Numerical Kernels for Monitoring and Repairing Plans Involving

Continuous and Consumable Resources

Enrico Scala

Dipartimento di Informatica, Universita’ di Torino, Via Pessinetto 117, Turin, Italy

Keywords:

Planning, On-line Planning, Repair, Monitoring, Plan Execution, Reactive AI.

Abstract:

In this paper we introduce a technique for monitoring and repairing a plan dealing with continuous and con-

sumable resources. The mechanism relies on the notion of numerical kernel. Concretely, a numerical kernel

establishes the sufﬁcient and necessary conditions for a plan to be valid in a speciﬁc state of the system. We

employ the mechanism in a continual planning agent and we evaluate experimentally the approach for the

Zenotravel domain. Results show good cpu-time w.r.t. a traditional replanning from scratch.

1 INTRODUCTION

In the last decade of research in AI, an increas-

ing amount of work has been devoted in extending

the automated planning (and especially the classi-

cal paradigm) in dealing with real world problems

((Hoffmann, 2003), (Gerevini et al., 2008)). One of

the most limitations in the classical setting is the lack

in the management of consumable and continuous re-

sources. In this perspective, the concept of numeric

planning has been introduced (Fox and Long, 2003).

However, while many efforts have been success-

fully devoted for the problem of off-line plan genera-

tion, just a little amount of work has been addressed

in studying the plan of actions for the on-line phase,

making exception for the works dealing with the tem-

poral dimension. In this context models as STP (Sim-

ple Temporal Problem) and DTP (Disjunctive Tem-

poral Problem) have been extensively adopted and

some extensions have been proposed ((Conrad and

Williams, 2011), (Policella et al., 2009), (Stergiou

and Koubarakis, 2000)).

The main contribution of this paper is a technique

for monitoring and repairing a plan involving contin-

uous and consumable resources modeled as numeric

state variables. The technique grounds on the notion

of numerical kernel, which is a set of inequality con-

ditions allowing to directly (without performing any

search and any propagation) assess the sufﬁcient and

necessary requirements that must be guaranteed in a

particular state of the system to be consistent with the

plan at hand. To demonstrate the utility of the tech-

nique, we propose a continual planning agent which

is able to deal with unexpected resource consump-

tion. To validate our approach, we experimented the

mechanism in two versions of the Zenotravel domain,

namely a well known test-bed developed by the plan-

ning community as a benchmark for testing the per-

formance of planners.

2 BACKGROUND

In the following we introduce the planning framework

our approach applies

. The following deﬁnitions ex-

tend the traditional STRIPS formalism analogously

to numeric expressions reported in (Fox and Long,

2003).Moreover, since the plan is the main object of

study for the monitoring and repair problem, we will

consider fully instantiated actions, as a plan typically

involve speciﬁc instances of actions instead of their

schema.

System State. To model resources our state of the

system extends the classical formulation with a vec-

tor of real values N which maps each resource in a

continuous value. Moreover, also other numeric con-

tinuous information can be modeled (e.g. the distance

among two sites).

Model of Actions. Since continuous resources can

be consumed and required by actions, the classical

STRIPS setting is not sufﬁcient. To this end, tran-

sitions in our system are represented by means of nu-

meric action. That is:

We assume the reader is familiar to the planning for-

malism.

531

Scala E..

Numerical Kernels for Monitoring and Repairing Plans Involving Continuous and Consumable Resources.

DOI: 10.5220/0004260205310534

In Proceedings of the 5th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2013), pages 531-534

ISBN: 978-989-8565-39-6

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

Deﬁnition 1 (Numeric Action). A numeric action is

a STRIPS action in which propositional preconditions

and effects are augmented with:

• numeric precondition, which is a conjunction of

comparisons, where each comparison is an in-

equality of the form { exp {<,≤,=,≥,>} exp’}.

• numeric effects, which consists of a set of opera-

tions, where each operation is deﬁned by means

of { f,{+ =,− =,=},exp}, and f is the resource

affected by the operation, i.e. f ∈ N

In the deﬁnition above exp,exp

′

are arith-

metic expressions involving a variety of resources

and numeric information from N. Informally,

the numeric operations model how a numeric ac-

tion affects the resources (e.g., power = power +

distance(A,B)/avg consumption), while the numeric

precondition expresses the resource requirements for

the action applicability (e.g., power > 5.0). The nu-

meric action is applicable in a state S if both the nu-

meric and classical preconditions are satisﬁed in S.

The application of this action in a state S will trans-

form S in S

′

according to (i) the operations deﬁned in

the numeric effects, and (ii) the traditional add/delete

list reported in the propositional effects.

The Plan. By focusing on the numeric part of the

state we can deﬁne a numeric plan as follows:

Deﬁnition 2 (Numeric Plan). A numeric plan is a to-

tal ordered set of numeric actions that, starting from a

state I, leads the agent in a state satisfying a goal G,

where: (i) I is a numeric state, i.e., a vector assign-

ing values to the numeric information of the domain,

(ii) G is the numerical goal for the agent containing

a conjunction of comparisons as the ones deﬁned for

the action preconditions.

3 MONITORING AND REPAIR

VIA NUMERICAL KERNEL

While the planning phase generally assumes deter-

ministic state transitions, the execution of the plan in

the real world can be prevented because of the pres-

ence of exogenous events as well as unexpected ac-

tion behaviors. Therefore a continuous monitoring is

often desirable for understanding if the observations

(i.e. the state) are consistent with the plan at hand.

In this context, the only checking of action pre-

conditions may, in general, does not sufﬁce. Albeit

the action can result applicable, the executability of

the rest part of the plan could depend by only certain

resources proﬁles. To capture such proﬁles, it is im-

portant to keep trace to the conditions expressed in the

goals and the model of actions present in the plan.

3.1 Numerical Kernel

Given a numeric plan π as the one in Deﬁnition 2, it

is possible to ﬁnd the set of conditions that must be

guaranteed for achieving G by means of π. We will

this set numerical kernel. More formally, from (Scala,

2012):

Deﬁnition 3 (Numerical Kernel). Given a plan π, a

goal G and a state S, a set of comparisons K over N

is said to be a numerical kernel for π and G when the

state S[π] satisﬁes G iff S satisﬁes K.

where S[π] is the state obtained from the recursive ex-

ecution of π starting from S. Thus, given a state of the

system, by verifying the condition expressed in the

numerical kernel, one can infer if the state is compat-

ible with the remaining plan for reaching the goal.

More precisely it is possible to ﬁnd a numerical

kernel for each step of the plan by backward propaga-

tion starting from G. That is,

the computation starts with the last numerical ker-

nel as it corresponds to the (trivial) numerical kernel

for an empty plan, i.e. the goal conditions. After

which, the remaining part of the kernels is constructed

by combining the information involved in the model

of the action (pre(a) and eff(a)) with the (previously

computed) next numerical kernel. In particular it is

necessary to iteratively combine the set of compar-

isons with the set of assignments listed in the opera-

tions set. Note indeed that every operation listed in

the model of the action can be transformed in an as-

signment operation. The procedure and the notion of

numerical kernels are introduced and described in de-

tail in (Scala, 2012).

Triangle table deﬁned in (Fikes et al., 1972) have

been introduced for a similar motivations by introduc-

ing the concept of propositional kernel. However, the

formulation of a kernel expresses precisely what is the

minimal set of propositionalatoms that must hold dur-

ing the plan execution. Here, the numerical kernel ex-

presses just the boundary of a state space to be valid

without specifying a particular state.

3.2 Monitoring and Repairing

To validate and motivate the utility of numerical ker-

nels, we developed a continual planning agent which

is in charge of dealing with continuousresources. The

continual planning paradigm (desJardins et al., 1999)

allows the agent to interleave the execution and the

planning for the purpose of repairing its plan of ac-

tions once unexpected conditions (in our case the ker-

nel violation) occur. Hence the usage of numerical

kernel is rather appropriate as the agent performs both

monitoring and repair, throughout the plan execution.

ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence

532

In order to handle plans involving both proposi-

tional and numerical aspects, we combined the notion

of the numerical kernel deﬁned in this paper with the

propositional kernels deﬁned for classical planning.

Algorithm 1: Monitoring and Repair.

Input: P: plan of numerical actions G: goal

Output: Success or Failure

K = KernelsComputation(plan,G)

i = 0

while plan is not empty do

SenseAndUpdate(S)

if S satisﬁes K[i] then

action = head-remove(plan)

execute(action)

i++

else

patch = repair(S,K[i])

if patch != null then

plan = plan.concat(patch)

else

plan = replan(S,G)

K = KernelsComputation(plan,G)

i=0

SenseAndUpdate(S)

if S satisﬁes G then

return Success

else

return Failure

The algorithm above reports the strategy of the

agent. It is quite similar to the main algorithm for the

continual planning reported in (Brenner and Nebel,

2009), making exception that here the agent has to

deal with continuous resources. For this reason, this

strategy includes an explicit way of repair, and a more

focused way to perform the monitoring.

The algorithm starts by computing the proposi-

tional and numerical kernel; then it retrieves the ker-

nel for this part of the plan and compares the condi-

tions against the current observed state of the system.

If both conditions are veriﬁed, namely the proposi-

tional and the numerical kernel, it extracts and exe-

cutes the next action from the plan; otherwise it looks

for a plan towards a state in which both propositional

kernel and numerical kernels are satisﬁed. The idea is

to re-establish the necessary conditions for the appli-

cation of the rest part of the plan.

Since the approach is not complete as the repair

mechanism is not allowed to search over all the search

space (the one generated from the current state to-

wards the goal), the agent can switch to replanning

from scratch when the patch via repair is not found.

As a common continual planning approach the algo-

rithm terminates once the plan is ﬁnished and the goal

reached. The strategy supports the most of the level

of expressiveness of PDDL 2.1 level 2, with the lim-

itation that propositional and numeric preconditions

must be conjunctively separated.

4 EXPERIMENTAL RESULTS

To verify the feasibility of the repair mechanism pre-

sented in the previous Section, we applied the strat-

egy to the ZenoTravel domain, which is a domain in-

troduced by the planning community for the Interna-

tional Planning Competition (IPC3

To make this domain more challenging, we pro-

posed an extended version in which refuel is possible

only in certain city. To obtain this we modiﬁed the

refuel action precondition for allowing its execution

only in city in which it is explictly stated the pres-

ence of the refuel station. This means that a plane has

to carefully choose the paths for moving persons all

around. We performed experiments in both domains,

that will be called the normal and hard domain.

Starting from the suite of numeric problems gen-

erated for the competition and for the purpose of vali-

dating our approach, we injected discrepancies on the

way in which the fuel is consumed. We focus on the

7 most difﬁcult cases where we injected 5 different

amounts of noise, which consequently has produced

different instances of repair problems. The total num-

ber of experiments is 35 cases for each domain. For

each case we measured the performance of the ap-

proach in terms of cpu-time spent

Figure 1 and 2 summarize the results for two dif-

ferent strategies. The one (line with rhombuses) is

the repair strategy (we turned off the replanning from

scratch), the other one (line with squares) is the re-

planning from scratch methodology (we turned off the

repair mechanism). Both strategies have been experi-

mented for the normal and the hard domain.

More precisely, the ﬁgures measure the computa-

tional effort for solving the impasse. In case of re-

pair, the solution corresponds to the bridge towards

the kernel, whereas for the replanning from scratch

the obtained new plan links the current state directly

with the goal. The x-axis enumerates the 7 cases con-

sidered whereas the y-axis the computational effort.

It is quite clear that for this class of problems, the

repair mechanism outperforms the replanning from

scratch in all cases we tested, and it is true even when

dealing with hard cases. In this latter cases the re-

planning performs rather bad since the performance

degrades rapidly. For instance in cases number 6 the

http://planning.cis.strath.ac.uk/competition/

Tests ran on an Intel Core Duo (1.66 Ghz) with 2 GB

of Ram. The implementation is in Java 1.6 and the planner

used is metric-ff (Hoffmann, 2003)

NumericalKernelsforMonitoringandRepairingPlansInvolvingContinuousandConsumableResources

533

Figure 1: ZenoTravel Domain (Normal); Cpu time.

Figure 2: ZenoTravel Domain (Hard); Cpu time.

replanning from scratch found a solution only after 30

seconds of cpu-time.

5 CONCLUSIONS

The paper proposes a new technique for monitoring

and repairing plans with actions dealing with contin-

uous and consumable resources. The technique ex-

ploits the notion of numerical kernel developed in

(Scala, 2012) accompanied to the well known propo-

sitional kernel. The strategy allows to focus the agent

just on a speciﬁc portion of the state which is the only

interesting for the monitoring and repair problem.

The repair via plan adaptation is a very attractive

area of research in AI since, despite opposite formal

complexity results (Nebel and Koehler, 1995), the re-

pair have been proved to be a viable solution in prac-

tice instead of a replanning from scratch. However

previous works have been largely applied just for the

classical paradigm (Gerevini and Serina, 2010).

For this reason, we applied the numerical kernel

notion for a repair problem via plan adaptation, and

we studied the performance of the system against a

blind replanning from scratch. Tests have been per-

formed on a well known benchmark domain deﬁned

by the planning community. Results showed that,

when the repair has to face unexpected resources con-

sumption, a focused repair may work very well w.r.t. a

replanning mechanism. As an immediate future work

we would like to test the developed repair strategy

on a larger set of domains, to understand the gener-

ality of the repair approach. Moreover, we would like

to study a more sophisticated way in the selection of

the kernels towards which perform the patch; in this

ﬁrst version in fact we employ a conservative setting

in which the repair is performed directly towards the

state that was expected. Moreover it would be inter-

esting to exploit the numerical kernel in the general

context of Case Based Planning.

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