Temporal Planning with Continuous Change J.Scott Penbrethy Daniel S. Weld Presented by - Parag

Temporal Planning with Continuous Change

J.Scott Penbrethy Daniel S. Weld

Presented by - Parag

An Example

Seattle

L.A.

Austin

1000 800

Mausam

Stanley

Initial configuration

Reach Austin in 5hr and 30 minutes. Fast fly : 600 miles/hr , 1 gallon/2 milesSlow fly: at 400 miles/hr, 1 gallon/3 milesBoarding time : 30 mins, Refuelling time: 1 hr. (750 gallons)

500 gallons

An Example

Seattle

L.A.

Austin

<Slow Fly> <Fast Fly>

Mausam, Stanley

<Fill fuel> <Board Stanley>

<Board Mausam> 30 min

2 hrs 30 mins

1 hr

1 hr, 20 mins

Final Configuration

An Example

Can the problem be solved using one of our earlier planners? Why or why not?

What all additional features do we need?

Temporal Planner

• Explicit Notion of Time - Deadline Goals

• Notion of Continuous Change - Metric changes

• Simultaneous Actions - Actions taking place at the same instant of time in the world.

Zeno : A Temporal Planner

What all kind of effects it can reason about?

• Deadline Goals

• Piecewise Linear Continuous Change

• Simultaneous Actions

Representation?

What kind of representation would we choose?

Can we do away with STRIPS like representation?Why or why not?

Representation?

How would we represent time?

What about continuous constraints?

e.g. the amount of Fuel in the plane changes continuously as it flies.

Representation?

Action and Goal Representation

• Typed First Order Logic

• Point Based Model of Time - what does it mean? Temporal functions and Relations use a time point as their first argument.

• All types except time assumed finite

An Example Action SchemaSchema Fast-Fly(m, l) at-time: [ts , te]

Example Preconditions: 1. Plane is at Seattle at ts ? at(ts,plane,seattle) 2. dist(m,l) = v2

3. While in flight, fuel has to be non-zero? Do we need some kind of quantification here?On which variable?

time, t є [ts,te], fuel(t,plane) > 0

Constraints:

Why do we need them? Any example constraint for above actionschema ?

An Example Action SchemaSchema Fast-Fly(m, l)

at-time: [ts , te]

te=ts+v2/600

An Example Action Schema

Post Conditions / Effects: 1. At(te,plane,l)

2. Conditional Effects: Why? An example?

Schema Fast-Fly(m, l)

at-time: [ts , te]

human o, time t, (t є (ts,te] Λ in(t,o) ) at(t,o,m)Λ at(te,o,l)

Partial Order Planning

• Search in the space of partial plans

• A partial plan represented as the triplet – <A,O,L> - A : set of Actions currently in the plan - O : ordering constraints on the set of actions in A e.g. Ae < Ap

- L : set of Causal links e.g. Ae Q Ap

An Example

Finish

Start

Left Shoe Right Shoe

Right SockLeft Sock

Left Sock on Right Sock on

Left Shoe on, Right Shoe on

Notion of a Threat

Let Ae < Ap be some ordering constraint.

Also, let (Ae Q Ap) L

Suppose that there is another action At which Deletes Q. And also, let us assume that ordering Ae < At < Ap is consistent with respect to theSet O. Then, we say that action At poses a threatTo the causal link (Ae Q Ap) .

In order to remove the threat, we try the following: - promotion - move At before the two actions. - demotion - move At after the two actions.

POP Algorithm1. Termination: If G = ø, return <A,O,L>

2. Goal Selection: Select a pair <Q,Aneed> from the agenda.

3. Action Selection: Let Aadd be an action which adds Q. Add AAdd to the set A, if it does not already contain Aadd .. Add the causal link Ae Q Ap to the set of causal links, L. Also add, Aadd < Aneed to the set O of ordering constraints. If no such action exists, return failure.

4. Update Goal Set: Update the agenda by removing <Q,Aneed> and adding any new preconditions introduced because of Aadd.

5. Causal Link Protection: Remove any threats to the links in L by promotion or demotion. Else return failure.

6. Recurse: Go back to step 1 with the new values of action set, link set and ordering constraints.

Input : (<AOL>,agenda)

Zeno: Plan Representation

Plan : A triplet of the form <S,L,C>

S : Set of steps in the partial plan L : Set of causal links C : Set of constraints

Initial State: <S,L,C> = ?

Searching for the Goal

How would it be different from the standard partial order Planner?

How do we take into account the continuous constraints?

Searching for the Goal

Searching for the Goal

Does our representation allow us to plan for deadline goals?

How?

Searching for the Goal

Simultaneous Actions?

Zeno AlgorithmIs C consitent?

Is G = ø ?Remove a goal <ψ,ig>

From G

Is ψ primitive?

Is ψ metric ?

Return P

Reduce <ψ,ig>Goto Start.

yes

yes

yes

yes

no

no

no

no

FAIL

Choose Source ei for <ψ,ig>Add link <ip,ψ > to L;

Resolve Threats.Go to start.

What are the non-deterministic steps in above algorithm?

Post <ψ,ig>Go to start

Goal ReductionReduce Goal <ψ,ig> time t, t є ig ψ .

Consider, ψ = fuel( t, plane) > 0

Split into to two sub-inverval goals.Or mark the interval as indivisible.

What is the intuition? When do we want to mark the interval as indivisible?

Goal Reduction

• Disjuction of a set of goals ?

• A universally quantified goal : type x, ψ. type time?

•Existential quantifiers?

- non-deterministically try each of the disjuncts.

- Enumerate all possible instances.

- treat as simple vairable names.

Metric Constraints

time t, t є [ts ,te], fuel(t,plane) > 0

Can we exploit linearity here?

- Apply mean value theorem.

- Need to post only two constraints - at the two end points of the interval.

Logical Goals

Example : time t, t є ig, ψ = At(t,Stanley,Austin)

How do we handle these?

Use standard techniques of promotion and demotionTo resolve the threats.

Non-deterministically find an effect concluding ψ. Letie be the interval over which ψ is concluded.

Add the ordering constraint ie precedes ip.

Add to L <ip, ψ> where ip covers both ie and ig.

Constraint Manager

• What kind of constraints do we need to solve? - linear equalities, linear inequalities, co-designation constraints, reasoning about time-intervals.

• Use standard techniques - Gaussian elimination, simplex method etc.

Experiments?

Comments about the paper?

Future Work/Extensions

• Heuristics for guiding the search.

• Fast implementation of temporal queries.

• Can the idea be extended to Graphplan?

TGP – Temporal Graphplan

• Weld & Smith – IJCAI – 99 • Use the graph structure to represent the flow of time.

• Each durative (temporal) action could span several layers of the graph.

• Strong mutual exclusion constraints : the propostions manipulated by the action locked for the duration of the action.

• Can be shown to generate optimal plans – in terms of time of execution.

Problems?

Could be very expernsive to search

- if the plan contains two or more actions of very different durations.

Mutual Exclusion Constraints are too strong.

- Example : action of lighting a matchstick.

Till what time does the effect of the action persist?

Another Approach

• Long & Fox – to appear at ICAPS 2003

• Use the graph structure to represent the logical flow and not the temporal flow.

• Use a separate entity to manage temporal constraints - somewhat like the constraint manager of Zeno.

• Split each durative action into two simple actions - the start action and the end action.

• Take care of invariants of a durative action.

Invariants

Matchstick Lighting action: Let Precondition : ¬Windy (invariant) Postconidition : Light

How do we make sure that ¬Windy holds throughoutThe duration of the action?

Introduce a new action <Inv> whose precondition is theInvariant.

Invariants

A-start A-endA-invariant

check

Pre: start

Add: As,initial effects

….. …..

Pre: inv,As

Add: AiDel: Ai.

+ final effects

Would it work?

A-start A-endA-invariantcheck

Pre: start

Add: As,initial effects

….. …..

Pre: inv,As

Add: As, AiDel: Ai.

+ final effects

Invariants

A-invariantcheck

Pre: inv,As

Add: As, Ai

…..

Remove the No-ops for the special effect As.

Pre: As,Ai

Graph RepresentationEach durative action represented by two actions –Start action and end action.

We do not worry about durations in the graph representation.

Invariants maintain the logical flow between the startAnd the end action.

Graph Representation

The action layers represent no lag in time.

The proposition layer represents the state of the system for a particular interval of time.

Graph constructed in the standard manner.

Searching Through the Graph

• Use the standard graph plan search.

• Additional constraints introduced because of actoin durations.

• Constraints represented in form of linear equations.

• Use a separate constraint solver to solve these constraints.

Constraint Satisfaction

• Basic idea :

- an equation for each durative action currently in plan.

- for each layer in the graph, there is a variable – represents

the duration for which it stays.

- the total time duration of the layers between the start action

and end action should be equal to the specified duration of

the action.

Comparisons

The described approach has the advantage of usingThe standard graph-plan structure for fast retrieval.

Zeno handles much more generic constraints for exampleNumeric constraints.

Can the benefits of two be combined?

References

• Temporal Planning with Continuous Change - J. Scott Penberthy & Daniel S. Weld

• Exploiting a Graphplan Framework in Temporal Planning – Derek Long & Maria Fox

• Fast Planning through Planning Graph Analysis – Arvim L. Blum & Merrick L. Frust

• An Introduction to least Commitment Planning – Daniel S. Weld

• Artificial Intelligence Modern Approach (Second Edition, Chapter 11) – Stuart Russell & Peter Norvig