Petri Nets for Object-Oriented Modeling

Transcription

1 Peri Nes for Objec-Oriened Modeling Sefan Wi Absrac Ensuring he correcness of concurren rograms is difficul since common aroaches for rogram design do no rovide aroriae mehods This aer gives a brief inroducion o Objec Colored Peri Nes which can be used for modeling he conrol flow of concurren rograms for heir simulaion and for deriving and roving secific roeries Emhasis will no rely uon he recise formal definiions bu raher on he conces and alicaions The aer is inended o give comuer scieniss who are no familiar wih Peri nes a shor overview of hem esecially of objecoriened nes 1 Inroducion Objec-oriened modeling requires formal echniques o handle he srucure of comlex sysems There are many differen aroaches such as he Objec Modeling Technique OMT or he Unified Modeling Language UML These are exensively used nowadays and hey rovide very good means for modeling he saic srucure of objec-oriened sysems ha means he classes and heir relaionshi They also roose mehods for describing dynamic asecs like dynamically generaed objecs in Objec Ineracion Diagrams or objec saes in Sae Diagrams Bu hese echniques for objec-oriened modeling lack formal mehods for describing concurrency Some echniques have been roosed bu hey do no rovide as much flexibiliy as Peri nes This aer inroduces Peri nes for objec-oriened modeling Peri nes are a formal aroach for modeling acions which are carried ou by concurren rocesses on a comuer Elemenary Peri Nes are defined in secion They are inroduced formally in order o give an imression of he necessary mahemaics ha is used o rove secific roeries of Peri nes The rinciles of exending hem o Higher Peri Nes are described in he hird ar In he main secion secion 4 Objec Colored Peri Nes are briefly inroduced which are a formal echnique for modeling concurren objecoriened sysems Finally a summary is given in secion 5 Elemenary Peri Nes A Peri ne describes he saic relaionshi beween concurren acions of rocesses as well as heir dynamic behavior during heir execuion The saic srucure of a Peri ne consiss of ransiions which deic he acions laces which are recondiions and oscondiions of he acions and a flow relaion ha models he inerdeendence of he ransiions and laces Formally hese can be defined as follows: Definiion A ne is a ule is he se of laces is he se of ransiions! " # and is he flow relaion Furhermore and mus be fulfilled 1

2 f A m K{ K{ m H Peri nes can be visualized in grahs Places are deiced as circles ransiions as squares and elemens of he flow relaion are drawn as arrows $ %'&() &+* &+- &/0 An examle is shown in figure 1 which resens he grah of a ne wih 1 $%'() 45*60 7 $%8 9&() '(:: ; '() &4*<= '() &4->: 9&4*? 5*: 9&4-? 5*: 5* &4<=0 and '( 5* The acions which are modeled by ransiions and can be carried ou concurrenly because he recondiion of one ransiion is no a oscondiion of he oher one 1 1 Figure 1: A Peri ne Since nes describe only he sa- ic srucure of acions hey are exended o lace/ransiion nes ha model he dynamic behavior This is done using okens which lie in laces if he corresonding condiion is fulfilled A ransiion swiches by aking okens from he recondiion laces and uing new okens in he oscondiion laces A comlee and formal definiion will be given now sulied from [4] and ' ' AB CD EFG Definiion A lace/ransiion ne is a ule Is ordered elemens are defined as follows: 1 4 and are a finie ne ie a ne wih a finie number of laces and ransiions AIHJ K LDM%>NO0 N is a caaciy funcion for each lace defined as where is an infinie caaciy CPH+ QR ST U VR WK LDXY%<Z[0 is a weigh U\] '^_` funcion ha aaches a weigh a Cb \] '^[W cz o each arc of he flow relaion If and only if hen E F H8 dk$le%>no0 is an iniial marking which indicaes he number of okens in each lace I resecs he caaciies ie f & a dh6e F g&ihjak g&+ The following definiion gives he swiching rule also called firing rule for lace/ransiion ne (see also [4]): l ' ' AB CD E F Definiion Le be a ne a 1 The recondiion se of a ransiion is defined as m nho %'&eqr 9&] ' a s0 Similarly he oscondiion se of is defined as m Ho %'&q[ = & a s0 A funcion H4 KuLeT%>NO0 is called a marking of if f & a vh g&yh Ak g& a A ransiion is enabled in if f & a ih g&iw!cb 9&] and f & a 9&+]xkCb U= &+ih!ay 9&+ This is denoed as z a 4 A ransiion swiches or fires o a follower marking of if K{ z and & a H g&} 9&+ { Cb g&] ']xkcb U= & This is denoed as z 4

3 ~ d ' 'A C 'E F An examle is given in figure which shows he Peri ne wih V%'&S(< &+* &4-)0 c V%'(G0 V%8 9&() '(:: '() &4*>: U'( &+-=0 A H:&(O KƒN &4* KƒN &4- K1N C H4 g&( '(>} KI [ < U'() &4*<} KI / U'(G &4-< K ˆ E F H& ( K / & * KŠˆ6 & - KI and The lef E ( E F side of figure shows he iniial marking he righ side shows C he marking afer swiching of ransiion 1 According o he arc weighs given by he ransiion akes E okens ( H[& ( & ( ou KPˆ? & & * of * laces KŒ / & & - - okens KŒ ino and one ino The new marking is hus Noe ha he number of inu okens may differ from he number of ouu okens ie a ransiion can absorb and generae okens Figure : Swiching a ransiion of an Elemenary Peri Ne On he lef he marking before he swiching is shown and on he righ he marking aferwards 1 Exensions of Elemenary Peri Nes Since he firs inroducion of Peri Nes by CA Peri in 196 differen exensions of Elemenary Peri Nes have been defined so-called Higher Peri Nes The basic idea of hese exensions is he usage of individual okens insead of elemenary ones In Colored Peri Nes each oken is of a cerain ye and each arc of he ne is inscribed wih a ye A ransiion can be enabled in a marking only if he okens in he recondiion se are of he yes he arcs require A se furher is he idea of using nes as okens insead of elemenary okens This resuls in a wo-level hierarchy of nes consising of one sysem ne and many oken nes These nes are called higher order Peri nes However hey are no furher regarded in his aer For objec-oriened modeling differen Peri nes have been defined Elemenary Objec Nes as described in [5] use Elemenary Peri Nes as okens and a swiching rule which regards boh he sae of he Sysem Ne and he Objec Ne (he okens) There is only one objec ne and hus he flexibiliy of Elemenary Objec Nes is resriced and no suiable for objec-oriened modeling Therefore hey are mainly used in workflow rocess modeling

4 Newer aroaches inroduce Higher Peri Nes for objec-oriened modeling which are used for formal modeling of concurren rocesses in racically used objec-oriened languages There are wo differen aroaches: Hierarchichal Objec-Oriened Peri Nes [1] and Objec Colored Peri Nes [] The laer will be described in more deail in he following secion and examles will be given Hierarchical Objec-Oriened Peri Nes are no followed in his aer since heir basic ideas are similar o Objec Colored Peri Nes for insance hey also use communicaion ransiions (called absrac ransiions ) for invoking mehods of oher nes 4 Objec Colored Peri Nes Objec Colored Peri Nes are a comound of saic class nes which describe he behavior of an objec class and dynamically generaed objec nes which are insances of class nes They have been defined in [] esecially for objec-oriened design of concurren rogram sysems Therefore hey reflec he asecs of objec-oriened rogramming such as dynamic generaion of objecs communicaion beween objecs and inheriance A class ne which is formally defined in [] is a Peri ne wih he following exensions comared o Elemenary Peri Nes: 1 I is a Colored Peri Ne ie he okens are of differen yes and he ransiions are enabled in a marking if he recondiion ses conain markings of he ye secified by he arcs from he recondiion laces o he ransiion Predefined yes are for anonymous okens and OID for objec idenifiers Objec idenifiers are used o reference o a secific objec ne There are differen ransiion yes: Normal ransiions and communicaion ransiions (IN OUT INV and REC) which are used for mehod invocaions (see below) A ransiion canno be of more han one ye Each lace is inscribed wih a ye and only okens of ha secific ye can be u ino he lace aend(x) x x aend ok ok aend() IN elem INT xs x::xs lis INT LIST aendok Figure : Funcion aend modeled as a service of a class ne OUT An examle of a class ne is given in figure similar o an examle from [] I shows he model of an aend mehod in a lis class ha can sore ineger values The mehod arameer \ is an ineger which is o be aended o he lis The model 4

5 consiss of an inu ransiion aend(x) an ouu ransiion aend() and a normal ransiion aend The inu ransiion is swiches every ime he mehod is invoked wih he arameer \ (see below) and laces one oken of ye INT ino he lace elem Noe ha he lace elem can only conain okens of INT The aend ransiion erforms he aend oeraion by uing he ineger value \ and he old lis \ ino he lis lace and aking he new lis \ from i I also us an anonym oken ie an unyed oken ino he lace aendok Finally he ouu ransiion aend() fires when he aend oeraion is done A call of he aend mehod is shown in figure 4 This class ne invokes he aend mehod by firing he idaendcall() ransiion which is marked as an invocaion I requires an INT oken ie an ineger value from lace elem and an objec idenifier of he ye OID ha idenifies he objec insance of he called elem INT aendok mylis OID y ok id id idaendcall(x) INV idaendret(x) REC Figure 4: Class ne calling he service aend mehod The conrol flow is given o he called mehod and when he call reurns he idaendret(x) ransiion which is of he receive ye swiches and us an anonymous oken ino he aendok lace This is indicaed by he doed arrow in figure 4 The dynamic asec of Objec Colored Peri Nes is he creaion and deleion of objec nes They are creaed by he funcion new which is a reserved funcion ha is globally defined I assigns a unique idenifier o he calling class ne and adds he new objec ne o he global se of nes An examle is shown in figure 5: The class ne A creaes a new insance of class ne B whenever ransiion Ž swiches An objec idenifier for he new objec ne is u ino lace Ž & - The creaed objec ne of class B looks like he class ne B and is addressed by a unique idenifier * ok ok a a 1 b b 1 ok ok a ok new B ok 1 a a b1 b OID Class Ne A ok ok Class Ne B Figure 5: Creaion of Objec Nes The lef side shows he class ne A which generaes a new insance of class ne B shown on he righ side Aar from dynamic generaion of objecs and mehod invocaion anoher objecoriened conce can be modeled using Objec Colored Peri Nes he inheriance relaion beween classes This is done by defining a subye relaion for he mehod signaures of a class For deails on his refer o [] 5

6 5 Summary In his aer an aroach for objec-oriened modeling wih Peri nes has been inroduced Objec Colored Peri Nes They rovide a formal mehod for describing boh saic and dynamic asecs of a sysem The saic srucure consiss of he class definiions along wih heir mehods and he class hierarchy The dynamic asecs are dynamically generaed objec nes and mehod invocaion Objec Colored Peri Nes can be used for simulaing he conrol flow of rogram sysems and for roving is correcness They are esecially good for objec-oriened rograms ha use concurrency because hey are mahemaically well undersood and hus rogram roeries can be derived from hem An examle is examinaion if deadlocks can occur in a aricular concurren rogram see [4] Using a Peri ne simulaor like Renew [] he modeling can be done comuer aided and furhermore roeries of modeled nes can be derived auomaically References [1] Jang-Eui Hong Doo-Hwan Bae Sofware modeling and analysis using a hierarchical objec-oriened Peri ne In Informaion Science Vol 10 no 1 4 ages Elsevier Science 000 [] Chrisoh Maier Daniel Mold Objec Coloured Peri Nes A Formal Technique for Objec Oriened Modelling In Gul A Agha Fiorella De Cindio Grzegorz Rozenberg ediors Concurren Objec-Oriened Programming and Peri Nes ages Sringer-Verlag Berlin Heidelberg New York 001 [] Renew The Reference Ne Worksho Available a h://wwwrenewde [4] Wolfgang Reisig Peri Nes An Inroducion Sringer-Verlag Berlin Heidelberg 1985 [5] Rüdiger Valk Peri Nes as Token Objecs An Inroducion o Elemenary Objec Nes In Jörg Desel Manuel Silva ediors Alicaion and Theory of Peri Nes 1998 ages 1 5 Sringer-Verlag Berlin Heidelberg New York