steps)) (at start crossing)(at end at_fuse_box)

)

Let current state S be P:{have_light@0; at_steps@0}; Q:{~have_light@15} t: 0(presumably after doing the light-candle action) Applying cross_cellar to this state gives

S’= P:{have_light@0; crossing@0}; :{have_light,<0,10>} Q:{at_fuse-box@10;~have_light@15} t: 0

(:durative-action burn_match:parameters ():duration (= ?duration 15):condition: (and (at start have_match)

(at start have_strikepad)):effect (and (at start have_light)

(at end (not have_light)))

)

Light-match

Light-match

Cross-cellar

1510

Time-stamp

Review

“Advancing” the clock as a device for concurrency control

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 problem—but 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).

Light-match

Cross-cellar

~have-light

1510

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 doneAt 0 or 15 (and the latter makes no sense)

Cross-cellar

Review

Search Algorithm (cont.) Goal Satisfaction: S=(P,M,,Q,t) G if <pi,ti> G either:

<pi,tj> P, tj < ti and no event in Q deletes pi.

e Q that adds pi at time te < ti. Action Application: Action A is applicable in S if:

All instantaneous preconditions of A are satisfied by P and M.

A’s 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 A’s instantaneous

effects. Persistent preconditions of A are put in Delayed effects of A are put in Q.

Flying

(in-city ?airplane ?city1)

(fuel ?airplane) > 0

(in-city ?airplane ?city1) (in-city ?airplane ?city2)

consume (fuel ?airplane)

Flying

(in-city ?airplane ?city1)

(fuel ?airplane) > 0

(in-city ?airplane ?city1) (in-city ?airplane ?city2)

consume (fuel ?airplane)

S=(P,M,,Q,t)

Search: Pick a state S from the queue. If S satisfies the goals, endElse non-deterministically do one of

--Advance the clock (by executing the earliest event in Qs

--Apply one of the applicable actions to S

[TLplan; Sapa; 2001]

Interference

Clearly an overkill

Regression Search is similar…

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)

[TP4; 1999]

A2:X

A3:W

A1:Y

Q

RWXy

We can either workOn R at tinf or R and QAt tinf-D(A3)

To work on have_light@<t1,t2>, we can either --support the whole interval directly with one action --or first split <t1,t2> to two subintervals <t1,t’> <t’,t2> and work on supporting have-light on both intervals




)

Let current state S be P:{at_fuse_box@0} t: 0

Regressing cross_cellar over this state gives

S’= P:{}; :{have_light,< 0 , -10>} Q:{have_light@ -10;at_stairs@-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 inLight-match to support have-light)




)

S’= P:{}; :{have_light,< 0 , -10>} Q:{have_light@-10;at_stairs@-10} t: 0

If we now decide to support the subgoal in QUsing light-match

S’’=P:{} Q:{have-match@-15;at_stairs@-10} :{have_light,<0 , -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 inLight-match to support have-light)

Cross_cellar

Have_light

Light-match

PO (Partial Order) Search

[Zeno; 1994]

Split theInterval intoMultiple overlappingintervals

Involves Posting temporal Constraints, andDurative goals

Involves LPsolving overLinear constraints(temporal constraintsAre linear too);Waits for nonlinear constraintsTo become linear.




)

at_fuse_box@G}




)

Cross_cellar GI

At_fusebox

Have_light@<t1,t2>

t1

t2

t2-t1 =10t1 < tGtI < t1

Have_light@t1




)

at_fuse_box@G}




)

Cross_cellar GI

At_fusebox

Have_light@<t1,t2>

t1

t2

t2-t1 =10t1 < tGtI < t1T4<tGT4-t3=15T3<t1T4<t3 V t1<t4

Have_light@t1

Burn_match

t3 t4~have-light

The ~have_light effect at t4 can violate the <have_light, t3,t1> causal link! Resolve by Adding T4<t3 V t1<t4




)

at_fuse_box@G}




)

Cross_cellar GI

At_fusebox

Have_light@<t1,t2>

t1

t2t2-t1 =10t1 < tGtI < t1T4<tGT4-t3=15T3<t1T4<t3 V t1<t4T3<t2T4<t3 V t2<t4

Have_light@t1

Burn_match

t3 t4~have-light

To work on have_light@<t1,t2>, we can either --support the whole interval directly by adding a causal link <have-light, t3,<t1,t2>> --or first split <t1,t2> to two subintervals <t1,t’> <t’,t2> and work on supporting have-light on both intervals

Notice that zenoallows arbitraryslack betweenthe two actions

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.

10/30 (Don’t print hidden slides)

Multi-objective search

Multi-dimensional nature of plan quality in metric temporal planning: Temporal quality (e.g. makespan, slack—the 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

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

Tempe

Phoenix

Los Angeles

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 <cost:5,Makespan:7> plan compare to <cost:4,Makespan:9>? Problem: We don’t know what the user’s utility metric

is as a function of cost and makespan.

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. <C:4,M9> dominated by <C:5;M:9>

A travel agent shouldn’t 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 problem—you are looking for multiple rather than a single solution.

Solution 2: Aggregate Utility Metrics Combine the various objectives into a single utility measure

Eg: w1*cost+w2*make-span Could model grad students’ preferences; with w1=infinity, w2=0

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

1. Find estimators for each of the factors2. Combine the estimates using the utility measure THIS IS WHAT IS DONE IN SAPA

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 simple—just 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

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..

Heuristics in Sapa are derived from the Graphplan-stylebi-level relaxed temporal planning graph (RTPG)

Progression; so constructed anew for each state..

Relaxed Temporal Planning Graph

Relaxed Action:No delete effects

May be okay given progression planningNo resource consumption

Will adjust later

PersonAirplane

Person

A B

Load(P,A)

Fly(A,B) Fly(B,A)

Unload(P,A)

Unload(P,B)

Init Goal Deadline

t=0 tg

while(true) forall Aadvance-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 SG then Terminate{solution}

S’ = Apply(advance-time,S) if (pi,ti) G such that ti < Time(S’) and piS then Terminate{non-solution} else S = S’end while; Deadline goals

RTPG is modeled as a time-stamped plan! (but Q only has +ve events)

Note: Bi-level rep; we don’t actually stack actions multiple times in PG—we just keep track the first time the action entered

Heuristics directly from RTPG

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 smalleror equal to their achievable times.

ADMISSIBLE

PersonAirplane

Person

A B

Load(P,A)

Fly(A,B) Fly(B,A)

Unload(P,A)

Unload(P,B)

Init Goal Deadline

t=0 tg

PersonAirplane

Person

A B

Load(P,A)

Fly(A,B) Fly(B,A)

Unload(P,A)

Unload(P,B)

Init Goal Deadline

t=0 tg

Heuristics from Relaxed Plan Extracted from RTPG

RTPG can be used to find a relaxed solution which is thenused 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.

PersonAirplane

Person

A B

Load(P,A)

Fly(A,B) Fly(B,A)

Unload(P,A)

Unload(P,B)

Init Goal Deadline

t=0 tg

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

The (Relaxed) Temporal PG

Tempe

Phoenix

Los Angeles

Drive-car(Tempe,LA)

Heli(T,P)

Shuttle(T,P)

Airplane(P,LA)

t = 0 t = 0.5 t = 1 t = 1.5 t = 10

Time-sensitive Cost Function

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

Tempe

Phoenix

L.A

Shuttle(Tempe,Phx): Cost: $20; Time: 1.0 hourHelicopter(Tempe,Phx):Cost: $100; Time: 0.5 hourCar(Tempe,LA):Cost: $100; Time: 10 hourAirplane(Phx,LA):Cost: $200; Time: 1.0 hour

cost

time0 1.5 2 10

$300

$220

$100

Drive-car(Tempe,LA)

Heli(T,P)

Shuttle(T,P)

Airplane(P,LA)

t = 0 t = 0.5 t = 1 t = 1.5 t = 10

Estimating the Cost Function

Tempe

Phoenix

L.A

time0 1.5 2 10

$300

$220

$100

t = 1.5 t = 10

Shuttle(Tempe,Phx): Cost: $20; Time: 1.0 hourHelicopter(Tempe,Phx):Cost: $100; Time: 0.5 hourCar(Tempe,LA):Cost: $100; Time: 10 hourAirplane(Phx,LA):Cost: $200; Time: 1.0 hour

1

Drive-car(Tempe,LA)

Hel(T,P)

Shuttle(T,P)

t = 0

Airplane(P,LA)

t = 0.5

0.5

t = 1

Cost(At(LA)) Cost(At(Phx)) = Cost(Flight(Phx,LA))

Airplane(P,LA)

t = 2.0

$20

Observations about cost functions

Because cost-functions decrease monotonically, we know that the cheapest cost is always at t_infinity (don’t 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 doesn’t allow you to say that the precondition must be true at an absolute time point—only 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. 9:40AM, 11:30AM, 3:15PM etc), we can model it by having a precondition “Have-flight” which is asserted at 9:40AM, 11:30AM and 3:15PM 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

ADDED

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)}

Probably other better ideas could be tried

Can’t use something like set-level idea here becauseThat will entail tracking the costs of subsets of literals

Termination Criteria

Deadline Termination: Terminate at time point t if: goal G: Dealine(G) t goal G: (Dealine(G) < 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) < , repeat the process of applying (set) of actions that can improve the cost functions k times.

cost

time0 1.5 2 10

$300

$220

$100

Drive-car(Tempe,LA)

H(T,P)

Shuttle(T,P)

Plane(P,LA)

t = 0 0.5 1 1.5 t = 10

Earliest time pointCheapest cost

Heuristic estimation using the cost functions

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

time

cost

0 t0=1.5 2 t = 10

$300

$220

$100

Cost(At(LA))

Earliest achieve time: t0 = 1.5Lowest cost time: t = 10

The cost functions have information to track both temporal and costmetric of the plan, and their inter-dependent relations !!!

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

Heuristic estimation by extracting the relaxed plan

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(RP+A),C(RP+A)) + 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

time

cost

0 t0=1.5 2 t = 10

$300

$220

$100

Tempe

Phoenix

L.A

f(t,c) = 100.makespan + Cost

End of 10/30 lecture

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

Partialization Example

A1 A2 A3

A1(10) gives g1 but deletes pA3(8) gives g2 but requires p at startA2(4) gives p at end We want g1,g2

A position-constrained plan with makespan 22

A1

A2

A3 G

p

g1

g2

[et(A1) <= et(A2)] or [st(A1) >= st(A3)][et(A2) <= st(A3)….

OrderConstrainedplan

The best makespan dispatch of the order-constrained plan

A1

A2 A3 14+

There could be multiple O.C. plansbecause of multiple possible causal sources. Optimization will involve Going through them all.

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

QR R

G

QR

{Q} {G}

t1 t2 t3

p.c plan o.c plan

Q R RG

QR

{Q} {G}

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

PQ

PQ

Xredundant

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

Modeling greedy approaches as value ordering strategies

Variation of [Kambhampati & Kedar,1993] greedy algorithm for temporal planning as value ordering: Supporting variables: Sp

A = A’ such that: etp

A’ < stpA in the p.c plan Ppc

B s.t.: etpA’ < etp

B < stpA

C s.t.: etpC < etp

A’ and satisfy two above conditions Ordering and interference variables:

pAB = < if etp

B < stpA ; p

AB = > if stpB > stp

A

rAA’= < if etr

A < strA’ in Ppc; r

AA’= > if strA > etr

A’ in Ppc; rAA’= other wise.

Key insight: We can capture many of the greedy approaches as specific value ordering strategies on the CSOP encoding

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

(Hasslum&Geffner)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(location1,location2,city): Cost = 2.0; Duration = 2.0;

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