Birch, Alexandra

Patron, Pedro

Plans executed in the real world have to adapt to the dierences between the expected and the observed state of the environment where is being executed. Plan- ning strategies dealing with this adaptation process can be compared by looking at the plan proximity between the original plan and the plans that these strategies generate to replace it. Plan proximity measures the dierence between these sequential plans and the ex- pected outcome states of these plans. We present ar- guments to support the claim that plan proximity is a more informed metric than plan stability in order to compare planning strategies solving these adaptation to the environment of execution

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     Edinburgh Research Explorer                                      Plan proximity: an enhanced metric for plan stabilityCitation for published version:Patron, P & Birch, A 2009, Plan proximity: an enhanced metric for plan stability. in Workshop on Verificationand Validation of Planning and Scheduling Systems, 19th International Conference on Automated Planningand Scheduling (ICAPS’09). pp. 74-75.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Publisher's PDF, also known as Version of recordPublished In:Workshop on Verification and Validation of Planning and Scheduling Systems, 19th International Conference onAutomated Planning and Scheduling (ICAPS’09)General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 05. Apr. 2019Plan proximity: an enhanced metric for plan stabilityPedro Patro´nOcean Systems LaboratoryHeriot-Watt University, UKp.patron@hw.ac.ukAlexandra BirchSchool of InformaticsUniversity of Edinburgh, UKa.c.birch-mayne@sms.ed.ac.ukAbstractPlans executed in the real world have to adapt tothe differences between the expected and the observedstate of the environment where is being executed. Plan-ning strategies dealing with this adaptation process canbe compared by looking at the plan proximity betweenthe original plan and the plans that these strategiesgenerate to replace it. Plan proximity measures thedifference between these sequential plans and the ex-pected outcome states of these plans. We present ar-guments to support the claim that plan proximity is amore informed metric than plan stability in order tocompare planning strategies solving these adaptationto the environment of execution.IntroductionPlan adaptation can be performed using differentstrategies. Different plan repair and re-planning tech-niques are being used in order to deal with a changingdynamic environment of execution.Plan stability was proposed as a metric for comparingtwo plans generated with different planning strategiessolving this adaptation process (Fox et al. 2006). Thismeasure considered the missing actions from the refer-ence and the extra actions on the test plan. However, itwas relaxed on the ordering of these actions in the planand it did not considered the outcomes of these plans.In this paper we propose an extension of the plan sta-bility measure that takes these two factors into accountproviding an enhanced vision of the difference betweentwo plans. Our metric looks at the combined plan differ-ences and the expected outcome state differences. Forboth cases, plan and outcome state, it takes the missingitems from the reference and the extra items appearingin the test into account.Plan DifferenceDefinition 1 A plan pi is an ordered list of actions oflength n that is expected to transform the state of theenvironment of execution from an initial state I to afinal state G.Copyright c© 2009, Association for the Advancement of Ar-tificial Intelligence (www.aaai.org). All rights reserved.A reference plan pi1 and a test plan pi2 can be com-pared by looking at the different number of actions thatthey contain. However, the point at which these actionsare performed is important during plan execution as itcan drastically change the outcome of the plan. Order-ing is, therefore, important when comparing two plans.The ‘diff’ algorithm (Hunt and McIlroy 1976) solvesthe longest common subsequence problem, which iscommonly used to find the lines that do not changebetween two files. This can be applied to comparetwo plans, finding the identically ordered items thatare present between two plans. In Table 1 we cansee an example of a reference and test plan with theirlongest common subsequence shown with aligned com-mon items. The action missing from the test sequence,and the extra actions that the test sequence contains,are also shown.Ref Test Missing ExtraY YA AB BC CZ ZD DTable 1: A sequence of actions for a reference (A BC D) and a test (Y A C Z) plan, with their longestsubsequence (A C) shown aligned, the missing actions(B D) and the extra actions (Y Z).Definition 2 Given a reference plan pi1 and a testplan pi2, the difference between pi1 and pi2, Dp(pi1, pi2)is the number of missing actions mp from the referenceplan pi1 and the number of extra actions ep from a testplan pi2 that do not appear in the longest common sub-sequence of actions.Dp(pi1, pi2) = mp + epThe plan difference is normalized using the sum ofthe number of actions of the reference plan n1 and thetest plan n2.Dˆp(pi1, pi2) =Dp(pi1, pi2)n1 + n2∈ [0; 1]74The Dˆp for Table 1 is 2 + 2/4 + 4 = 0.5. Table 2shows an example where plan stability and plan dif-ference provide different interpretations. The plan sta-bility measure would consider these two plans to beidentical, as they contain the same actions. The plandifference measure, however, takes the ordering of theactions into account and returns a result of 0.5.Ref Test Missing ExtraA AB BA ATable 2: A sequence of actions for a reference (A B)and a test (B A) plan, with their longest subsequence(B), the missing actions (A) and the extra actions (A).State DifferenceDefinition 3 A state s is a set of proposition factsdescribing a configuration of the environment of execu-tion. Proposition facts are propositions of the domainmodel whose variables have been all grounded to objectsin the environment of execution.The ordering of these propositions does not changethe state. If m is the total number of proposition factsin the domain of execution, a state can be representedas a binary string of length m containing 1’s on theposition of the string that corresponds to the proposi-tion facts that its configuration represents. From theserepresentation, the difference between two states canbe calculated using the Hamming distance (Hamming1950). The Hamming distance between two strings ofequal length is the number of positions for which thecorresponding symbols are different.Definition 4 Given a reference state s1 and a teststate s2 for a common environment of execution, thestate difference can be calculated as the Hamming dis-tance between the string representation of s1 and s2.Ds(s1, s2) =m∑i=1xi where xi ={0 if s1(i) = s2(i)1 otherwiseThe state difference is normalized using the stringlength m.Dˆs(s1, s2) =Ds(s1, s2)m∈ [0; 1]Plan ProximityThe proximity between a reference plan and a test planis the combination of the plan difference and the esti-mated final state difference.Definition 5 Given a reference plan pi1 and a testplan pi2, their plan proximity PP (pi1, pi2) is the normal-ized balanced sum of the plan difference Dp(pi1, pi2) andthe state difference of the estimated final states that theyare expected to produce Ds(G1, G2).PPα(pi1, pi2) = 1 −α · Dˆp(pi1, pi2)−(1− α) · Dˆs(G1, G2) ∈ [0; 1]where α ∈ [0; 1] represents a balance factor betweenplan and state difference. The logical default value of αwould be 0.5. However, it will depend on the particu-lar evaluation of plans for the different environments ofexecution. Plan proximity can be interpret as the per-centage of similarity between two plans. It satisfies thedistance function properties. In consequence, togetherwith the set of possible plans for an environment of ex-ecution, makes up a metric space.Dynamic Planning ProblemsThe dynamic planning problem represents the problemof adapting plans in a dynamic changing environmentof execution.Definition 6 A dynamic planning problem is a tuple(I,G, pi0, I ′, G′) where pi0 is a plan that achieves the goalG from the initial state I and I ′ is a new initial statefrom which the new goal state, G′ , must be achieved.The Plan Proximity metric space can be used to com-pare different planning strategies aiming to solve thedynamic planning problem. Plans coming out of thesestrategies can be compared and verified against an orig-inal or reference plan (Patro´n, Lane, and Petillot 2009).Definition 7 Given a dynamic planning problem(I,G, pi0, I ′, G′), a planning strategy, P1 , achievesgreater plan proximity than a planning strategy, P2 ,if the plans produced by P1 and P2 , pi1 and pi2 respec-tively, satisfy PP (pi0, pi2) < PP (pi0, pi1).ConclusionThis paper presents a novel approach towards compar-ing different planning strategies solving the dynamicplanning problem. We use the term ‘plan proximity’to refer to a measure of the similarity between a refer-ence plan and a test plan. This distance-based metricconsiders actions missing from the reference plan, ex-tra actions added in the test plan, sequential orderingof the plans and the expected outcomes states of theseplans. By taking all these factors into account, it canbe claimed that plan proximity is more informed thanplan stability in providing an enhanced view of compar-ing planning strategies in a dynamic environment.ReferencesFox, M.; Gerevini, A.; Long, D.; and Serina, I. 2006.Plan stability: Replanning versus plan repair. In Int.Conf. on Automated Planning and Scheduling.Hamming, R. W. 1950. Error detecting and errorcorrecting codes. Bell System Technical Journal 292:147–160.Hunt, J. W., and McIlroy, M. D. 1976. An algo-rithm for differential file comparison. Computing Sci-ence Technical Report, Bell Laboratories 41.Patro´n, P.; Lane, D.; and Petillot, Y. 2009. Continu-ous mission plan adaptation for autonomous vehicles:balancing effort and reward. In 4th Workshop on Plan-ning and Plan Execution for Real-World Systems, 19thInt. Conf. on Automated Planning and Scheduling.75

Plan proximity: an enhanced metric for plan stability

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Plan proximity: an enhanced metric for plan stability

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