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

An Example Seattle L.A. Austin Mausam Stanley Initial configuration Reach Austin in 5hr and 30 minutes. Fast fly : 600 miles/hr, 1 gallon/2 miles Slow fly: at 400 miles/hr, 1 gallon/3 miles Boarding time : 30 mins, Refuelling time: 1 hr. (750 gallons) 500 gallons

An Example Seattle L.A. Austin Mausam, Stanley 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 Schema Schema Fast-Fly(m, l) at-time: [t s, t e ] Example Preconditions: 1. Plane is at Seattle at t s ? at(t s,plane,seattle) 2. dist(m,l) = v 2 3. While in flight, fuel has to be non-zero? Do we need some kind of quantification here? On which variable?  time, t є [t s,t e ], fuel(t,plane) > 0

Constraints: Why do we need them? Any example constraint for above action schema ? An Example Action Schema Schema Fast-Fly(m, l) at-time: [t s, t e ] t e =t s +v 2 /600

An Example Action Schema Post Conditions / Effects: 1. At(t e,plane,l) 2. Conditional Effects: Why? An example? Schema Fast-Fly(m, l) at-time: [t s, t e ]  human o,  time t, (t є (t s,t e ] Λ in(t,o) )   at(t,o,m)Λ at(t e,o,l)

Partial Order Planning Search in the space of partial plans A partial plan represented as the triplet – - A : set of Actions currently in the plan - O : ordering constraints on the set of actions in A e.g. A e < A p - L : set of Causal links e.g. A e  Q A p

An Example Finish Start Left Shoe Right Shoe Right Sock Left Sock Left Sock on Right Sock on Left Shoe on, Right Shoe on

Notion of a Threat Let A e < A p be some ordering constraint. Also, let (A e  Q A p )  L Suppose that there is another action A t which Deletes Q. And also, let us assume that ordering A e < A t < A p is consistent with respect to the Set O. Then, we say that action A t poses a threat To the causal link (A e  Q A p ). In order to remove the threat, we try the following: - promotion - move A t before the two actions. - demotion - move A t after the two actions.

POP Algorithm 1. Termination: If G = ø, return 2. Goal Selection: Select a pair from the agenda. 3. Action Selection: Let A add be an action which adds Q. Add A Add to the set A, if it does not already contain A add.. Add the causal link A e  Q A p to the set of causal links, L. Also add, A add < A need to the set O of ordering constraints. If no such action exists, return failure. 4. Update Goal Set: Update the agenda by removing and adding any new preconditions introduced because of A add. 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 : (,agenda)

Zeno: Plan Representation Plan : A triplet of the form S : Set of steps in the partial plan L : Set of causal links C : Set of constraints Initial State: = ?

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

Does our representation allow us to plan for deadline goals? How?

Searching for the Goal Simultaneous Actions?

Zeno Algorithm Is C consitent? Is G = ø ? Remove a goal From G Is ψ primitive? Is ψ metric ? Return P Reduce Goto Start. yes no FAIL Choose Source e i for Add link to L; Resolve Threats. Go to start. What are the non-deterministic steps in above algorithm? Post Go to start

Goal Reduction Reduce Goal   time t, t є i g  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 [t s,t e ], 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 i g, ψ = At(t,Stanley,Austin) How do we handle these? Use standard techniques of promotion and demotion To resolve the threats. Non-deterministically find an effect concluding ψ. Let i e be the interval over which ψ is concluded. Add the ordering constraint i e precedes i p. Add to L where i p covers both i e and i g.

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 throughout The duration of the action? Introduce a new action whose precondition is the Invariant.

Invariants A-start A-end A-invariant check Pre: start Add: As, initial effects ….. Pre: inv,As Add: Ai Del: Ai. + final effects Would it work?

A-start A-end A-invariant check Pre: start Add: As, initial effects ….. Pre: inv,As Add: As, Ai Del: Ai. + final effects Invariants A-invariant check Pre: inv,As Add: As, Ai ….. Remove the No-ops for the special effect As. Pre: As,Ai

Graph Representation Each 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 start And 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 using The standard graph-plan structure for fast retrieval. Zeno handles much more generic constraints for example Numeric 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