Modeling Deontic States in Petri Nets* - Euridis

Modeling Deontic States in Petri Nets *Jean-Francois Raskin + Yao-Hua Tan ++ Leendert W.N. van der Torre ++AbstractPetri nets can be used to model and analyze bureaucratic procedures. Obligations and permissions areimportant aspects of bureaucratic procedures, hence these deontic aspects should also be represented inthese Petri nets. In this paper we show how to represent deontic states in Petri nets. These states can berepresented by adding a preference relation to Petri nets. This preference relation can be seen as aninstrument to restrict undesirable behavior that is represented in the Petri net. A rational person issupposed to minimize the total sum of its penalties and therefore not to perform sub-ideal behavior. Thepreference relation introduced in this paper can represent so-called Contrary-To-Duty obligations in Petrinets, and is closely related to the preference ordering in the diagnostic framework for deontic reasoningDIODE.1 IntroductionPetri nets [PET81] are a popular graphical tool for the modeling and analysis of discrete dynamicsystems. These nets are appropriate for modeling distributed systems, because they can be used torepresent parallelism and synchronization. Therefore, Petri nets are very suitable to modelprocedures and processes within and between organizations. Several Petri net based modeling toolshave been developed. For example, Van der Aalst developed the ITCPN model to model logisticprocesses in organizations (see [AAL92]). Lee developed the CASE/EDI tool to model bureaucraticprocedures (see [LEE91, LEE92]). This tool has been applied to model inter organizationalprocedures in international trade such as contract negotiation, the exchange of bill-of-lading forletter-of-credit, custom clearance etc. (see [BLW95]). CASE/EDI can also be used to dynamicallysimulate Petri nets. Hence, it can also be used to check the procedures represented in a Petri net forconsistency and dead-locks. The modeling of bureaucratic procedures is quite different from themodeling of logistic processes, because in bureaucratic procedures deontic aspects such asobligations, prohibitions and permissions play an important role. For example, if a contract isrepresented in a Petri net, the net should represent that if goods have been delivered by the seller,then subsequently the buyer is obliged to pay the bill for the goods. Also the net should be capableof representing that if the buyer does not pay, then the buyer is obliged to return the goods. Thissecond obligation, to return the goods, is conditional on the violation of the first obligation to pay thedelivered goods. Such an obligation, which is evoked when another obligation is violated, is called a* This research was partially sponsored by the ESPRIT III Basic Research Project No. 6156 DRUMS II and theESPRIT III Basic Research Working groups No. 8319 MODELAGE.+ Computer Science Institute of the Facultes Universitaires Notre-Dame de la Paix, Rue du Grandgagnage21, B5000 Namur, Belgium. E-Mail: jfr@info.fundp.ac.be++ Erasmus University Research Institute for Decision and Information Systems (EURIDIS). ErasmusUniversity Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands. E-Mail:{ytan,ltorre}@euridis.fbk.eur.nl. Tel: (+31)10-4082601. Fax:(+31)10-4526134.Http://www.euridis.fbk.eur.nl/Euridis/welcome.html.

contrary-to-duty (CTD) obligation. CTD obligations tell you what you should do, given that youalready have violated an obligation. We say that a behavior by which no obligations are violated isideal, and when there are obligations violated, we call behavior sub-ideal. Clearly, CTD obligationsapply to sub-ideal behavior only.The traditional formalism for modeling deontic reasoning is deontic logic, which is a modal logicwhere the modal operator O(p) means that p is obliged. It was already observed in [CHI63] thatCTD obligations can lead to paradoxes in deontic logics. These paradoxes are due to the fact that indeontic logic it is very difficult to model the distinction between ideal versus varying sub-idealbehavior. Recently, it was discovered that this distinction can be modeled if we add techniques fromnon-monotonic logic to deontic logic, in particular preferential semantics (see [JP85, HOR93,PS94]). In [TT94a, TT94b] a diagnostic framework for deontic reasoning (DIODE) was introducedin which a preferential semantics is proposed to model sub-ideal states.In this paper we show how to represent the deontic notions of ideal and sub-ideal behavior in Petrinets. We extend standard Petri nets with a preference relation on its possible executions. Thispreference ordering is analogous to the preference ordering that was introduced in DIODE.2 The Petri net formalismA Petri net is a directed graph which consists of two disjunct sets of nodes. These nodes are calledplaces (represented as circles) and transitions (represented as bars). Places and transitions areconnected by arcs. It is not allowed to connect two places or two transitions. Arcs can have a valuewhich indicates how many tokens are required to fire a transition. However, in the examples in thispaper we only consider arcs that require exactly one token. The dynamic behavior of the modeledsystem is represented by tokens flowing through the net. A token is represented by a dot. Each placemay contain several tokens (the so-called marking of the place). A transition is enabled if all its inputplaces (i.e., an arc exists from the place to the transition) contain the specified number of tokens,which corresponds to the value of the arc. A transition may fire whenever it is enabled. Whenever atransition is fired, it has the effect that the specified number of tokens of its input places is removedand at the same time the specified number of tokens is added to its output places. Note that we say"may" in the previous sentence, because when two transitions are enabled at the same marking, wehave the choice to fire one of the two, and the second will not necessary still be enabled in themarking which results from firing the first transition. In this way, choice is modeled in Petri nets. Toillustrate the distinction between ideal versus sub-ideal behavior, consider the following Petri net:2

p1t1p2t3t5t2t6t4p3Places: p1: borrowed book; p2: damaged book; p3: returned book;Transitions: t1: to damage the book; t2: to repair the book; t3: 1 week too late;t4: 1 week too late; t5: to return the book; t6: to return the book.Figure 1Figure 1 models the possible behaviors of a borrower. A marking for this net is denoted by a tuplen 1,n 2,n 3where n iindicates the number of tokens at place p i . In the initial marking 1, 0, 0displayed in the figure, there is only a token at place p 1, which represents that the borrower has abook. At this marking he has the choice to return the book, returning it too late, or to damage thebook. If he decides to be too late, he has again the same choices. If he damages the book he has thechoice to repair it before returning it, or to return it damaged, he may also be even later. In the rest ofthis paper, we will consider the executions of this Petri net which start in marking 1, 0, 0 and resultin marking 0,0,1 , which represents that the book is returned. When executing this Petri net, thereis the choice to perform ideal behaviors or sub-ideal behaviors. For example, if the borrower returnsthe book in time and not damaged, we can say that he performs the ideal behavior. On the otherhand, if he returns the book one week too late, he does not perform the ideal behavior. Thedistinction between ideal and sub-ideal cannot be represented in standard Petri nets as Figure 1. Onecan model the choice, but nothing in the Petri net formalism indicates that the one execution is betterthan the other. 1 In this paper we show how the standard Petri net formalism can be extended with apreference relation such that it can represent this distinction. In the set of all possible executions of asystem, intuitively, the preferred ones are those which contain a minimum of sub-ideal behaviors. Inother words, those executions should be preferred that contain as few sub-ideal transitions aspossible. We first start with the basic definitions of the Petri net formalism, see [PET81, GEN86,AAL92] for the details. A Petri net is a kind of directed labeled graph, in which the places areconnected by transitions.1 In some extensions of Petri nets preferences are added to the transitions, but the preferred transition hasto fire. Hence, in these approaches the notion of choice is not modeled.3

Definition 1 (Petri net) A Petri net structure N = P,T, Pre, Post consists of two disjunct nonemptyfinite sets of places P and transitions T, and two functions from P × T to the set of positiveinteger numbers ℵ: Pre:P × T →ℵ and Post:P × T →ℵ.The state of the graph is called a marking.Definition 2 (Marking) Let N = P,T,Pre, Post be a Petri net. A marking M:P →ℵ is anassignment of tokens to the places of a Petri net. The marking M can also be written as a vectorM = M 1, M 2,...,M nwhere n = P and each M i∈ℵ, i=1,...,n. We write N, M for a Petri net Nwith marking M.The marking of a graph determines which transitions are enabled.Definition 3 (Enabled transition) Let N = P,T,Pre, Post be a Petri net with marking M,M(p) the number of tokens contained in p ∈P and t ∈T a transition. The transition t is enabled inN, M if and only if (iff) ∀p ∈P:M( p)≥ Pre( p,t).The dynamic behavior of a Petri net is expressed by changing markings. A marking changes when atransition fires, and a transition may fire when it is enabled.Definition 4 (Firing a transition) Let N = P,T,Pre, Post be a Petri net with marking M,M(p) the number of tokens contained in p ∈P and t ∈T an enabled transition. Firing the transition tin N,M results in a new marking M , written as M|t > M, given byM(p i) = M(p i)− Pre(p i,t) + Post(p i,t). Hence, Pre( p i,t) denotes the number of tokens needed inp ifor the firing of transition t and Post(p i,t) denotes the number of tokens added to place p iwhentransition t has fired.The procedural semantics of a marked Petri net is given by the set of its possible executions(sequences of transitions) between two markings. We give a recursive definition:Definition 5 (Execution) Let N = P,T,Pre, Post be a Petri net with marking M, s= s 1,...,s mwith s i∈T a finite sequence of transitions of T, and T* the set of all finite sequences that can becomposed from transitions of T. The sequence s is an execution of the marked net N, M iff it isfireable in the marked net N, M , resulting in some marking M , written as M|s > M. M|s > M iff:1. either s = λ (the empty sequence), then M = M ,2. or s = s' t , with s′∈T * and t ∈T , and there is an M' such that M|s′> M′ and M′|t > M.The set of possible executions of a Petri net N from M to M is defined by the setE(N,M,M) = s ∈T * :M|s > M{ }.4

3 The extended Petri net formalismIn order to represent the distinction between ideal and sub-ideal states in a Petri net, we can partitionthe set of transitions of a Petri net into two subsets, that represent ideal and sub-ideal transitionsrespectively. Given this partition, we can define a preference ordering on executions of a net thatcompares the sub-ideal transitions of the executions. For example, we can define that an execution sis preferred to another execution s' iff s contains less sub-ideal transitions than s'. 2 This preferenceordering will be denoted by the symbol ≥ n. As an example, consider the set S = { t 1,t 3,t 4 } thatrepresents the sub-ideal transitions of our example in Figure 1. Between marking M= 1, 0, 0 andmarking M'= 0,0,1 , we have t 3,t 5> nt 1,t 4,t 6, because the first execution contains less sub-idealbehaviors than the second one. This is intuitively correct, we prefer an execution in which onereturns a book too late but undamaged to an execution in which one returns the book too late anddamaged.However, the executions t 3,t 5and t 1,t 6are equivalent for ≥ n. This is unintuitive, becauseone prefers that a book is returned too late to a book that is returned in time with damages. The orderrelation ≥ ndoes not take into account the differences which can exist between sub-ideal behaviors,because it takes only into account the number of sub-ideal behaviors. However, the violations do nothave the same seriousness. As a solution, the transitions can be partitioned in more than twosubsets, which expresses the deontic notion of ideal and varying sub-ideal, see [TT94a,TT94b].This can be modeled by assigning a weight to each transition. This weight can, for example, be aninteger which is large if the violation corresponding to the sub-ideal behavior is serious. Given thispartition in ideal and varying sub-ideal transitions, the new problem is how to compare executions.A simple solution is to say that a sequence of transitions s is preferred to a second sequence oftransitions s´ iff the sum of the weights of the transitions of s is less than that of the transitions of thesequence s´. This preference relation will be denoted by ≥ p. Intuitively, the weights represent finesfor the violations and the preference relation prefers a minimal total sum of fines. For example, theweight function of our example can be given by w(t 1) = 10,w(t 3) = 1, w(t 4) = 1 , which states thatdamaging a book is ten times worse than returning a book one week too late. Wehave t 3,t 5> pt 1,t 6, which is intuitively correct.However, also with this definition new problems occur. For example, the executionst 1,t 2,t 5, t 1,t 6are equivalent for ≥ p. This is not intuitively correct, because we want to preferexecutions where one repairs a damaged book to executions where one does not repair it. Thesepreferences are related to the deontic notion of contrary-to-duty obligations, as discussed in theintroduction. For example, the obligation to repair the damaged book is a CTD obligation. To dealwith this kind of transitions, which we call repairing transitions of sub-ideal behavior, we define afiner partition of the set T, and give a new definition to the weight function w. Hence, we defineanother subset of transitions, that contains the repairing transitions of sub-ideal behaviors and a2 Another often used solution is to define a subset ordering on the sub-ideal transitions, see [TT94a].5

negative weight will be assigned to the transitions of this set. Intuitively, the negative fines can beconsidered as rewards for good behavior. This preference ordering is presented formally in thefollowing section.3.1 The extended fine systemAn extended Petri net is a Petri net with varying sub-ideal and repairing transitions.Definition 6 (Extended Petri net) Let N = P,T,Pre, Post be a Petri net and Z the set ofintegers. An extended Petri net EN = P,T,S, R,w, Pre, Post is the extension of N with two disjunctsets S ⊆ T and R ⊆ T that represent sub-ideal and repairing behavior, and a fine functionw:T → Z, defined as follows :⎧⎪t ∈T / { S ∪ R}:w(t)= 0.⎨t ∈S:w(t) ≥ 0.⎩⎪ t ∈R:w(t) ≤ 0.A repairing transition is a transition that has a negative weight in order to recover from a sub-idealsituation that was brought about by sub-ideal behavior. We can now give a formal definition of theextended preference relation on executions.Definition 7 (Preference ordering on executions) Let EN = P,T,S, R,w, Pre, Post be anextended Petri net, M and M two markings, ≥ p:E(EN, M, M) × E(EN, M, M) a preference relationdefined on the set E(EN, M, M) of the possible executions of EN from the marking M to themarking M , s,s′∈E(EN, M, M) two executions, and the function lg(s) the length of the tuples= s 1,...,s m(here equal to m). s is preferred to s´, written ass ≥ ps' , iff :lg(s)∑i=1w(s i) ≤lg(s′)∑j=1w(s j')The preference ordering gives preferred executions.Definition 8 (Preferred execution) Let EN = P,T,S, R,w, Pre, Post be an extended Petrinet, M and M two markings, s ∈E(EN, M, M) an execution from M to M and the function lg(s)the length of the tuple s. The execution s is a preferred execution from M to M iff:lg(s)lg(s')⎛⎞⎜∀s' ∈E(EN, M, M): ∑ w(s i) ≤ ∑w(s j') ⎟⎝⎠i =1j=16

In the following example we use some new graphical notations in addition to the usual notations ofpthe Petri net formalism. A place is represented in the usual way by : i. If t i∈T \ { S ∪ R}, t iistrepresented in the usual way by : i (the weight is not represented, because it is always equal tow(t )jt j with w(t j) ≥ 0.zero). If t j∈S, t jcorresponds to a sub-ideal behavior and is represented by :w(t k) t kIf t k∈R, t kcorresponds to a repairing behavior and is represented by : with w(tk ) ≤ 0. As anillustration of extended Petri nets, consider the extension of the example in Figure 2.p110t1p2t21 1t3 -5t5 t6t4p3with S = { t 1,t 3,t 4 },R={ t 2 }, and the definition of the fine system w:w(t 1) = 10,w(t 2) =−5, w(t 3) = 1, w(t 4) = 1p1: borrowed book; p2: damaged book; p3: returned book; t1: To damage the book; t2: Torepair the book; t3: 1 week too late; t4: 1 week too late; t5: To return the book; t6: Toreturn the book.Figure 2With our extended fine system w, we obtain the intuitive result : t 1,t 2,t 5> pt 1,t 6. Furthermore,we have that t 5> pt 1,t 2,t 5. This states that we prefer that the borrower does not damage the bookeven if he repairs it.The fine function given in definition 6 is rather basic, one could impose extra constraints on it inorder to get certain desirable properties. For example, one could argue that the fine function shouldreflect the property that it is better not to do a sub-ideal behavior than doing it first and repairing itafterwards. A simple way to obtain this property is to impose on the fine function the followingconstraint. If one has a transition t 1that represents sub-ideal behavior and a transition t 2that repairsthe sub-ideal behavior, then −w(t 2) < w(t 1).7

3.2 Preferred reachable statesThe preferences on transitions model what ought to be done. Besides these ought-to-do obligationsalso ought-to-be obligations can be defined, which are preferences on the markings (the states).Preferences on markings can be derived from preferences on transitions and vice versa, in thissection we show how preferences on markings can be derived in our extended Petri nets. We havedecided to define the preference relation on the transitions because it is more expressive: twotransitions between the same places can have different preferences. The preference relation onmarkings is defined on all reachable markings.Definition 9 (Reachable markings) Let EN = P,T,S, R,w, Pre, Post be an extended Petri netwith marking M, and T* the set of all sequences that can be composed from transitions of T. The setof reachable markings for the marked net EN, M is R(EN, M) = { M:∃s ∈T * :M|s > M}.A reachable marking M 1is preferred to a second reachable marking M 2, iff a preferred execution,which leads from M to M 1, has a weight less than the weight of a preferred execution which leadsfrom M to M 2(where M is the initial marking).Definition 10 (Preference ordering markings) Let EN = P,T,S, R,w, Pre, Post be anextended Petri net with marking M, s 1,s 2∈E(EN, M, M) two executions, ≥ M:R(EN, M) × R(EN, M) a preference relation defined on the set R(EN,M) of the reachable markingsof EN, M 1, M 2∈R(EN, M) two markings and lg(s) the length of the tuple s. M 1is preferred to M 2,written as M 1≥ MM 2, iff:lg(s 1 )∑i=1lg(s 2 )∑w(s 1,i) ≤ w(s 2, j)j=1Where s 1is a preferred execution of EN, M to EN, M 1ands 2is a preferred execution of EN, M to EN, M 2.4 ConclusionsIn this article we have shown how to represent ideal and sub-ideal deontic states in Petri nets byextending these nets with a preference relation. This preference relation was defined by introducing afine function on transitions of the Petri net, from which a preference ordering on possibleexecutions was derived (which represent its procedural semantics). The fine system can be seen asan instrument to reduce undesirable behavior. A rational person is supposed to minimize the totalsum of its penalties and therefore not to perform sub-ideal behavior. We have also shown that byordering the transitions of the net, we can define a preference relation on the reachable states of an8

extended marked Petri net. Algorithms to compute the preference relation in Petri nets, based ongraph-search-path techniques, are given in [RAS95]. The preference relation introduced in thispaper, which can represent Contrary-To-Duty obligations in Petri nets, is closely related to thepreference ordering in the DIODE framework, see [TT94a], [TT94b].References[AAL92] W.N.P. van der Aalst, Timed Colored Petri Nets and their Application to Logistics, PhDthesis, Technical University Eindhoven, 1992.[BLW95] R.W.H. Bons, R.M. Lee, R.W. Wagenaar and C.D. Wrigley, Modeling InterorganizationalTrade Procedures Using Documentary Petri nets, Proceedings of the HawaiiInternational Conference on System Sciences 95, Hawaii, 1995.[CHI63] R.M. Chisholm. Contrary-to-duty imperatives and deontic logic. Analysis, 24:33-36,1963.[FOR84] J.W. Forester. Gentle murder, or the adverbial Samaritan. Journal of Philosophy, 19:75-93, 1984.[GEN86] H. J. Genrich, Predicate Transition Nets, Computer Sciences, vol. 254, Springer Berlin,1986.[HOR93] J.F. Horty. Nonmonotonic techniques in the formalization of commonsense normativereasoning. Proceedings of the Workshop on Nonmonotonic Reasoning, pages 74-84, Austin,Texas, 1993.[JP85] A.I.J. Jones and I. Porn. Ideality, sub-ideality and deontic logic. Synthese, 65:275-290,1985.[JS94] A.J.I. Jones and M. Sergot. Deontic logic in the representation of law: towards amethodology. Artificial Intelligence and Law, Vol. 1, Nr. 1, 1992.[LEE91] R. M. Lee, CASE/EDI: EDI Modeling, User Documentation, Technical Report, Euridis,Erasmus University Rotterdam, 1991.[LEE92] R. M. Lee, Dynamic Modeling of Documentary Procedures: A Case for EDI, Report No.92.06.01, Euridis, Erasmus University Rotterdam, 1992.9

[MW94] J.-J. Ch. Meyer and R. J. Wieringa, Deontic Logic in Computer Science, NormativeSystem Specification, part I : Tutorial introduction, 1994.[PET81] J. L. Peterson, Petri Net Theory and the Modeling of Systems, Prentice-Hall, 1981.[PS94] H. Prakken and M.J. Sergot. Contrary-to-duty imperatives, defeasibility and violability.Proceedings of the Second Workshop on Deontic Logic in Computer Science, Oslo, 1994.[RAS94a] J.-F. Raskin. Deontic aspects of Petri nets, Technical Report, November 94, Euridis,Erasmus University Rotterdam.[RAS94b] J.-F. Raskin. Modeling Sub-ideal Deontic State in Petri Nets, technical report, December94, Euridis, Erasmus University Rotterdam.[TOR94] L.W.N. van der Torre, Violated obligations in a defeasible deontic logic, Proceedings ofthe 11th European Conference on Artificial Intelligence (ECAI'94), Amsterdam, 1994.[TT94a] Y.H. Tan and L.W.N. van der Torre. Representing Deontic Reasoning in a DiagnosticFramework. Proceedings of the Workshop on Legal Application of Logic Programming,International Conference on Logic Programming (ICLP'94), Genoa, Italy, 1994.[TT94b] Y.H. Tan and L.W.N. van der Torre. Multi Preference Semantics for a Defeasible DeonticLogic. Proceedings of the Seventh International Conference on Legal Knowledge-Based Systems(JURIX'94). Amsterdam, the Netherlands, 1994.10