Time Constraints in Planning

Time Constraints in Planning

Sudhan Kanitkar

([email protected])

References

• Fahiem Bacchus, Michael Ady “Planning with Resources and Concurrency A Forward Chaining Approach”

• Ch. 13 Time for Planning

• Ch. 14 Temporal Planning• http://www.cs.toronto.edu/~fbacchus/tlplan-manual.html

Agenda

• TLPlan – Practical Approach– Functions, Timestamped States, Queues– Algorithm, Example– Changes needed in the domain

• A more theoretical but expressive approach described in the textbook.

TLPlan

• Functions– Similar to state variable representation

discussed earlier

• Timestamped States

• Queues

Functions

• In traditional planning States are represented as databases (sets) of predicate instances and operators as making changes to these databases.

• It is needed to add/delete all the predicates • (drive ?t ?l ?l’)

.

.

(forall (?o) (int ?o ?t)

(and (add (at ?o ?l’)) (del (at ?o ?l))))

)

Functions

• Instead of having predicates for all facts we use functions.

• Functions seem to analogous to variables in programming languages

• They represent values

• Predicate (at ?x ?l) just describes the location of object x.

• Instead model the location of the object using a function (loc ?x)

Functions (Cont’d)

• (loc ?x) acts just like a variable which describes the location of the object x.

• In the drive predicate we make the following changes

• (drive ?t ?l ?l’)

.

.

(forall (?o) (in ?o ?t)

(add (= (loc ?o) ?l)))

)

Recall State-Variable Representation

Functions – More Examples

• Predicate (refuel ?t) refuels the truck t• (capacity ?t) is the fuel capacity of the truck• (fuel ?t) is the current level of fuel• (fuel-used) is a total fuel used globally• (refuel ?t)

(and

(add (= (fuel-used)

(+ (fuel-used)

(- (capacity ?t) (fuel ?t)))))

(add (= (fuel ?t) (capacity ?t)))

)

)

Forward chaining Planners

• Forward chaining has proved to be useful for high-performance planners.

• Domain independent heuristics for search• Drawback: They explore only totally ordered

sequences of action.• Hence, modeling concurrent actions with

linear sequences become problematic – e.g. Two trucks in two different locations can

travel simultaneously in parallel.– Plans generated by GraphPlan

Why make time explicit ?

• Model the duration of action

• Model the effects and conditions of an action at various points along duration

• Handle goals with relative and absolute temporal constraints

• To be able to use events happening in the future which are not immediate effects of actions

Principle

• In classical planners the effects of an action are visible immediately and hence validating the preconditions of further action

• This approach suppresses the visibility of effects for the duration of action

• Hence the further actions which use these effects as preconditions cannot be used.

Timestamps

• Associate with each state a timestamp• Timestamp starts with a fixed start time in the

initial state• Denotes the actual time the state will occur

during the execution of a plan• Timestamp of a successive state changes only

when no other action can be applied and it is necessary to wait for an action that takes some time to finish.

• The effects which are not delayed still become available instantaneously

Queue

• State also has an event queue

• Queue has updates scheduled to occur at some time in the future

• These updates are predicates and time at which they become effective

• Each state inherits the pending events of its parent state

Actions

• s is the current state

• a is an action which is applicable to s only if it satisfies all the preconditions of s.

• Applying a to s generates a new successor state s+

• An action can have two kinds of effects– Instantaneous effects– Delayed effects

Example• (def-adl-operator (drive ?t ?l ?l’)

(pre (?t) (truck ?t) (?l) (loc ?l) (?l’) (loc ?l’) (at ?t ?l) ) (del (at ?t ?l)) (delayed-effect (/ (dist ?l ?l’) (speed ?t)) (arrived-driving ?t ?l ?l’) (add (at ?t ?l’)) ))

Instantaneous Effect

Delayed Effect

Why two types of effects ??

• Instantaneous effects make sure that objects in question are not reused

• Delayed effects ensure that the timing constraints are satisfied

(add (at ?t ?l’))Delayed Effect

delayed-action

• Parameters– delta: the time further from the current time

that the action is time stamped with

• Instantaneous effects change the database of s immediately

• Delayed effects are added to the queue of the state to be applied later

unqueue-event Action

• A mechanism is needed which will remove events from the queue when the time is up and update the database

• A special action

• Advances the world clock

• Remove all actions scheduled for current time from the queue and update the database

Planning AlgorithmState & Queue pair

Advance to new state

Record Previous State

Record Action

New timestamp

Apply all updates with current timestamp from the queue

Two types of Updates

Non-deterministic:Operator or unqueue-event

Notes on Algorithm

• The non-deterministic choice operator is realized by search.

• The choice of which action to try is made by heuristic or domain specific control

• Temporal Control Formula from previous class

• Instead of a plan the final goal state is returned

• The sequence of actions leading to the goal can be determined using action and prev pointers

TLPlan support

• Following actions can be defined for TLPlan– (delayed-action delta tag formula)– (wait-for-next-event)

• TLPlan Manual link– http://www.cs.toronto.edu/~fbacchus/tlplan-manual.html

– Look for section titled “Support for Concurrent Planning”

l1

l0

Example

v0

c0

Goal: Get cargo at location l0

(at c0 l0)

Plan:

move(v0,l0,l1)

load(c0,v0,l1)

move(v0,l1,l0)

unload(c0,v0,l0)

State:

(at v0 l0)

(at c0 l1)

Queue:State:

(at c0 l1)

Queue:

(at v0 l1)

State:

(at c0 l1)

(at v0 l1)

Queue:State:

(at v0 l1)

(in c0 v0)

State:

(in c0 v0)

Queue:

(at v0 l0)

State:

(at v0 l0)

(in c0 v0)

State:

(at v0 l0)

(at c0 l0)

Thanks: Joe Souto http://www.cse.lehigh.edu/~munoz/AIPlanning/classes/Graphplan.ppt

Importance of control Formula• 0 (move v0 l0 l1 f2 f1)• 20 (event (moving-truck • v0 l0 l1 f2 f1))• 20 (load c0 v0 l1 s1 s0)• 20 (move v0 l1 l0 f1 f0)• 40 (event (moving-truck • v0 l1 l0 f1 f0))• 40 (unload c0 v0 l0 s0 s1)

• 0 (move v0 l0 l1 f1 f0)• 0 (move v1 l1 l0 f2 f1)• 20 (event ...• 20 (move v0 l1 l0 f1 f0)• 20 (load c0 v1 l0 s2 s1)• 20 (load c1 v1 l0 s1 s0)• 20 (unload c0 v1 l0 s0 s1)• 20 (donate l2 l0 f2 f1 f0 f0

f1)• 20 (load c0 v1 l0 s1 s0)

Note Redundant actions

Changes in Domain File(define (domain mprime-strips) (:types space vehicle cargo) (:predicates (at ?v ?l) (conn ?l1 ?l2) (has-fuel ?l ?f) (fuel-neighbor ?f1 ?f2) (in ?c ?v) (has-space ?v ?s) (space-neighbor ?s1 ?s2) (not-equal ?l1 ?l2) ) ......

(declare-described-symbol (predicate cargo-at 2) (predicate vehicle-at 2)

(predicate conn 2) (predicate has-fuel 2) (predicate fuel-neighbor 2) (predicate in 2) (predicate has-space 2) (predicate space-neighbor 2) (predicate not-equal 2)

)......

Changes in Domain File(:action move :parameters ( ?v - vehicle ?l1 ?l2 - location ?f1 ?f2 – fuel) :precondition (and (at ?v ?l1) .. (fuel-neighbor ?f2 ?f1)) :effect (and (not (at ?v ?l1)) .. (has-fuel ?l1 ?f2)))

(def-adl-operator (move ?v ?l1 ?l2 ?f1 ?f2) (pre (?v ?l1) (vehicle-at ?v ?l1) (?l2) (conn ?l1 ?l2) (?f1) (has-fuel ?l1 ?f1) (?f2) (fuel-neighbor ?f2 ?f1)) (del (vehicle-at ?v ?l1)

(has-fuel ?l1 ?f1)) (delayed-action 20 (moving-truck ?v ?l1 ?l2 ?f1 ?

f2) (add (vehicle-at ?v ?l2) (has-fuel ?l1 ?f2) )))

Changes in Domain File

• Add operator to unqueue events

(def-adl-operator (event)

(wait-for-next-event)

)

• Add to the top of the domain file

(enable concurrent-planning))

Changes in Problem Filedefine (problem strips-mprime-. .-c4)(:domain mprime-strips)(:objects f0 f1 f2 - fuel ..

c0 c1 - cargo)(:init (not-equal l0 l1) (not-equal l0 l2) . .)(:goal(and (at c0 l0) .. (at c1 l2)))

(define (state0)(not-equal l0 l1)(not-equal l0 l2)(not-equal l1 l0)..

)(define goal0

(cargo-at c0 l0)(cargo-at c1 l2)

)

Break

• After the break we will look at the one theoretical approach

Formal Representation

• Formal representation of a temporal planning domain has following objects

• Symbols

• Relations– Rigid Relations – Flexible Relations

• Constraints– Temporal Constraints– Binding Constraints

Symbols

• Constant Symbols– Objects which remain constant over time or

state changes– Objects of classes such as robot, crane

• Variable Symbols– Objects whose value changes over time or

state changes– e.g. temporal variables ranging over R

Relations

• Rigid Relations– Relations which do not change over time or

state transitions– e.g. adjacent(loc1,loc2)

• Flexible Relations – Also called Fluents– Relations which invalidate/validate over a

period of time– e.g. at(robot1,loc1)

Constraints

• Binding constraints

• Temporal constraints– If t1 and t2 are two temporal variables and r is a

constraint defined on them – r = 2P

– P = {<,>,=}– 2P={Φ,{<},{=},{>},{<,=},{>,=},{>,<},P}

Temporally Qualified Expression

• A temporally qualified expression (tqe) is an expression of the form p(ζ1,…, ζk)@[ts,te]

- p is a flexible relation- ζ1,…, ζk are constants or object variables

- ts,te are temporal variables such that ts<te

- A tqe asserts that for the time range ts≤t<te the relation p(ζ1,…, ζk)holds

Temporal Database

• A temporal database is a pairΦ = (F,C)- F is a finite set of tqes- C is a finite set of temporal and object

constraints

Textbook. Pg: 312

Enabling Conditions

• In the temporal database shown previously there are two instances of tqe free(l)@[t,t’).

• This tqe holds w.r.t to database only if one of the following holds:– {l=loc3, τ0 ≤ t, t’ ≤ τ5}– {l=loc2, τ6 ≤ t, t’ ≤ τ7}

• These two sets of constraints are called enabling conditions for the tqe to be supported by F

• One of them has to be consistent with C for the database to support the tqe.

Definitions• A set F supports a tqe e = p(ζ1,…,ζk)@[t1,t2] iff there is in F a tqe p(ζ1’,…,ζk’)@[τ1,τ2] and a substitution σ such that σ(p(ζ1,…,ζk)) = σ(p(ζ1’,…,ζk’)) and

• An enabling condition for e in F is conjunction of the temporal constraints τ1 ≤ t1 and t2 ≤ τ2 with binding constraints of σ.

• θ(e/F) is set of all the possible enabling conditions for e in F.

• θ(ε/F) is set of all the possible enabling conditions for a set of tqes ε in F. In this case F is said to support ε.

• A temporal database Φ=(F,C) supports a set of tqes ε if all the enabling conditions c Є θ(ε/F) are consistent with C.

• Φ=(F,C) supports another database (F’,C’) when F supports F’ and there is an enabling condition c Є θ(F’/F) such that C’U c is consistent with C.

Temporal Planning Operators

• It’s a tuple

• o = (name(o), precond(o), effects(o), const(o))

• name is an expression of form o(x1,…xk, ts, te) such that o is an operator, x1,…xk are object variables, ts, te are temporal variables

• precond(o) and effects(o) are tqes

• const(o) is a conjunction of constraints

Temporal Planning Operator

- Action is a partially instantiated operator

- If preconditions and constraints of an action hold then action will run from ts to te.

-effects describe the new tqes that result from an action

Textbook. Pg: 315

Applicability of an Action

• An action a is applicable to a temporal database (F,C) if and only if precond(a) is supported by F and there is an enabling condition c in θ(a/F)for the a such that C U const(a) U c is consistent with the set of constraints

• Γ(Φ,a) = {(F U effects(a), C U const(a) U c | c Є θ(a/F)}

• Note that actions are applied to database and the result is a set databases since action can be applied differently at different times.

Domain Axioms

• The operators described till now do not express the negative effects of the actions

• The action thus keeps on increasing the size of the database where we might have conflicting statements appearing.

• Domain axioms is the mechanism used to overcome this shortcoming.

• Domain axiom is a conditional expression of the form

p = cond(p) disj(p)- cond(p) is a set of tqes- disj(p) is a disjunction of temporal and object

constraints

Domain Axiom (Cont’d)

• Consider a scenario which has two robots r and r’ an two locations l and l’

- {at(r,l)@[ts,te),at(r’,l’)@[ts’,te’)}

(r ≠ r’) v (l = l’) v (te ≤ ts’) v (te’≤ ts)

- {at(r,l)@[t1,t1’),free(l’)@[t2,t2’)}

(l ≠ l’) v (t1’ ≤ t2) v (t2’≤ t1)

Domain Axiom Support

• Let p be an axiom and Φ=(F,C) be a temporal database such that cond(p) is supported by F and θ(p/F) is set of enabling conditions for cond(p) in F.

• Φ is consistent with p iff for each enabling condition c1 in θ(p/F) there is atleast one disjunct c2 in disj(p) such that C U c1 U c2 is consistent set of constraints.

• This means that for every for every tqe to be supported by F, there is needs to be atleast one disjunct in disj(p) which is consistent with Φ or C.

• A consistency condition for Φ w.r.t a set of axioms X is:

• A set of all such conditions is denoted by θ(X/F)

Domain Axioms- Actions

• So for a set of axioms to be applicable the consistency condition needs to satisfied

• As result we get a new set of databases as

• Earlier it was mentioned that effect of applying an action a to Φ is a set of databases.

• Many of these databases may not be consistent with X• So we now restrict that definition to only those databases

which are consistent with X as follows:

Temporal Planning Domain

• A temporal Planning domain is the tripleD = (ΛΦ , O, X)

- ΛΦ is set of all temporal databases that can be defined

- O is a set of temporal planning operators

- X is a set of domain axioms

Temporal Planning Problem

• Is the triple P = (D, Φ0, Φg)

– D is the planning domain– Φ0 = (F,C) is the initial state of the domain

– Φg = (G,Cg) is the goal state of the domain

• The statement of the problem is given by– P = (O, X, Φ0, Φg)

TPS Procedure

Note the similarity with Plan-space Planning approach

TPS Procedure

• It maintains the data structure Ω.• Ω = { Φ, G, K, π }• Φ = { F, C } is the current temporal database• G is a set of tqes corresponding to current open

goals• K = { C1,…,C2 } is the set of pending enabling

conditions of actions, consistency conditions of axioms

• π is a set of actions corresponding to current plan

• A tqe in F can support a tqe e Є G if there is an enabling condition θ(e/F). Updates are– K K U {θ(e/F)}– G G – {e}

• Updates owing to action a for this goal– π π U {a}– F F U effects(a)– C C U const(a)– G (G – {e}) U precond(a)– K K U {θ(a/Φ)}

Flaws – Open Goals

Flaws - Axioms and Threat

• Unsatisfied Axioms– These flaws are possible inconsistencies of instances

of Φ w.r.t to the axioms of X.

– A resolver is a set of consistency conditions θ(X/Φ)– K K U {θ(X/Φ)}

• Threats– Over the period of time we have kept on adding new

constraints which are required to be solved to K. – For every Ci in K, the resolver is a constraint c such

that:– C C U c

– K K - {Ci}

Thank You

[email protected]

Thank You

Example

• move(r,l,l’)@[ts,te]

precond: at(r,l)@[t1,ts)

free(l’)@[t2,te)

effects: at(r,routes)@[ts,te)

at(r,l’)@[te,t3)

free(l)@[t4,t5)

const: ts < t4 < t2

adjacent(l,l’)

Temporal Constraints

Binding Constraints

l1

l0

Example

v0

c0

Goal: Get cargo at location l0

Plan:

move(v0,l0,l1)

load(c0,v0,l1)

move(v0,l1,l0)

unload(c0,v0,l0)

State:

(at v0 l0)

(at c0 l1)

Queue:State:

(at c0 l1)

Queue:

(at v0 l1)

State:

(at c0 l1)

(at v0 l1)

Queue:State:

(at v0 l1)

(in c0 v0)

State:

(in c0 v0)

Queue:

(at v0 l0)

State:

(at v0 l0)

(in c0 v0)

State:

(at v0 l0)

(at c0 l0)

l1

l0

Example

v0

c0

Goal: Get cargo at location l0

move(v0,l0,l1)

load(c0,v0,l1)

move(v0,l1,l0)

unload(c0,v0,l0)