Automated Synthesis of Liveness Enforcing Supervisors Using Petri Nets

Transcription

1 Auomaed Synhesis of Liveness Enforcing Suervisors Using Peri Nes Technical Reor of he ISIS Grou a he Universiy of Nore Dame ISIS Ocober, 000 Revised in January 00 and May 00 Marian V. Iordache John O. Moody Panos J. Ansaklis Dearmen of Lockheed Marin Dearmen of Elecrical Engineering Federal Sysems Elecrical Engineering Universiy of Nore Dame 80 Sae R. 7C, MD 00 Universiy of Nore Dame Nore Dame, IN 66 Owego, NY Nore Dame, IN 66 iordache.@nd.edu john.moody@lmco.com ansaklis.@nd.edu Inerdiscilinary Sudies of Inelligen Sysems

2 AUTOMATED SYNTHESIS OF LIVENESS ENFORCING SUPERVISORS USING PETRI NETS Marian V. Iordache, John O. Moody, Panos J. Ansaklis Absrac Given an arbirary Peri ne srucure, which may have unconrollable and unobservable ransiions, he liveness enforcemen rocedure resened here deermines a se of linear inequaliies on he marking of a Peri ne. When he Peri ne is suervised so ha is markings saisfy hese inequaliies, he suervised ne is roved o be live for all iniial markings ha saisfy he suervision consrains. Also he suervision is roved o be maximally ermissive for a large class of Peri nes, which includes he fully conrollable and observable Peri nes. Moreover, he suervisor suors secificaions requiring only some of he Peri ne ransiions o be live. The maximal ermissiviy yically alies also for his case. The rocedure allows auomaed synhesis of he suervisors. The sufficien condiions for which our heoreical resuls are guaraneed o aly can be auomaically verified. Conens Inroducion. Scoe of he Paer Relaed Work Paer Srucure Review of Some Peri Ne Basic Proeries Deadlock and Liveness Proeries of Peri Nes. Inrinsic Proeries Condiions for Deadlock Prevenion and Liveness Enforcemen A Characerizaion of Peri Nes Based on Subnes which Can Be Made Live, in View of Deadlock Prevenion and Liveness Enforcemen Preliminaries o he Liveness Enforcing Mehod. A Transformaion of Peri Nes o PT-ordinary Peri Nes Transformaion of Peri nes o asymmeric choice Peri nes Peri Ne Suervisors Based on Place Invarians Fully Conrollable and Observable Peri Nes Peri Nes wih Unconrollable and Unobservable Transiions Dearmen of Elecrical Engineering, Universiy of Nore Dame, Nore Dame, IN 66 ( iordache., ansaklis.@nd.edu) Lockheed Marin Federal Sysems, 80 Sae R.7C, MD 00, Owego NY ( john.moody@lmco.com)

3 . Sihon Conrol Based on Place Invarians Case : All Transiions are Conrollable and Observable Case : Transiions Unconrollable and/or Unobservable are Presen The Liveness Enforcing Mehod 9. Inroducion o he Mehod Generaing Marking Consrains The enforced lace invarians Consrains which do need conrol lace enforcemen Consrucing he consrains of (L, b) and(l 0,b 0 ) Imlicily conrolled sihons Iniial consrain ransformaion Transforming Consrains o Admissible Consrains The Comuaion of a T-minimal Acive Subne The Liveness Enforcing Procedure Remarks Illusraive Examles Proeries 6. Basic Proeries of he Mehod Inroducion and Noaions Proeries Main Resuls Success and Permissiviy Resuls Exending Permissiviy Terminaion Resuls Final Remarks and Direcions for Furher Research Addiional Consrains Finie Caaciy Peri Nes The Terminaion Problem Summary of Resuls Inroducion. Scoe of he Paer Liveness is a desirable qualiy of concurren sysems. Due o muual inerdeendencies, such sysems may reach saes of local or oal deadlock. Deadlock means ha some acions (or all, for oal deadlock) are imossible o ursue. A sysem is live when deadlock (boh local and global) is imossible. Raher han roviding a mehod o verify wheher a sysem is live, we rovide a mehod which synhesizes a suervisor such ha he suervised sysem is live. We consider discree even sysems modeled as Peri nes. We noe ha i is more naural o model concurren sysems as Peri nes raher han as finie auomaa. Furher on, we do no resric he Peri ne models. They are allowed o be unbounded, generalized and

4 wih unconrollable and unobservable ransiions. The aroach is no deenden on he iniial marking. Insead, he se of iniial markings for which liveness is enforced is characerized as he feasible se of a sysem of linear marking inequaliies. Thus he liveness suervisor roduced by our aroach is defined for a se of iniial markings, raher han for a single iniial marking. Moreover, when he suervisor is maximally ermissive, enforcing liveness is imossible for all iniial markings for which he suervisor is no defined. Thus he mehod can also be used for liveness verificaion. In some alicaions, some Peri ne ransiions may model undesirable behavior. Such ransiions should no be live. Thus, raher han suervising for liveness, we suervise for T -liveness, which is a conce generalizing liveness: insead of requiring all ransiions o be live, only he ransiions in he se T are o be live. Noe ha liveness is a secial case of T -liveness. The liveness enforcing rocedure can be described as follows. Given a fully conrollable and observable Peri ne and a se T, he rocedure will rovide a suervisor for T -liveness which yically is maximally ermissive, or i will deec ha T - liveness is imossible, in which case he suervisor will enforce T x -liveness, for T x T. Given a Peri ne wih unconrollable and/or unobservable ransiions and a se T, he rocedure will rovide a suervisor enforcing T x -liveness, T x T ;ift x T hen he rocedure is unable o enforce T -liveness. A sufficien condiion which guaranees he suervisor o be maximally ermissive can be easily esed. In aricular, in he case of fully conrollable and observable Peri nes, he suervisor is guaraneed o be maximally ermissive in he case of liveness enforcemen. The disadvanages of our rocedure are ha erminaion is no guaraneed and ha when he rocedure erminaes he comuaions may be comlex. However all comuaions are erformed offline. Thus he suervisor generaed by he rocedure is aroriae for real-ime roblems. I is ossible o guaranee erminaion, bu his may come a he exense of ermissiviy. We give wo varians of he rocedure wih guaraneed erminaion. However hese wo varians are only useful for bounded Peri nes. The mehod resens he condiions necessary o insure liveness enforcemen as a se of linear marking inequaliies. This feaure can be used direcly in oimizaion roblems, e.g., a linear rogram can be used o deermine he minimum number of resources a sysem requires such ha deadlock can be avoided. An ineresing roery of our mehod is ha i solves a roblem which canno be solved wih finie auomaa based aroaches. Indeed, by considering all ossible iniial markings, an auomaon wih an infinie number of saes is obained. Noe ha his is no he case for he aroaches which consider a given iniial marking and a bounded Peri ne. Alicaions which may benefi from considering he iniial marking unknown are in he area of Flexible Manufacuring, as he iniial marking corresonds o he number of available resources.. Relaed Work Previous resuls abou enforcing liveness in Peri nes usually consider resriced classes of Peri nes. A necessary and sufficien condiion for he exisence of liveness suervisors aears in [8]. A mehod for liveness enforcemen in a class of conservaive ordinary Peri nes has been given in []; he aroach is no maximally ermissive. The aroach of [] has been recenly exended o generalized Peri nes in []. Polynomial comlexiy has been roved, however he considered Peri nes are conservaive and he aroach is no maximally ermissive. A liveness enforcing aroach for a resriced class of ordinary Peri nes is given in [9]. Anoher liveness enforcing aroach aears in [0]; i is based on he coverabiliy grah, and hence he iniial marking is required. In [7] he auhors consider enforcing liveness based on he unfolding of a Peri ne. Unfolding is an efficien echnique of searching he reachabiliy grah. The aroach of [7]

5 is limied o bounded Peri nes and he iniial marking mus be known. Our aroach is mos relaed o he deadlock revenion rocedure we resened in [0, 8], and is imrovemen in [9]. While our former rocedure revened deadlock bu was no guaraneed o enforce liveness, he rocedure of his aer is guaraneed o enforce liveness. The liveness enforcemen rocedure of his reor is ieraive, a every ieraion correcing new deadlock siuaions. Using ieraions o correc deadlock siuaions has also been used in []. In our rocedure we emloy suervisory conrol based on lace invarians [, ], which is an esablished mehod in he suervisory conrol of Peri nes. We also use a ransformaion o almos ordinary Peri nes and a ransformaion o asymmeric choice nes. The firs ransformaion was insired by a similar ransformaion in []. A ransformaion o free choice nes, which is a aricular class of asymmeric choice nes, has been used in [7]. In [7] i is shown ha liveness enforcing olicies of a free choice equivalen of a Peri ne can be used o enforce liveness in he original Peri ne. Our ineres for asymmeric choice nes sems from a generalizaion of he Commoner s Theorem for asymmeric choice nes []. Thus liveness in an asymmeric choice Peri nes can be relaed o sihon conrol. Our aroach involves he comuaion of a secial class of minimal sihons. Mehods of sihon comuaion have been given for insance in [,, 6].. Paer Srucure The documen is organized as follows. Secion reviews basic Peri ne roeries and describes he noaions which are used hroughou he aer. Secion resens some deadlock and liveness roeries. We emhasize he suervisory conrol asec of enforcing liveness and revening deadlock and we derive significan consequences of a known resul. Thus we derive Corollary. which is he basis for beer deadlock ess, such as Proosiion. and Proosiion.. In Theorem. we rove a fundamenal resul for our mehod. A consequence of Theorem. is Proosiion.6, which gives a necessary condiion and a sufficien condiion for T -liveness in a class of Peri nes. In secion we resen reliminaries o our mehodology. The suervisory echnique used by our mehod, suervisory conrol based on lace invarians [, ], is oulined in secion.. Transformaions from generalized Peri nes o ordinary Peri nes and o asymmeric choice Peri nes are resened in secions. and.. The sihon conrol aroach (largely a aricular case of he suervision based on lace invarians) is given in secion.. Secion defines he liveness enforcemen rocedure and he oeraions which are involved. The rocedure for liveness enforcemen is saed in secion.. Illusraive examles are given in secion.6. Secion 6 gives he formal characerizaion of he rocedure. The analysis of he rocedure is comlex, so in secion 6. we rovide some basic roeries characerizing he rocedure or he oeraions involved by i. These roeries are also used o derive our main resuls. The main resuls are given in secion 6.. Theorem 6. roves ha he rocedure does enforce liveness and Theorem 6. roves ha for a large class of Peri nes he rocedure is no more resricive han any oher liveness enforcing suervisor. Secion 6.. conains resuls ha show ha by (ossibly) comromising he erformance of he rocedure, erminaion can be guaraneed. We conclude wih some significan secial cases and remarks in secion 6.. This is an almos self-conained reor. The liveness enforcemen rocedure of his reor is essenially a modificaion of he deadlock revenion mehod resened in our revious echnical reor [9]. Thus a significan ar of he maerial of [9] resembles or is included in his reor. The new maerial of his reor is included in he following secions. Secion. includes Theorem., which is essenial for our liveness enforcemen aroach. Secion. describes he asymmeric choice ransformaion. Secion is he adaaion for liveness enforcemen of he similar secion in [9], exce for secion., which conains new

6 maerial. Comared o [9], some resuls of secion 6 needed new roofs or resaemens due o he new form of he rocedure; also, new echnical resuls have been added. The main heoreical resuls for he liveness enforcing rocedure are given in secion 6., where he erminaion resuls are essenially he same as in [9]. Review of Some Peri Ne Basic Proeries We assume he reader o be familiar wih Peri ne fundamenals. Peri ne surveys may be found in [6], [] and []. In his secion we inroduce our convenions and noaions. A Peri ne srucure is a quadrule N =(P, T, F, W) wherep is he se of laces, T he se of ransiions, F (P T ) (T P )isheseofransiion arcs and W : F N \{0} is a weigh funcion. A marking µ of he Peri ne srucure is a ma µ : P N. A Peri ne srucure N wih iniial marking µ 0 is called a Peri ne, and will be denoed by (N,µ 0 ). For simliciy, we may denoe someimes by Peri ne a Peri ne srucure. I is useful o consider a marking boh as a ma and as a vecor. These requiremens are no necessarily conflicing, because vecors can be seen as mas defined on an arbirary finie se domain [6], insead of {,,...m}, as is cusomary. The marking vecor is defined o be [µ( ),µ( ),...µ( n )] T,where,,... n are he laces of he ne enumeraed in a chosen (bu fixed) order and µ he curren marking. The same symbol µ will denoe a marking vecor. The marking vecor of a Peri ne may be regarded as he sae variable of he Peri ne. An equivalen way of saying ha lace has he marking µ() isha has µ() okens. Figure could be used o illusrae he grahical reresenaion of Peri nes. A oken is reresened by a bulle. The marking vecor in figure (b) is [0,, ] T. An arc weigh is indicaed near he arc when i is no one. For insance, in figure (b) W (, )=andw(, )=. The rese of a lace is he se of incoming ransiions o : = { T :(, ) F }. Theosse of a lace is he se of oucoming ransiions from : = { T :(, ) F }. is a source lace if = and a sink lace if =. Similar definiions aly for ransiions. They are also exended for ses of laces or ransiions; for insance, if A P, A =, A =. A We use µ[ o denoe ha µ enables he ransiion and µ[ >µ o denoe ha µ enables and if fires, hen he marking becomes µ. The marking µ is reachable from µ if here is a sequence of markings µ,...µ k, µ k = µ, and a sequence of ransiions i,... ik s.. µ[ i >µ [... ik >µ.these of reachable markings of a Peri ne (N,µ) (i.e. he se of markings reachable from he iniial marking µ) will be denoed by R(N,µ). In a Peri ne N =(P, T, F, W) wihm laces and n ransiions, he incidence marix is an m n marix defined by D = D + D, where he elemens d + ij and d ij of D+ and D are d + ij = W ( j, i )if( j, i ) F and d + ij = 0 oherwise; d ij = W ( i, j )if( i, j ) F and d ij = 0 oherwise. The incidence marix allows an algebraic descriion of he marking change of a Peri ne: µ k = µ k + D u k () where u k is called firing vecor, and is elemens are all zero exceing u k,i =,wherei corresonds o he ransiion i ha fired. We will denoe by firing vecor also a vecor x associaed wih a sequence of ransiions ha have fired, whose enries record how ofen each ransiion aears in he sequence. If x is A

7 he firing vecor of he ransiion sequence ha led he Peri ne from he marking vecor µ 0 o µ k : µ k = µ 0 + D x () A vecor x is called lace invarian if x T D = 0. A vecor x is called ransiion invarian if D x =0. The suor of a ransiion invarian x is x = { j T : x(j) 0}. APerine(N,µ 0 )issaidobedeadlock-free if for any reachable marking µ here is an enabled ransiion. (N,µ)isindeadlock if no ransiion is enabled a marking µ. Le (N,µ 0 ) be a Peri ne. A ransiion is said o be live if µ R(N,µ 0 ) µ R(N,µ) such ha is enabled by µ. A ransiion is dead a marking µ if no marking µ R(N,µ) enables. (N,µ 0 )issaid o be live if every ransiion is live. A nonemy se of laces S P is called a sihon if S S and ra if S S. In aricular, S = P may be sihon. An emy sihon wih resec o a Peri ne marking µ is a sihon S such ha µ() = 0. The aribue emy refers o he fac ha S has no okens. A sihon has he roery ha S if for some marking i is emy, i will be so for all subsequen reachable markings. A ra has he roery ha if a some marking i has one oken, hen for all subsequen reachable markings i will have a leas one oken. See figure for sihon examles. In figure (a), {, } and {, } are ras. S is a minimal sihon if here is no oher sihon S (by definiion, S ) such ha S S. Deadlock and Liveness Proeries of Peri Nes This secion inroduces cerain liveness and deadlock roeries, focusing on heir relaion o srucural roeries of Peri nes and suervision. Throughou his secion all ransiions are considered o be conrollable and observable.. Inrinsic Proeries APerineN =(P, T, F, W) isordinary if f F : W (f) =. We will refer o slighly more general Peri nes in which only he arcs from laces o ransiions have weighs equal o one. We are going o call such Peri nes PT-ordinary, because all arcs (, ) from a lace o a ransiion saisfy he requiremen of an ordinary Peri ne ha W (, ) =. Definiion. Le N =(P, T, F, W) be a Peri ne. We call N PT-ordinary if P, T, if (, ) F hen W (, ) =. The mehodology of our work deends on a well known necessary condiion for deadlock [6], namely ha a deadlocked ordinary Peri ne conains a leas one emy sihon. I can easily be seen ha he roof of his resul also is valid for PT-ordinary Peri nes. Proosiion. A deadlocked PT-ordinary Peri ne conains a leas one emy sihon. An examle is shown in figure (a). Proosiion. shows ha deadlock migh be revened if i can be ensured in a nonblocking way ha no sihon ever loses all is okens. The condiion in Proosiion. is only necessary. The examle of figure (c) illusraes ha he condiion of Proosiion. is no sufficien and figure (b) ha he resul is no alicable o Peri nes more general han PT-ordinary.

8 (a) (b) (c) Figure : (a) A deadlocked PT-ordinary Peri ne. An emy sihon is {,, }. (b) A deadlocked Peri ne wih no emy sihon which is no PT-ordinary. (c) A deadlock-free Peri ne (for he marking dislayed) wih an emy sihon {, }. Definiion. Le N be a Peri ne and M I be a se of iniial markings. A sihon S is said o be conrolled wih resec o M I if µ 0 M I, µ R(N,µ 0 ): µ(). A conrolled sihon conains for all reachable markings a leas one oken. A ra conrolled sihon is a sihon ha includes a ra. Recalling he ra roery, for all markings such ha he ra has one oken, he sihon is conrolled. We define an invarian conrolled sihon as a sihon S of a Peri ne N wih he roery ha N has a lace invarian x such ha for all i =,,... P, ifx(i) > 0hen i S. I is easy o show ha for all iniial markings µ 0, such ha x T µ 0, he sihon S is conrolled. In aricular, a sihon which conains a conrolled sihon is conrolled. Therefore in a Peri ne such ha all minimal sihons are conrolled, all sihons are conrolled. Also, by Proosiion., a PT-ordinary Peri ne is deadlock-free if all is sihons are conrolled. This is may no be rue for more general Peri nes. Proosiion. has been generalized in [] for Peri nes which are no PT-ordinary (bu see also Proosiion. and commens in [8]). We do no use ha resul. Insead we ransform generalized Peri nes o PT-ordinary Peri nes (refer o secion.) and hen use Proosiion.. Anoher drawback of Proosiion. is ha i is no effecive for Peri nes which are no reeiive. We define he reeiive Peri nes in secion. and hen we give new resuls which are adequae for he Peri nes which are no reeiive. Loss of liveness is a less severe form of deadlock, where some acions can no longer haen while ohers may sill be ossible. Deadlock imlies loss of liveness. An emy sihon is a necessary and no a sufficien condiion for deadlock, while for loss of liveness i is a sufficien bu no a necessary condiion. Commoner s Theorem saes ha in an ordinary free choice ne N, if here are dead ransiions for a marking µ, hen here is a reachable marking µ R(N,µ) such ha a sihon is emy ([6].0). We include laer in he secion a generalizaion o asymmeric choice nes as Theorem.. S. Condiions for Deadlock Prevenion and Liveness Enforcemen Definiion. Le N =(P, T, F, W) be a Peri ne, M he se of all markings of N and U M. A suervisory olicy Ξ is a funcion Ξ:U T ha mas o every marking a se of ransiions ha he Peri ne is allowed o fire. The markings in M\U are called forbidden markings. 6

9 We denoe by R(N,µ 0, Ξ) he se of reachable markings when (N,µ 0 ) is suervised wih Ξ. I is known ha if (N,µ 0 ) is live, hen (N,µ)wihµ µ 0 may no be live. The same is rue for deadlock-freedom, as shown in figure. The following resul shows ha if liveness is enforcible a marking µ or if deadlock can be revened a µ, hen his is also rue for all markings µ µ. We say ha deadlock can be revened in a Peri ne N if here is an iniial marking µ 0 and a suervisory olicy Ξ such ha (N,µ 0 ) suervised by Ξ is deadlock-free. Similarly, we say ha liveness can be enforced in N if here is an iniial marking µ 0 and a suervisory olicy Ξ such ha (N,µ 0 )suervised by Ξ is live. Proosiion. If a suervisory olicy Ξ which revens deadlock in (N,µ 0 ) exiss, hen for all µ µ 0 here is a suervisory olicy which revens deadlock in (N,µ). The same is rue for liveness enforcemen. Proof: Le µ µ 0. A suervisory olicy for (N,µ )isξ defined as follows: { Ξ(µ) T f (µ) for µ R(N,µ 0 ) Ξ (µ + µ µ 0 )= oherwise where T f (µ) denoes he ransiions enabled by he marking µ, aar from he suervisor. 6 6 (a) (b) Figure : A Peri ne which is live for he iniial marking µ 0 shown in (a) and no even deadlock-free for he iniial marking µ µ 0 shown in (b). Definiion. [] A Peri ne is said o be (arially) reeiive if here is a marking µ 0 andafiring sequence σ from µ 0 such ha every (some) ransiion occurs infiniely ofen in σ. The following lemma seems o be necessary for he sufficiency roof of Theorem., which is a known resul. The auhors are unaware of a reference in which Lemma. or he sufficiency roof of Theorem. aear. We rove he lemma as we need i in order o rove a number of oher resuls, including Corollary.. A relaed roof aears in [6] a age 70. Lemma. Le N =(P, T, F, W) beaperineofincidencemarixd. Assume ha here is an iniial marking µ I which enables an infinie firing sequence σ. Le U T be he se of ransiions which aear infiniely ofen in σ. There is a nonnegaive ineger vecor x such ha Dx 0, x(i) 0 i U and x(i) =0 i T \ U, where i denoes he ransiion corresonding o he i h column of D. 7

10 Proof: Consider firing σ and le µ 0 be he marking reached afer all ransiions which aear finiely ofen in σ have fired. Le σ = σ 0 σ σ...σ k... such ha each σ k is finie, for all k each of he ransiions in U aears in σ k,andµ I [σ 0 >µ 0.Thenleµ, µ,... be defined as follows: µ k [σ k >µ k for all k. Le V n be a nonemy se of he form V n = {y N n : y i V n,y y i,y y i or y y i }. Nexiis roved by inducion ha V n is finie (i.e. i canno have infiniely many elemens). Assume ha any V n is finie. Then, le y s,n V n ; V n C k,u,wherec k,u = {y N n : y(j k )=u, y(i k ) >y s,n (i k ), y i V n,y k,u y i,y y i or y y i }, is defined for 0 u<y s,n (j k )andk =,...n(n ) corresonds o he ossibiliies in which i k j k,0 i k,j k n can be chosen. The inducion assumion imlies ha each C k,u is finie, because he comonen j k of he vecors is fixed and only he remaining n can be varied. So V n is finie. Le M be recursively consruced as follows: iniially M 0 = {µ 0 }; for all i, M i = M i {µ i } if y M: y µ i or y µ i and else M i = M i. The revious aragrah showed ha n 0 N: k >n 0, M k = M n0. Le M = M n0 and M = {y N n : y x M,y y x }. Boh are finie ses. Here i is shown ha i, j, 0 i<j, such ha µ i µ j leads o conradicion. Assuming he conrary, k >0 y x Msuch ha µ k+n0 y x and µ k+n0 y x. If y N n, y x Mand y x y, henforu such ha u y x and u y x eiher y u or boh y u and y u; foru such ha u y and u y eiher y x u or boh y x u and y x u. Le M () be consruced in a similar way as M, bu saring from M () 0 =(M {y}) \{u M: u y}, wherey = µ +n0, and using µ n0+i insead of µ i for M () i. For he same reason he consrucion ends in finiely many ses. Also, M () M and n 0, such ha k >0 y x Msuch ha µ k+n0, y x and µ k+n0, y x. So we can coninue in he same way wih M (),...M (j), also subses of M. However hese oeraions canno be reeaed infiniely ofen: j M, because M (j) j conains a leas one elemen from M\ M (i). (This is so because y u, y u, u M (i) y/ M (i), i= also u M (i) \M (i ) v M (i ) : v u, hence u M (i) : y u imlies v M: y v.) So, M (j+) canno be consruced for some j, which imlies µ +n0,j u, u M (j), which is a conradicion. Therefore j, k, j<k, such ha µ j µ k. Le q j and q k be he firing coun vecors: µ j = µ 0 + Dq j and µ k = µ 0 + Dq k ;lex = q k q j.thenµ k µ j 0 Dx 0, and by consrucion x 0, x(i) > 0 i U and x(i) =0 i T \ U. Theorem. [] A Peri ne is (arially) reeiive if and only if a vecor x of osiive (nonnegaive) inegers exiss, such ha D x 0, x 0. In general i may no be ossible o enforce liveness or o reven deadlock in an arbirary given Peri ne. This may haen because he iniial marking is inaroriae or because he srucure of he Peri ne is incomaible wih he suervision urose. The nex corollary characerizes he srucure of Peri nes ha allow suervision for deadlock revenion and liveness enforcemen, resecively. I shows ha Peri nes in which liveness is enforcible are reeiive, and Peri nes in which deadlock is avoidable are arially reeiive. Par (b) of he corollary also aears in [8]. Corollary. Le N =(P, T, F, W) be a Peri ne. (a) Iniial markings µ 0 exis such ha deadlock can be revened in (N,µ 0 ) if and only if N is arially reeiive. (b) Iniial markings µ 0 exis such ha liveness can be enforced in (N,µ 0 ) if and only if N is reeiive. 8

11 Proof: (a) If deadlock can be avoided in (N,µ 0 )henµ 0 enables some infinie firing sequence σ, andby definiion N is arially reeiive. On he oher hand, if N is arially reeiive, hen by Theorem. here is a nonnegaive vecor x, x 0suchhaDx 0. Le σ x be a firing sequence associaed o a firing vecor q = x and le q denoe he firing vecor afer he firs ransiion of σ x fired, q afer he firs wo fired, and so on o q k = q. The incidence marix D can be wrien as D = D + D,whereD + and D corresond o he weighs W (, ) and W (, ), resecively. If he rows of he D are d T, dt,..., dt P, hen a marking which enables σ x is µ 0 ( i )= min(0, min j=...k dt i q j ) i =... P () A leas one deadlock revenion sraegy exiss for µ 0 : o allow only he firing sequence σ x,σ x,σ x,... o fire. This infinie firing sequence is enabled by µ 0 because µ 0 + Dx µ 0 and µ 0 enables σ x. (b) The roof is similar o (a). Le Ξ denoe a suervisory olicy. Le R(N,µ 0, Ξ) denoe he se of reachable markings from iniial marking µ 0,when(N,µ 0 ) is suervised by Ξ. A vecor x S R n has maximum suor if no oher vecor in S has more nonzero enries han x. Theminimum suor is similarly defined. Corollary. Consider a Peri ne N =(P, T, F, W) which is no reeiive. Then a leas one ransiion exiss such ha for any given iniial marking i canno fire infiniely ofen. Le T D be he se of all such ransiions. There are iniial markings µ 0 and a suervisory olicy Ξ such ha µ R(N,µ 0, Ξ), no ransiion in T \ T D is dead. Proof: There is an ineger vecor x 0wihmaximum suor such ha Dx 0, which means ha for all ineger vecors w 0 such ha Dw 0, w x. Indeed if y 0, z 0 are ineger vecors and Dy 0, Dz 0, hen D(z + y) 0andsoy + z 0and y, z y + z. If j T can be made live, here is a marking ha enables an infinie firing sequence σ such ha j aears infiniely ofen in σ. Therefore by Lemma. y 0 such ha Dy 0andy(j) > 0. Since x has maximum suor, y x and so j x. This roves ha all ransiions ha can be made live are in x. Therefore T D is nonemy. Nex, he roof shows ha all ransiions in x can be made live, which imlies ha T \ T D = x. Le σ x be a firing sequence associaed wih x, i.e. every i T aears x(i) imes in σ x. Then here is a marking µ 0 given by equaion () which enables he infinie firing sequence σ x,σ x,σ x,... Also, we may choose Ξ o resric all ossible firings o he former infinie firing sequence, so all ransiions in x can be made live. In Corollary., T D is nonemy. Oherwise, since all ransiions from T \ T D could simulaneously be made live, his would imly ha N is reeiive, which is a conradicion. A secial case is T \ T D =, when he Peri ne is no even arially reeiive, and so deadlock can no be avoided for any marking. I was already shown ha only reeiive Peri nes can be made live. The corollary above shows ha he se of ransiions of a arially reeiive Peri ne can be uniquely divided in ransiions ha can be made live and ransiions ha canno be made live. So he liveness roery of arially reeiive Peri nes is ha all ransiions ha can be live are live. 9

12 . A Characerizaion of Peri Nes Based on Subnes which Can Be Made Live, in View of Deadlock Prevenion and Liveness Enforcemen We denoe by an acive subne a ar of a Peri ne which can be made live by suervision for aroriae markings. In he following definiion we use he noaions from Corollary.. Definiion. Le N =(P, T, F, W) be a Peri ne, D he incidence marix and T D T be he se of all ransiions which canno fire infiniely ofen given any iniial marking. N A =(P A,T A,F A,W A ) is an acive subne of N if P A = T A, F A = F {(T A P A ) (P A T A )}, W A is he resricion of W o F A and T A is he se of ransiions wih nonzero enry in some nonnegaive vecor x which saisfies Dx 0. The maximal acive subne of N is he acive subne N A =(P A,T A,F A,W A ) such ha T A = T \ T D. A minimal acive subne has he roery ha he vecor x defining i has minimum suor. Definiion.6 Given an acive subne N A ofaperinen, a sihon of N is said o be an acive sihon (wih resec o N A ) if i is or includes a sihon of N A. An acive sihon is minimal if i does no include anoher acive sihon (wih resec o he same acive subne.) Proosiion. A sihon which conains laces from an acive subne is an acive sihon wih resec o ha subne. Proof: Using he noaions from Definiion., le S be a sihon such ha S P A. S S imlies ha S T A S T A. If T A and for some P :, hen P A, by Definiion.. Hence S T A (S P A ) and so S T A =(S P A ) T A. Noe also ha (S P A ) T A S T A. Therefore S S imlies (S P A ) T A (S P A ) T A, which roves ha S P A is a sihon of N A. The significance of he acive subnes for deadlock revenion can be seen in he following resuls. Firs we rove a echnical resul. Lemma. Le N A =(P A,T A,F A,W A ) be an acive subne of N. Given a marking µ of N and µ A is resricion o N A,if T A is enabled in N A,hen is enabled in N. Proof: By definiion, here is an nonnegaive ineger vecor x 0 such ha Dx 0(D is he incidence marix) and x(i) > 0for i T A and x(i) =0for i T \T A. This imlies ha here are markings such ha he ransiions of T A can fire infiniely ofen, wihou firing oher ransiions (see roof of Corollary..) If is no enabled in N, here is such ha / P A (he oeraors are aken wih resec o N,no N A,) since is enabled in N A. Noe ha / P A imlies T A =. If =, canno fire infiniely ofen, which conradics he definiion of T A, since T A. If x, he ransiions of T A canno fire infiniely ofen wihou firing x, which again conradics he definiion of T A. Therefore is also enabled in N. Noe ha in a reeiive Peri ne all sihons are acive wih resec o he maximal acive subne. The nex resul is a generalizaion of he well known Proosiion.. Proosiion. Le N A be an arbirary, nonemy, acive subne of a PT-ordinary Peri ne N.Ifµ is a deadlock marking of N, hen here is a leas one emy minimal acive sihon wih resec o N A. 0

13 Proof: Since µ is a deadlock marking and N =(P, T, F, W) is PT-ordinary, T : µ() = 0. The acive subne is buil in such a way ha if he marking µ resriced o he acive subne enables a ransiion, henµ enables in he oal ne (Lemma..) Therefore, because he oal ne (N,µ) is in deadlock, he acive subne is oo. In view of Proosiion., le s be an emy minimal sihon of he acive subne. Consider s in he oal ne. If s is a sihon of he oal ne, hen s is also a minimal acive sihon; herefore he ne has a minimal acive sihon which is emy. If s is no a sihon of he oal ne: s \ T A. Le S be he se recursively consruced as follows: S 0 = s, S i = S i { ( S i \ S i ):µ() =0}, where µ is he (deadlock) marking of he ne. In oher words S is a comleion of s wih laces wih null marking such ha S is a sihon. By consrucion S is an acive sihon and is emy for he marking µ. Hence an emy minimal acive sihon exiss. The racical significance of Proosiion. is ha i rovides a suor for doing deadlock revenion, since deadlock is no ossible when all acive sihons wih resec o a nonemy acive subne canno become emy. A less resricive condiion is given in he nex resul. Proosiion. Deadlock is unavoidable for he marking µ if for all minimal acive subnes N A here is an emy acive sihon wih resec o N A. Proof: For any emy (acive or no) sihon, all ransiions in he osse of ha sihon are emy. Therefore for all acive minimal subnes, some of heir ransiions are dead. If deadlock is avoidable, afer some ransiions firings a marking can be reached which enables σ x σ x...,whereσ x is a finie firing sequence. Le q be he firing coun vecor for σ x. Then Dq 0. If he acive subne for q is minimal, we le x = q, bu if i is no, here is x such ha x q, x 0,x 0, Dx 0 and he acive subne associaed o x is minimal. Bu i mus be an acive sihon wih regard o ha acive subne, herefore no all of he ransiions of x can fire, which imlies ha no all of he ransiions of σ x can fire, which is a conradicion. The revious resul suors maximally ermissive deadlock revenion. Deadlock is avoidable in a PTordinary Peri ne as long as i can be insured ha for all allowed markings, here is a minimal acive subne such ha all minimal acive sihons have a oken. The usage of Proosiion. for maximally ermissive deadlock revenion has been demonsraed in secion 6.. of [9]. An asymmeric choice ne is a Peri ne N = (P, T, F, W) wih he roery ha, P, = or. The following new resul can be seen as he corresonden for T-liveness of a revious resul for liveness in []. However, noe ha even for liveness he nex resul is sronger, as i relaes he dead ransiion o an emy sihon. Theorem. Consider a PT-ordinary asymmeric choice Peri ne N and a marking µ such ha a ransiion is dead. Then here is µ R(N,µ) such ha S is an emy sihon for he marking µ and S. Proof: In an asymmeric choice Peri ne, imlies or. Therefore given n laces such ha i j =0, i, j {,,...n}, wehave i i... in,wherei,...i n are disinc and i j {,,...n} for all j =...n. Le = {,... n }, where he noaion is chosen such ha... n. We rove firs ha µ R(N,µ)and j {,...n} such ha µ x R(N,µ ): µ x ( j ) = 0. Assume he conrary. Le µ = µ and i be he leas number in {,...n} such ha µ i, R(N,µ ): µ i, ( i )=0(i exiss, for is dead and

14 N is PT-ordinary). Then µ i, R(N,µ i, ): µ, ( i ). If µ i, R(N,µ i, ): µ i, ( i ), hen le µ = µ i,,leibehe leas ineger in {,...n} such ha µ i, R(N,µ ): µ i, ( i ) = 0 and reea he oeraion above. Noe ha i is increasing, and so afer a mos n such ses we find ha µ i, R(N,µ i, ): µ i, ( i ) = 0. (Oherwise we would have a reachable marking enabling.) From µ i, ( i ) andµ i, ( i )=0 we infer ha µ i, R(N,µ i, )and i i such ha µ i, enables i. Noe ha i j j = i...n, so µ i, ( j ) j = i...n. By he choice of i, µ i, ( j ) j =...i. Therefore µ i, enables. Conradicion. Therefore, µ R(N,µ)and j {,...n} such ha µ x R(N,µ ): µ x ( j ) = 0. We recursively use his roery o consruc S. Noe ha all ransiions in j are dead for µ. Le S 0 = and S = { j }. We recursively consruc S by generaing S,...S n+ and he markings µ,...µ n+. S i for i issuch ha all ransiions in S i are dead for some marking µ i. The consrucion in a ieraion is as follows. Le µ i+ R(N,µ i ) such ha (S i \ S i ) µ x R(N,µ i+ ) : µ x () = 0. Then we le S i+ = S i { x : µ x R(N,µ i+ ):µ x () =0}. There is n such ha S n+ = S n,forhe x (S i\s i ) Peri ne has a finie number of laces. We le S = S n and µ = µ n. Since j S, S. By consrucion S is a sihon, S is emy for µ,andµ R(N,µ). In general we may no wan all ransiions o be live. For insance some ransiions of a Peri ne may model fauls and we wan o insure ha some oher ransiions are live. This is he reason for he following definiion. Definiion.7 Le (N,µ 0 ) be a Peri ne and T a subse of he se of ransiions. The Peri ne is said o be T-live if all ransiions T are live. Noe ha a live ransiion is no he oosie of a dead ransiion. Tha is, a ransiion may be neiher live nor dead. Indeed, a ransiion is live if here is no reachable marking for which i is dead. Noe also ha T-liveness corresonds o liveness when he se T equals he se of ransiions. Definiion.8 Le N be a Peri ne, T a subse of he se of ransiions and N A =(P A,T A,F A,W A ) an acive subne. We say ha N A is T-minimal if T T A and T A Tx A for any oher acive subne Nx A =(Px A,Tx A,Fx A,Wx A ) such ha T Tx A. In general he T-minimal acive subne is no unique. However, as shown in he nex Proosiion, any T-minimal acive subne can be used o characerize T-liveness. Proosiion.6 Given a PT-ordinary asymmeric choice Peri ne N,leT be a se of ransiions and N A a T -minimal acive subne which conains he ransiions in T. If all he minimal sihons wih resec o N A are conrolled (i.e. hey canno become emy for any reachable marking), he Peri ne is T -live (and T A -live). If he Peri ne is T -live, here is a T -minimal acive subne N A such ha all minimal acive sihons wih resec o N A are conrolled. Proof: Assume ha no acive sihon becomes emy. If here is a reachable marking such ha a ransiion T A is dead (and T T A ), by Theorem. here is a reachable marking such ha a sihon S is emy and S. However S imlies S P A, and by Proosiion. S is an acive sihon. Conradicion, for S is emy.

15 Le Ni A denoe a T -minimal acive subne, i =...k,wherek is he number of T -minimal acive subnes. If here is a reachable marking µ such ha an acive sihon S i is emy, le T i = S Ti A,whereT i A is he se of ransiions of Ni A. Because S i is acive, T i is nonemy; because S i is emy, he ransiions of T i are dead. Assume ha here is an infinie firing sequence σ x such ha all ransiions of T aear infiniely ofen in σ x and afer a ar of σ x is fired, (le µ x be he marking reached) all T -minimal acive subnes Ni A have an emy acive sihon S i. Le σ be he remaining ar of σ x which is enabled by µ. All ransiions of T aear infiniely ofen in σ. Therefore, by Lemma., here is x 0 such ha Dx 0(D is he incidence marix) and T x. However, x does no conain all ransiions of any of he T -minimal subnes Ni A: T i x \Ti A,fori =...k. This imlies ha x defines anoher T -minimal acive subne, which is a conradicion. Preliminaries o he Liveness Enforcing Mehod. A Transformaion of Peri Nes o PT-ordinary Peri Nes We are ineresed in using a ransformaion o PT-ordinary Peri nes because Proosiions. and. in secion aly o PT-ordinary Peri nes. We use a modified form of he similar ransformaion from [], and we call i he PT-ransformaion. Le N =(P, T, F, W) be a Peri ne. Transiions j T such ha W (, j ) > forsome j may be sli (decomosed) in several new ransiions: The ransiion j is sli in m = n( j ) ransiions: j,0, j,, j,,... j,m,wheren( j )=max{w(, j ): (, j ) F }. Also,m new laces are added: j,, j,,... j,m. The connecions are as follows: (i) j,i = j,i, j,i = j,i and j,i = j,i,fori =...m (ii) j,i = { j : W (, j ) >i}, fori =0...m (iii) j,0 = j Noe ha j resembles very much j,0 : j,0 has all he connecions of j lus one addiional ransiion arc. Afer he sli is erformed, we denoe j,0 by j. The PT-ransformaion consis in sliing all ransiions such ha W (, ) > forsome. In his way he ransformed Peri ne is PT-ordinary. A few roeries are aaren: j,i = j,i = i =...m () j,i = i =...m () We use he convenion ha a sli ransiion j is also a ransiion of he PT-ransformed ne, since we denoe j,0 by j. Le P T be he se of laces of he ransformed ne. To a marking µ of he original ne we associae in he ransformed ne a marking µ T such ha µ T () =µ() P and µ T () =0 P T \ P. Firing of an unsli ransiion j in he original ne corresonds o firing he same ransiion in he ransformed ne. Firing of a sli ransiion j in he original ne corresonds in he ransformed ne o firing he sequence j,m... j,, j. For similar iniial markings µ and µ T (see above) he firing sequence σ T corresonds o a firing sequence σ, such ha every sli ransiion j in σ is relaced in σ T by is comonens j,m... j,, j, and firing σ in N roduces a similar marking µ o he marking µ T reached by firing σ T in he ransformed ne.

16 6 6 6,,,,,, 6 (a) (b) Figure : Illusraion of he PT-ransformaion. (a) Original ne and (b) ransformed ne.,,,, (a) (b) (c) (d) Figure : Illusraion of he ransiion sli: (a) iniial configuraion; (b) he effec of he PT-ransformaion; (c) iniial configuraion; (d) he effec of he AC-ransformaion. Figure shows an examle in which he ransiion is sli in, and, and he ransiion is sli in,,, and. Firing in he original ne corresonds o firing, and in he ransformed ne, and firing in he original ne corresonds o firing,,, and in he ransformed ne. Anoher examle is he Peri ne of figure 7(a), which is changed as shown in figure 7(b) afer i is PT-ransformed. The ransiion is relaced by, and,and by, and.. Transformaion of Peri nes o asymmeric choice Peri nes Le N =(P, T, F, W) beaperineandn =(P,T,F,W ) be he ransformed Peri ne, where P P, T T. The idea of he ransformaion is as follows. Given he ransiion, i and j such ha i j and j i,remove from eiher he osse of i or ha of j by adding an addiional lace and ransiion. The idea is illusraed in figure (c-d). Noe ha he oeraions corresond o a modified form of ransiion sli oeraions (secion.). Algorihm of he AC-Transformaion Inu: N and oionally M P ; he defaul value of M is M = P. Ouu: N Iniialize N o be idenical wih N.

17 For every T wih > do. Consruc U = {( i, j ) P P : i, j, i j and j i }.. if U is emy, hen coninue wih he nex ieraion.. Le Q :=.. For every ( i, j ) U (a) A lace { i, j } M is seleced. If wo choices are ossible: i. = i (or = j )if i (or j ) has been reviously seleced for anoher elemen of U. ii. oherwise is chosen such ha aears in oher elemen of U. Ifboh i and j saisfy his roery, selec { i, j } such ha =max{ i, j }. iii. if none of i and j aears in anoher elemen of U, selec { i, j } such ha =max{ i, j }. (b) If a lace could be seleced (i.e. if { i, j } M ) henq := Q {}. For all Q, delee from N he ransiion arc (, ) and add a new lace and a new ransiion such ha = {}, = { }, = {}, W (, )=W (, )=andw (,)=W(, ). We call he ransformaion o asymmeric choice Peri nes AC-ransformaion. The oeraion in he se of he algorihm is a ransiion sli. The ransiion sli of he AC-ransformaion is slighly differen from he ransiion sli of he PT-ransformaion in secion.. The second argumen of he algorihm, M, is used by he liveness enforcemen rocedure in order o selec he ransiions which he algorihm slis. Indeed, in general here are many ways in which o choose which ransiions o be sli such ha he ransformed ne is wih asymmeric choice. I will be seen ha he liveness enforcemen rocedure selecs M such ha he lace invarians creaed in revious ieraions are no modified by he AC-ransformaion.. Peri Ne Suervisors Based on Place Invarians We ouline here resuls from [, ] for suervisors based on linear consrains, in he aricular case of fully conrollable and observable Peri nes. The resuls of his secion sill aly for Peri nes wih unconrollable and unobservable ransiions, if he desired consrains are admissible... Fully Conrollable and Observable Peri Nes The conrol roblem considered here is o enforce a se of n c linear consrains o reven reaching undesired markings in a Peri ne. The consrains are wrien in a marix form: L µ b (6) where L is an ineger n c n marix (n c - he number of consrains, n - he number of laces of he given Peri ne), b is an ineger column vecor and µ denoes a marking vecor. Le µ c be a vecor of n c nonnegaive slack variables, defined as: µ c = b L µ (7)

18 Le µ c0 be he slack variables ha corresond o he iniial marking µ 0,haisµ c0 = b Lµ 0. Le q be he firing vecor associaed wih he ransiions ha led he Peri ne from µ 0 o µ and D he incidence marix, ha is µ = µ 0 + Dq. Soweseehaµ c = b L (µ 0 + D q), which also can be wrien as: µ c = µ c0 +( LD ) q (8) Therefore µ c may be regarded as a marking of some addiional conrol laces, where he exended (suervised) Peri ne has a marking vecor µ =[µ T,µ T c ] T, and an incidence marix D =[D T,Dc T ] T, and where D c = LD. In he suervised ne, iniial markings µ 0 such ha L µ 0 >bcanno be considered, since equaion (7) shows ha in his case µ c0 will no be nonnegaive. When he consrains are iniially saisfied, he iniial marking of he conrol laces may be chosen according o equaion (7), and herefore he consrains will remain saisfied for any reachable marking, since he D c ar of he incidence marix revens any firings which would aem o make any of he elemens of µ c negaive. The way he consrains are enforced revens only forbidden markings o be reached, so he suervisor is maximally ermissive. The nex heorem summarizes he consrucion above: Theorem. Le a lan Peri ne wih conrollable and observable ransiions, incidence marix D and iniial marking µ 0 be given. A se of n c linear consrains Lµ b areobeimosed.ifb Lµ 0 0 hen a Peri ne conroller (suervisor) wih incidence marix D c = LD and iniial marking µ c0 = b Lµ 0 enforces he consrain Lµ b when included in he closed loo sysem D =[D T,Dc T ] T. Furhermore, he suervision is maximally ermissive. Proof: See [, ]. Because D c = LD, every row of [L, I] is a lace invarian of he incidence marix of he closed loo sysem, D... Peri Nes wih Unconrollable and Unobservable Transiions Unconrollable and/or unobservable evens of he lan corresond o unconrollable and/or unobservable ransiions in he Peri ne model of he lan. Unconrollable evens canno be inhibied and unobservable evens canno be observed. As he Peri ne suervisor is imlemened in he form of conrol laces conneced o he lan Peri ne, we need o make sure ha no conrol lace ever aems o inhibi an unconrollable ransiion enabled in he lan Peri ne, and no conrol lace marking is varied by firing unobservable ransiions. The consrains Lµ b which saisfy his requiremen are called admissible consrains. Noe ha he admissibiliy of a consrain may deend on he iniial marking of he Peri ne. (For insance, all consrains are admissible in he rivial case wih null iniial marking.) In his aer we are ineresed in consrains which are admissible for all iniial markings. I can easily be seen ha Lµ b is admissible for all iniial markings if and only if he following equaions of [] are rue: LD uc 0 (9) LD uo = 0 (0) where D uc and D uo denoe he columns of he incidence marix which corresond o unconrollable and unobservable ransiions, resecively. From he viewoin of his aer all linear consrains ha have 6

19 marices L ha saisfy he condiions above are admissible. Such consrains may be enforced as in secion... Consrains Lµ b which do no saisfy (9) and (0) may be ransformed o a new se of consrains L µ b such ha (i) L saisfies (9) and (0), and (ii) µ N n : L µ b Lµ b. Unless µ N n : L µ b Lµ b, his aroach of enforcing Lµ b may no be maximally ermissive. Noe ha enforcing linear consrains is maximally ermissive in he case of fully conrollable and observable Peri nes (Theorem.). Algorihms which ransform linear consrains o admissible linear consrains are given in [].. Sihon Conrol Based on Place Invarians Proosiion. showed ha in a PT-ordinary Peri ne deadlock is no ossible if all sihons are conrolled. This suggess ha all sihons should be made conrolled sihons. An easy way o make a sihon conrolled is o creae a lace invarian o conrol he sihon. This is done below by adding an addiional lace o he original Peri ne. Early references of his aroach for sihon conrol are in [, ]. This secion resens i as a secial case of he suervision mehod based on lace invarians (secion.). The oeraions described here do no deend on he fac ha he srucure hey are alied o is a sihon, so hey are described in more general erms... Case : All Transiions are Conrollable and Observable Le N =(P, T, F, W) be a Peri ne. Given a se of laces S, µ() is he desired conrol olicy. This consrain can be enforced using he mehodology of invarian based suervision of [, ], oulined in secion., which yields an addiional lace C, called conrol lace. The lace invarian creaed is x, such ha x(i) =for i S, x(i C )= andx(i) = 0 for all oher indices, where i C is he row index of C in he closed loo incidence marix. This invarian corresonds o he equaion µ(c) = µ( k ) () k S where he consan resuls from he iniial marking of he conrol lace. There are several aricular cases: (a) C = and C : no ransiion increases he marking of S and here are ransiions which decrease he marking of S. In his case C alone makes u a minimal sihon which canno be conrolled (see also [],.87-88). (b) C S (in aricular C = ): no ransiion can make S oken free. Also, C S if and only if S is a ra. Therefore when S is also a sihon, i is (ra) conrolled for all iniial markings µ 0 ha saisfy µ 0 (). S 0 (c) C = and C = : he marking of S canno vary, and so here is a lace invarian x such ha x(i) = for all i S and x(i) = 0 oherwise. Case (a) deecs ransiions ha canno be made live when S is a sihon (Corollary.). Case (b) shows he case when S does no need conrol. Noe ha he mehod deends only on srucural roeries of he Peri ne. Tha is, i does no deec wheher S does no need conrol for some iniial markings, bu i deecs only he case when S does no need conrol for all iniial markings µ 0 such ha µ 0 (). Therefore he S S 7

20 mehod when alied o a sihon ha is no a ra, bu includes a ra, always roduces a conrol lace. The reason ha his is correc is ha here are nonzero iniial markings of he sihon such ha he included ra has null marking; hence he sihon is no ra conrolled for such markings. 7 8 C 6 7 C C 7 (a) (b) Figure : Sihon Conrol Examles. Connecions o conrol laces are dashed. In figure (a) here is a single minimal sihon, {,,,, 6, 7 }. The sihon includes a ra {,, 6, 7 }, bu i is no ra conrolled because he marking of he ra is 0. The conrol lace C revens firing, which would emy he sihon. In figure (b) he original Peri ne has wo minimal sihons, {,, } and {,,,, 6 }. Their conrol laces are C and C, resecively. C is an examle of case (a). Also, he conrol lace C ha resuls by conrolling he minimal sihon {,C } saisfies C = and C =. By Theorem., he way in which he consrain µ 0 () was enforced is maximally ermissive. Therefore, because he enforcemen of his consrain on a sihon by definiion makes he sihon conrolled, here is no oher more ermissive way o conrol a sihon. This is no he only way o rovide maximally ermissive conrol of a sihon; however, any oher way is equivalen. An imoran qualiy of his echnique is ha he closed loo remains a Peri ne. S.. Case : Transiions Unconrollable and/or Unobservable are Presen Le D be he incidence marix of a Peri ne, and le D uo and D uc be D resriced o he columns of unobservable and resecively unconrollable ransiions. In order ha he consrain l T µ b be admissible, he suervisor enforcing i should no need o deec unobservable ransiions or inhibi enabled unconrollable ransiions, and so he consrain is required o saisfy l T D uo =0andl T D uc 0. There are mehods ha allow o ransform a consrain in a anoher consrain, in general more resricive, which saisfies he las wo requiremens. Two such mehods can be found in []. Ye we will choose o use a differen mehod in secion..6. When a desired consrain µ() is inadmissible, i can be ransformed o a consrain S of he form l T µ b. In boh secion..6 and [], b = (in [] consider he consrucion of Lemma.0). Therefore he admissible form of he consrain µ() is α µ(). The algorihm of he secion..6 is guaraneed o find a soluion o his roblem if any of he form l T µ b exiss. Noe ha he ransformaion o admissible consrains is no always ossible. There are cases when his is imossible because of limied informaion due o unobservable ransiions and/or limied abiliy o conrol firing ransiions can make imossible he ask o design a suervisor which guaranees ha he S S 8