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Planning for Temporally Extended Goals Fahiem Bacchus

Planning for Temporally Extended Goals Fahiem Bacchus
Planning for Temporally Extended Goals Fahiem Bacchus

Planning for Temporally Extended Goals

Fahiem Bacchus Dept.of Computer Science University of Waterloo

Waterloo,Ontario

Canada,N2L3G1

Froduald Kabanza Dept.de Math et Informatique Universite de Sherbrooke

Sherbrooke,Quebec

Canada,J1K2R1

Abstract

In planning,goals have been traditionally been viewed as specifying a set of desirable?nal states.Any plan that trans-forms the current state to one of these desirable states is viewed to be correct.Goals of this form are limited as they do not allow us to constrain the manner in which the plan achieves its objectives.

We propose viewing goals as specifying desirable sequences of states,and a plan to be correct if its execution yields one of these desirable sequences.We present a logical language,

a temporal logic,for specifying goals with this semantics.

Our language is rich and allows the representation of a range of temporally extended goals,including classical goals,goals with temporal deadlines,quanti?ed goals(with both universal and existential quanti?cation),safety goals,and maintenance goals.Our formalism is simple and yet extends previous approaches in this area.

We also present a planning algorithm that can generate correct plans for these goals.This algorithm has been implemented, and we provide some examples of the formalism at work.The end result is a planning system which can generate plans that satisfy a novel and useful set of conditions.

Introduction

One of the features that distinguishes intelligent agents is their?exibility:generally they have the ability to accomplish a task in a variety of ways.Such?exibility is necessary if the agent is to be able to accomplish a variety of tasks under a range of conditions.Yet this?exibility also poses a problem: how do we communicate to such an agent the task we want accomplished in a suf?ciently precise manner so that it does what we really want.

In the area of planning,methods and algorithms are studied by which,given information about the current situation,an intelligent agent can compose its primitive abilities so as to accomplish a desired task or goal.The afore mentioned problem then becomes the problem of designing suf?ciently expressive and precise ways of specifying goals.

Much of the work in planning has dealt with goals speci?ed as conditions on a?nal state.For example,we might specify

ning systems and theories speci?cally designed to deal with temporal constraints(and sometimes other metric resources) have also been developed[V er83,Wil88,AKRT91,CT91, Lan93,PW94].

The main difference between these previous works and what we present here,lies in our use of a temporal logic that supports a unique approach to computing plans,an approach based on formula progression.The method of formula pro-gression lends itself naturally to the speci?cation and uti-lization of domain dependent search control knowledge.As shown in our previous work[BK95],the approach of do-main dependent search control offers considerable promise, and has motivated our approach to dealing with temporally extended goals.The other works that have constructed tem-poral planners have utilized complex constraint management techniques to deal with temporal information.

In[Kab90,GK91,BKSD95]similar temporal logics and similar notions of formula progression have been utilized. In this case the main difference is that here we address classical plans,i.e.,?nite sequences of actions,while these works have concentrated on generating reactive plans,i.e., mappings from states to actions(sometimes called universal plans).Reactive plans have to specify an on-going interac-tion between an agent and its environment,and thus pose a quite distinct set of problems.

To generate plans that achieve the goals expressed in our formalism we present a planning algorithm that uses the logical mechanism of formula progression.This notion was previously utilized in our TLP LAN system[BK95].In fact we have implemented the planning algorithm by extending the TLP LAN system.TLP LAN is planning system whose key feature is that it is able to utilize domain dependent search control information..This control is expressed in a temporal logic that is a limited form of the logic presented here,and it is utilized by the planner via the mechanism of formula progression.

The planning algorithm we develop is sound and complete and as we will demonstrate it is able to generate a range of interesting plans.Further work is required,however, to evaluate the planner’s performance on realistic planning problems.

In the rest of the paper we will?rst provide the details of the logic we propose for expressing goals.This logic is a temporal logic that is based on previous work by Alur et al.[AFH91].We then present our approach to planning, provide examples to demonstrate the range of goals that our system can cope with,and discuss the heuristic adequacy of our approach to planning.Finally,we close with some conclusions and discussion of future work.

Expressing goals in MITL

We use a logical language for expressing goals.The logic is based on Metric Interval Temporal Logic developed by Alur et al.[AFH91],but we have extended it allow?rst-order quanti?cation.Syntax

We start with a collection of-ary predicate(including equal-ity and the predicate constants TRUE and FALSE)function and constant symbols,variables,and the connectives(not)and (and).We add the quanti?ers and and the modal

operators(next)and U(until).From this collection of symbols we generate MITL,the language we use to express goals.MITL is de?ned by the traditional rules for gener-ating terms,atomic formulas,and Boolean combinations, taken from ordinary?rst-order logic.In addition to those formula formation rules we add:(1)if is a formula then so is;(2)if and are formulas and is an interval then so is U(the syntax of intervals is de?ned below); and(3)if is an atomic formula in which the variable is free,and is a formula then so are,and .

Notice that in our language we use bounded quanti?cation. The atomic formula is used to specify the range over which the quanti?ed variable ranges.The precise semantics are given below.

The syntax of intervals is as one would expect.The al-lowed intervals are all intervals over the non-negative real line,and we specify an interval by giving its two endpoints, both of which are required to be non-negative numbers.To allow for unbounded intervals we allow the right endpoint to be.For example,speci?es the interval of num-bers such that,speci?es the interval

,and speci?es the interval

(i.e.,the point).

Although non-negative intervals are the only ones allowed in the formulas of MITL,in the semantics and algorithms we will need to utilize shifted intervals and to test for negative intervals.For any interval let be the set of numbers such that,be the set of numbers such that,and be true iff all numbers in are less than0.For example,is the new interval ,is the new interval,and

is true.

Finally,we introduce(implication),and(disjunction) as standard abbreviations.We also introduce the temporal modalities eventually and always as abbreviations with TRUE U,and.We will also abbreviate intervals of the form and,e.g.,

will be written as and as.Finally,we will often omit writing the interval,e.g.,we will write U as U.

Semantics

We intend that goals be expressed as sentences of the lan-guage MITL.As hinted in the introduction such formulas are intended to specify sets of sequences of states.Hence,it should not be surprising that the underlying semantics we as-sign to the formulas of MITL be in terms of state sequences.

A model for MITL is a timed sequence of states,

.In particular,a model is an in?nite se-quence of states,and each state is a?rst-order model over a ?xed domain.That is,each state assigns a denotation for each predicate and function symbol over the domain. Furthermore,there is a timing function that maps each state in to a point on the non-negative real line such that for all,and for all real numbers there exists an such that.This means that time is only required to be non-decreasing,not strictly increasing. Time can stall at a single point for any?nite number of states. Eventually,however,time must increase without bound. Let be a variable assignment,i.e.,a mapping from the variables to elements of;,,and be formulas of MITL;and be an MITL model.The semantics of MITL are then de?ned by the following clauses.

,when is atemporal(i.e.,contains no temporal modalities)and quanti?er free,iff.

iff.

U iff there exists with

such that and for all with

we have.

iff for all such that

we have.

iff there exists such that

and.

It is not dif?cult to show that any formula of MITL that has no free variables,called a sentence of MITL,has a truth value that is independent of the variable assignment.Given a sentence of MITL we say it is true in a model,, iff.

Since sentences of MITL are either true or false on any individual timed sequence of states,we can associate with every sentence a set of sequences:those sequences on which it is true.We express goals as sentences of MITL,hence we obtain our desired semantics for goals:a set of acceptable sequences.

Discussion

Intuitively,the temporal modalities can be explained as fol-lows.The next modality simply speci?es that something must be true in the next state.Its semantics do not depend on the time of the states.It is important to realize,however, that what it requires to be true in the next state may itself be a formula containing temporal modalities.MITL gets its expressive power from its ability to nest temporal modalities. The until modality is more subtle.The formula U

,for example,requires that be true in some state whose time is between5and7units into the future,and that be true in all states until we reach a state where is true.The eventually modality thus takes on the semantics that requires that be true in some state whose time lies in the

Inputs:A state,with formula label,and a time duration to the successor state.

Output:A new formula representing the formula label of the successor state.

Algorithm Progress(,,)

Case

1.contains no temporal modalities:

if TRUE

else FALSE

2.:Progress Progress

3.:Progress

4.:

5.U:

if FALSE

else if Progress

Progress U else Progress U

6.:Progress

7.:Progress

Table1:The progression algorithm.

Since actions map states to new states,any?nite sequence of actions will generate a?nite sequence of states:the states that would arise as the plan is executed.Furthermore,we will assume that part of an action’s speci?cation is a speci?cation of its duration,which is constrained to be greater than or equal to0.This means that if we consider to commence at time0,then every state that is visited by the plan can be given a time stamp.Hence,a plan gives rise to a?nite timed sequence of states—almost a suitable model for MITL. The only dif?culty is that models of MITL are in?nite sequences.Intuitively,we intend to control the agent for some?nite time,up until the time the agent completes the execution of its plan.Since we are assuming that the agent is the only source of change,once it has completed the plan the?nal state of the plan idles,i.e.,it remains unchanged. Formally,we de?ne the MITL model corresponding to a plan as follows:

De?nition1Let plan be the?nite sequence of actions .Let be the sequence of states such that,and is the initial state. is the sequence of states visited by the plan.Then the MITL model corresponding to and is de?ned to be

,i.e.,with the?nal state idled, where duration,, ,and the time of the copies of increases without bound.

Therefore,every?nite sequence of actions we generate corresponds to a unique model in which the?nal state is idling.Given a goal expressed as a sentence of MITL we can determine,using the semantics de?ned above,whether or not the plan satis?es the goal.

De?nition2Let be a plan,be a goal expressed as a formula of MITL,be the initial state,and be the model corresponding to and.is a correct plan for given iff.Generating Plans

We will generate plans by adopting the methodology of our previous work[BK95].In particular,we have constructed a forward-chaining planning engine that generates linear se-quences of actions,and thus linear sequences of states.As these linear sequences of states are generated we incremen-tally check them against the goal.Whenever we can show that achieving the goal is impossible along a particular se-quence we can prune that sequence and all of its possible extensions from the search space.And we can stop when we?nd a sequence that satis?es the goal.The incremental checking mechanism is accomplished by the logical progres-sion of the goal formula.

Formula Progression The technique of formula progres-sion works by labeling the initial state with the sentence representing the goal,call it.For each successor of the initial state,generated by forward chaining,a new formula label is generated by progressing the initial state’s label us-ing the algorithm given in Table1.This new formula is used to label the successor states.This process continues.Every time a state is expanded during planning search each of its successors is given a new label generated by progression. Intuitively a state’s label speci?es a condition that we are looking for.That is,we want to?nd a sequence of states starting from this state that satis?es the label.The purpose of the progression algorithm is to update this label as we extend the state sequence.It takes as input the current state and the duration of the action that yields the successor state.

The logical relationship between the input formula and output formula of the algorithm is characterized by the fol-lowing proposition:

Proposition3Let M be any MITL model. Then,we have for any formula of MITL,M if and only if M Progress. This proposition can easily be proved by utilizing the def-inition of MITL semantics.

Say that we label the start state,,with the formula ,and we generate new labels using the progression al-gorithm.Furthermore,say we?nd a sequence of states,

,starting at state that satis?es’s label. Then a simple induction using Proposition3shows that the sequence leading from to followed by the sequence, i.e.,,satis?es.The progression al-gorithm keeps the labels up to date:they specify what we are looking for given that we have arrived where we are. From this insight we can identify two important features of the formula progression mechanism.First,if we?nd any state whose idling satis?es its label,we have found a correct plan.

Proposition4Let be a sequence of states generated by forward chaining from the initial state to. For each state let its label be.Let the labels of the states be computed via progression,i.e.,for each state in the sequence

Progress

Inputs:A state,and a formula.

Output:True if the state sequence,where time increases without bound,satis?es.False otherwise.

Algorithm Idle(,)

Case

1.contains no temporal modalities:

if return TRUE

else return FALSE

2.:return Idle Idle

3.:return Idle

4.:return Idle

5.U:

if return FALSE

else if return Idle

else return Idle Idle

6.:return Idle

7.:return Idle

Table2:The idling algorithm.

Then iff

.

The proof of this proposition follows directly from Propo-sition3.

Since is a formula specifying the goal,this propo-sition shows that the plan leading to satis?es the goal. Hence,if we have a method for testing for any state and any formula MITL whether or not, we have a termination test for the planning algorithm that guarantees soundness of the algorithm.We will describe an appropriate method below.

Furthermore,as long as the search procedure used by the algorithm eventually examines all?nite sequences of states the planning algorithm will also be complete.

The second feature of formula progression is that it allows us to prune the search space without losing completeness.As we compute the progressed label we simplify it by processing all TRUE and FALSE subformulas.For example,if the label TRUE is generated we simplify this to.If any state receives the label FALSE we can prune it from the search space,thus avoiding searching any of its successors.From Proposition3we know that this label speci?es a requirement on the sequences that start at this state.No sequence can satisfy the requirement FALSE,hence no sequences starting from this state can satisfy the goal and this state and its successors can be safely pruned from the search space. Termination As indicated above,we can detect when a plan satis?es the goal if we can detect when an idling state satis?es its label.This computation is accomplished by the algorithm given in Table2.

Proposition5Idle returns TRUE if and only if .That is,Idle detects if an idling state satis?es a formula.

The Planning Algorithm Given the pieces developed in the previous sections we specify the planning algorithm pre-sented in Table3.The algorithm labels the initial state with the goal and searches among the space of state-formula pairs. We test for termination by running the Idle algorithm on the Inputs:An initial state,and a sentence MITL specifying the goal.

Returns:A plan consisting of?nite sequence of actions. Algorithm Plan(,)

1.Open.

2.While Open is not empty.

2.1Remove an element of Open.

2.2if Idle Return().

2.3Successors Expand().

2.4For all Successors

2.4.1Progress duration.

2.4.2if FALSE

2.4.2.1Parent().

2.4.2.2Open Open.

Table3:The planning algorithm.

state’s formula.To expand a state-formula pair we apply all applicable actions to its state component,returning all pairs containing a successor state and the action that produced that state(this is accomplished by Expand()).We then compute the new labels for those successor states using the Progress algorithm.

It should be noted that we cannot treat action sequences that visit the same state as being cyclic.If we are only looking for a path to a?nal state,as in classical planning,we could eliminate such cycles.Goals in MITL,however,can easily require visiting the same state many times.Nevertheless, we can view visiting the same state-formula pair as a cycle, and optimize those cycles using the standard techniques. Intuitively,when we visit the same state-formula node we have arrived at a point in the search were we are searching for the same set of extensions to the same state. Proposition6The planning algorithm is sound and com-plete.That is,it produces a plan that is correct for given (De?nition2),and so long as nodes are selected from Open in such a manner that every node is eventually selected,it will?nd a correct plan if one exists.

This proposition follows from the soundness of our termina-tion test(Proposition4).

We have implemented the planning algorithm as an ex-tension of the TLP LAN system[Bac95].This allowed us to utilize many of the features already built into the TLP LAN system,including full support of the A DL formalism[Ped89] for specifying actions.

Example and Empirical Results

Types of Goals

The domain we used is a variant of the classical S TRIPS robot rooms domain[FN71].The con?guration of the rooms is illustrated in Figure1.In this domain there are objects and a robot,which can be located at any of the2locations in the corridor,or,or any of the4rooms. The robot can move between connected locations,it can

Precondition Deletes

open opened

close closed

grasp holding

release handempty

move at

holding

at

We must appropriately rewrite in terms of bounded

quanti?cation.Also it is not dif?cult to see that multiple variables

in can be handled by additional quanti?ers.Similar remarks hold

for encoding restore.

in the initial state.Notice that the semantic distinction between restore and don’t-disturb goals is made clear by our formalism.Restore goals use while don’t-disturb goals use.That is,restore goals allow the violation of during the plan,as long as these conditions are eventually restored in the?nal state.

Both of these conditions are limited special cases.MITL can express much more than this.For example,say that we want to constrain the robot to close doors that it opens. We cannot place a don’t-disturb condition closed,as this would prohibit the robot from moving into rooms where the doors are closed.If we specify this as a restore condition, the robot might leave a door opened for a very long time until it has?nished the rest of its plan.In MITL,however we can use the formula

connects

(1)

at closed open

at closed

This formula speci?es that if the robot opens a closed door(closed open),then it must go through the door(at)and then it must close the door (closed).Hence,the robot is forced to be tidy with respect to doors:it only opens doors for the purpose of mov-ing through them,and it closes the doors it opens behind it.

Timing Deadlines:MITL is also capable of expressing goals with timing conditions.For example requires the condition be achieved within ten time units. Empirical Results

We have tested different goals from each of the cate-gories mentioned above.Most of the plans were generated from the initial state in which at obj1,at obj2, at robot,handempty,object obj1,object obj2,and all of the doors are opened.

G1:From this initial state we set the goal to be at robot at obj1.This corre-sponds to the classical goal at robot at obj1. The planner generates the plan:move, grasp obj1,move,release obj1,move, move.It took the planner22sec.,expanding636 worlds to?nd this plan.

G2:From the same initial state we set the goal to be object at handempty.Now the plan-ner generates the plan:move,move, grasp,move,release.In this case it has generated a plan for a quanti?ed goal.This plan takes the planner3sec.,expanding126worlds to?nd the plan.

that has been able to produce polynomial time behavior in these domains.

As a simple illustration of the power of this using search control consider the following trivial search control formula: at at at

object holding holding

holding

If we conjoin this formula with any other goal,the planner will prune sequences in which(1)the robot grasps an object and then immediately releases it,and(2)the robot moves away from a location and then immediately moves back.For this domain these sequences serve no purpose even in plans where the robot must visit the same state more than once. Conjoining this formula with the example goals given above we obtain the following speedups.

Time New-Time

2212

212693

7718

428438

1207.75

The columns give the planning time and the number of worlds expanded,before and after we add the search control formula. Note in particular,the speedups obtained on the harder prob-lems.Furthermore,it should be noted that this is only the simplest and most obvious of control formulas for this do-main.

References

[AFH91]Rajeev Alur,Tomas Feder,and Thomas Henzinger.

The bene?ts of relaxing punctuality.In Tenth Annual

ACM Symposium on Principles of Distributed Comput-

ing(PODC1991),pages139–152,1991.

[AKRT91]J.Allen,H.Kautz,Pelavin R.,and J.Tenenberg.Rea-soning about Plans.Morgan Kaufmann,San Mateo,

CA,1991.

[Bac95]Fahiem Bacchus.Tlplan(version2.0)user’s manual.

Available via

the URL ftp://logos.uwaterloo.ca:/pub/bacchus/tlplan-

manual.ps.Z,1995.

[BK95]Fahiem Bacchus and Froduald https://www.doczj.com/doc/498543096.html,ing tem-poral logic to control search in a forward chaining

planner.In Proceedings of the3rd European Work-

shop on Planning,1995.Available via the URL

ftp://logos.uwaterloo.ca:/pub/tlplan/tlplan.ps.Z. [BKSD95]M.Barbeau,F.Kabanza,and R.St-Denis.Synthesizing plant controllers using real-time goals.In Proc.Thir-

teenth International Joint Conference on Arti?cial In-

telligence(IJCAI’95),pages791–798,1995.

[CT91]K.Currie and A.Tate.O-plan:the open planning architecture.Arti?cial Intelligence,52:49–86,1991.

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