I R I S A P U B L I C A T I O N I N T E R N E N o ON THE CONSTRUCTION OF PULLBACKS FOR SAFE PETRI NETS ERIC FABRE ISSN

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1 I R I P U B L I C A T I O N I N T E R N E 1750 N o S INSTITUT DE RECHERCHE EN INFORMATIQUE ET SYSTÈMES ALÉATOIRES A ON THE CONSTRUCTION OF PULLBACKS FOR SAFE PETRI NETS ERIC FABRE ISSN I R I S A CAMPUS UNIVERSITAIRE DE BEAULIEU RENNES CEDEX - FRANCE

3 INSTITUT DE RECHERCHE EN INFORMATIQUE ET SYSTÈMES ALÉATOIRES Campus d Bauliu Rnns Cdx Franc Tél. : (33) Fax : (33) On th construction of pullbacks for saf Ptri nts Eric Fabr * Systèms communicants Projt Distribcom Publication intrn n 1750 August 1st, pags Abstract: Th product of saf Ptri nts is a wll known opration : it gnralizs to concurrnt systms th usual synchronous product of automata. In this short not, w considr th dfinition of pullbacks of saf PNs, anothr catgorical construction. Pullbacks gnraliz th product to nts which intract both by synchronizd transitions and by a shard sub-nt. Ky-words: qualizr saf Ptri nt, catgory thory, Winskl morphism, compltnss, pullback, (Résumé : tsvp) * ric.fabr@irisa.fr Cntr National d la Rchrch Scintifiqu (UMR 6074) Univrsité d Rnns 1 Insa d Rnns Institut National d Rchrch n Informatiqu t n Automatiqu unité d rchrch d Rnns

4 Construction ds pullbacks pour ls résaux d Ptri saufs Résumé : L produit d résaux d Ptri saufs (évntullmnt à labls) st un opération bin connu : on put la voir comm un généralisation du produit synchron d automats à ds systèms concurrnts. Dans ctt not, on s intérss à la construction d pullbacks d résaux saufs, un autr construction catégoriqu. Ls pullbacks généralisnt l produit d résaux n prmttant un intraction non sulmnt par la synchronisation d transitions, mais aussi par partag d placs t d transitions. Mots clés : résau d Ptri sauf, théori ds catégoris, morphism d Winskl, complétud, pullback, égalisur

5 Pullbacks of saf Ptri nts 3 Contnts 1 Introduction 4 2 Notations 4 3 Equalizr in Nts Equalizr and coqualizr in Sts Equalizr Coqualizr Candidat qualizr in Nts Cohrnc of th dfinition Univrsal proprty Synthsis Product Pullback Spcial cas PI n 1750

6 4 E. Fabr 1 Introduction W considr th catgory Nts of saf Ptri nts (PN) as dfind by Winskl in [2]. Saf Ptri nts provid a natural and widsprad modl for concurrnt systms. A product N was dfind in [2] for saf PNs, that can b considrd as a gnralization of th usual synchronous product of automata. In practic, this product is ssntially intrsting whn spcializd to labld nts : roughly spaking, it would thn synchroniz transitions of two nts as soon as thy carry th sam labl. As a nic proprty, N is th catgorical product in Nts. Pushing forward this ida, it can b intrsting to driv a notion of pullback for PNs. Whil th product assums that nts intract through common vnts, th pullback gos furthr and also allows intractions by shard placs and transitions. Th notion of pullback has bn xtnsivly xplord for othr modls of concurrncy (transition graphs, graph grammars, tc.) [6], or for othr catgoris of Ptri nts [3] (proposition 11). But th choic of nt morphisms plays a crucial rol, and apparntly th construction of pullbacks in th catgory Nts of [2] is still missing. This catgory rmains of grat intrst howvr, bcaus it allows foldings (and so unfoldings!), and alrady has a product. Lt us mntion som contributions to th topic. B. Konig provids in [8] a dfinition for spcific pullback diagrams. M. Bdnarczyk t al. prov in [7] that Nts is finitly complt, so all pullbacks xist. But th rsult is obtaind in a much mor gnral stting, and is hard to spcializ to th cas of saf nts (dixit on of th authors). Finally, lt us strss that [7] mntions in its introduction (p.3) that th xistnc of a pullback construction for saf Ptri nts has bn rportd... although thy hav not bn abl to locat any rfrnc! It is thrfor usful that w try to provid a dirct dfinition. W procd in svral stps. W first considr unlabld nts. It is a wll known fact that th labling is ssntially a dcoration that can b rincorporatd at no cost in nt oprations (s [5]), which w do at th nd of th papr. Scondly, w rcall that a pullback opration can b drivd from a product and an qualizr (s [1], chap. V-2, thm. 1, and [6], sc. 5). Sinc all products xist in Nts, w simplify th construction (and proofs) by attmpting to build qualizrs, which is th hart of th contribution. W finally gathr all pics to giv a comprhnsiv dfinition of th pullback of labld Ptri nts, first in th gnral cas, thn in th spcific cas whr morphisms ar partial functions. 2 Notations Nt. W dnot Ptri nts by N = (P, T,, P 0 ), rprsnting rspctivly placs, transitions, initially markd placs and th flow rlation. For ach plac p P, w assum p p 1, and for ach transition t T, t 1 and t 1. For labld nts, w tak N = (P, T,, P 0, λ, Λ) whr λ : T Λ is th labling function. Morphism. A morphism φ : N 1 N 2 btwn nts N i = (P i, T i, i, P 0 i ) is a pair (φ P, φ T ) whr C1. φ T is a partial function from T 1 to T 2, and φ P a rlation btwn P 1 and P 2, C2. P2 0 = φ P (P1 0 ) and p 2 P2 0,!p 1 P1 0 φ P : p 1 p2, φ P C3. if p 1 p2 thn th rstrictions φ T : p 1 p 2 and φ T : p 1 p 2 ar total functions, Irisa

7 Pullbacks of saf Ptri nts 5 C4. if t 2 = φ T (t 1 ) thn th rstrictions φ op P : t 2 t 1 and φ op P : t 2 t 1 ar total functions. whr φ op P dnots th opposit rlation to φ P. Notic that points 3 and 4 ntail that th pair (φ P, φ T ) prsrvs th flow rlation (on its domain of dfinition). Obsrv also that point 3 implis that if φ P is dfind at p 1 P 1, thn φ T is dfind at all transitions t 1 T 1 connctd to p 1. In th squl, w will simply writ φ for φ P or φ T, and φ(x) to dnot placs in rlation with at last on plac in X. By Dom(φ), w rprsnt th lmnts of N 1 (placs or transitions) whr φ is dfind, i.. φ 1 (P 2 T 2 ). Rmark. Notic that condition C2 bcoms a consqunc of C3 and C4 whn on assums th xistnc of fak initial transitions t i,0 in ach N i, such that t i,0 = Pi 0 and t 2,0 = φ(t 1,0 ). W shall us this trick in th squl to simplify proofs (focusing on C3, C4 and omitting to chck C2). Saf Ptri nts with th abov dfinition of morphisms dfin th catgory Nts. For labld nts, w naturally considr labl-prsrving morphisms, which yilds th catgory λnts. Dcomposition of th pullback. Lt N 0, N 1, N 2 b nts, and f i : N i N 0, i = 1, 2 b nt morphisms, so N 0 forms a kind of intrfac btwn N 1 and N 2, or bttr a common constraint 1. W look for a trminal nt N = (P, T,, P 0 ), associatd to morphisms g i : N N i, i = 1, 2, such that (fig. 1) : f 1 g 1 = f 2 g 2 (1) By trminal, w man th univrsal proprty of th pullback : whnvr thr xists anothr tripl (N 3, h 1, h 2 ) satisfying th sam commutativ diagram, thr xists a uniqu morphism ψ : N 3 N prsrving th commutativity, namly h i = g i ψ. W dnot th pullback by N 1 N 0 N 2, or by N 1 N 2 for short. N 3 g 1 ψ g 2 N π 1 π 2 N 1 N 2 f 1 f N 2 0 Figur 1: Commutativ diagram of th pullback N = N 1 N 2. It is wll known that th pullback opration can b dcomposd into a product, followd by an qualization. Considr th product nt N 1 N N 2, and th associatd canonical projctions π i : N 1 N N 2 N i, i = 1, 2. In gnral, N 1 N N 2 and th π i do not satisfy th pullback condition, i.. f 1 π 1 f 2 π 2. Howvr, by qualizing thm, on gts th dsird rsult. (N, ) qualizs f 1 π 1 and f 2 π 2 iff (f 1 π 1 ) = (f 2 π 2 ), and for any othr candidat (N 3, h) thr xists a uniqu ψ : N 3 N such that h = ψ (fig. 2). It is straightforward to chck that (N, π 1, π 2 ) yilds th dsird pullback. For dtails, w rfr th radr to [1], chap. V-2, thm. 1, or to [6], sc. 5 whr this construction is also usd. 1 Th trm intrfac suggsts that all intractions ar capturd by th intrmdiary nt N 0, which is not th cas. Th two nts N 1, N 2 may still hav intractions outsid N 0 by mans of synchronizd transitions, as it will b obvious in th pullback dfinition givn at th nd of ths nots. PI n 1750

8 6 E. Fabr N 3 ψ N h N x 1 N 2 f 1 f 2 o π1 o π2 N 0 3 Equalizr in Nts Figur 2: Equalizing f 1 π 1 and f 2 π 2. Considr two nts N i = (P i, T i, i, P 0 i ), i = 1, 2 rlatd by two morphisms f, g : N 1 N 2. W want to build th qualizr (N, ) of f and g, i.. a nt N and a morphism : N N 1 satisfying f = g, and such that for any othr candidat pair (N 3, h) thr xists a uniqu morphism ψ : N 3 N satisfying h = ψ (fig. 3). N 3 ψ N h N 1 f g N 2 Figur 3: A pair (N, ) qualizing f and g. 3.1 Equalizr and coqualizr in Sts W rcall hr two classical rsults that will b instrumntal in th squl Equalizr W considr th catgory of sts with partial functions as morphisms (or quivalntly pointd st with total functions). Lt T 1, T 2 b two sts rlatd by partial functions f, g : T 1 T 2. Th qualizr of f and g is th pair (T, ) whr T = {t 1 T 1 : f(t 1 ) = g(t 1 ) or both f and g ar undfind at t 1 } (2) and is th canonical injction of T into T 1 (w ll us th shorthand t 1 T instad of t T, t 1 = (t)). In th stting of pointd sts, whr functions point to th spcial valu ɛ of a st to man undfind, (2) taks th simplst form f(t 1 ) = g(t 1 ). Givn anothr candidat pair (T 3, h), th uniqu morphism (partial function) ψ : T 3 T is obtaind by ψ = 1 h (it is asy to chck that Im(h) T ) Coqualizr W now considr th catgory of sts with total functions. Th coqualizr diagram corrsponds to fig. 3 with all arrows rvrsd. Lt S 2, S 1 b two sts rlatd by total functions F, G : S 2 S 1, and dnot by (S, E) th coqualizr of F and G. Th construction is a bit mor complx. Irisa

9 Pullbacks of saf Ptri nts 7 S 2 F G S 1 a b c d E S _ a=b=c _ d Figur 4: Coqualizing th total functions F and G. Dfin th rlation R on lmnts of S 1 by p 1 R p 1 p 2 S 2, {p 1, p 1 } = {F (p 2), G(p 2 )} (3) and considr th quivalnc rlation gnratd by R. W dnot by [p 1 ] th class of p 1 for. Thn S = {[p 1 ] : p 1 S 1 } (4) and th function E : S 1 S is simply th quotint opration, i.. E(p 1 ) = [p 1 ]. Givn anothr candidat pair (S 3, H), th uniqu morphism (total function) Ψ : S 3 S is obtaind by Ψ = H E 1, or in othr words by [p 1 ] S, Ψ([p 1 ]) = H(p 1 ). Indd, it is asy to chck that H is ncssarily class invariant. 3.2 Candidat qualizr in Nts Lt (N, ) dnot th dsird qualizr, with N = (P, T,, P 0 ) and : N N 1. Transitions. On transition sts, f, g : T 1 T 2 ar partial function, so w adopt dfinition (2) for T and on T. Placs. On plac sts, th dfinition is a bit mor complx. Th morphism dfinition in Nts actually stats in C4 that φ op : t 2 t 1 and φ op : t 2 t 1 ar total functions, for t 2 = φ(t 1 ), which orints us to co-qualizrs in Sts. So lt t b a transition of T, with t 1 = (t) T 1. Assum first that f, g ar dfind at t 1, and f(t 1 ) = g(t 1 ) = t 2 T 2. W tak for op in t 1 th coqualizr of f op, g op : t 2 t 1. Eq. (3) thus dfins R t 1, th quivalnc rlation t 1 and plac classs [p 1 ] t 1. And similarly in th post-st of t 1. Whn f, g ar both undfind at t 1, w tak for op in t 1 (or t 1 ) th coqualizr of functions f op, g op from th mpty st. So op is simply th idntity. In summary, th plac st P of N is a subst of 2 P 1 givn by P = {[p 1 ] t 1 : t 1 T, p 1 t 1 } {[p 1 ] t 1 : t 1 T, p 1 t 1 } (5) and th rlation on placs is simply givn by p p 1 iff p 1 p. Obsrv that a plac p 1 P 1 not connctd to a transition of T has no countrpart in P. Lmma 1 Lt t 1, t 1 T. Assum p 1, p 1 t 1 t 1, thn p 1 t 1 p 1 p 1 t 1 p 1 (6) PI n 1750

10 8 E. Fabr t 1 t 2 p 1 p 1 f g p 2 t 1 t 2 Figur 5: Idntity of quivalnc classs. Proof. Assum p 1 p 1 and p 1 R t 1 p 1. This mans f, g ar dfind at t 1, f(t 1) = t 2 = g(t 1), and for xampl 2 p 2 t f g 2 : p 1 p 2 p 1. Lt t 2 = f(t 1 ) = g(t 1 ), by C4 on f or g, on has p 2 t 2, whnc p 1 R t 1 p 1. This provs [p 1 ] t 1 [p 1 ] t 1. On can show in th sam way th rvrs inclusion, which provs th lmma. Naturally, th lmma holds also for th othr arrow orintations, i.. for p 1, p 1 t 1 t 1 and for p 1, p 1 t 1 t 1. Initial placs. In q. (5), w assum th xistnc of (fak) transitions t i,0 with t i,0 = P 0 i and f(t 1,0 ) = g(t 1,0 ) = t 2,0. So initial placs in P ar givn by P 0 = {[p 1 ] t 1,0 : p 1 P 0 1 } (7) For p 1 P 1 and t 1 T 1, notic that th quivalnc class [p 1 ] t 1 (or quivalntly [p 1 ] t 1 ) may both contain markd placs of P1 0 and unmarkd placs of P 1 \ P1 0. Such a class is not takn as an initial plac of N. S th xampl of p in fig. 6. Convrsly, assum an quivalnc class [p 1 ] t 1 (for xampl) satisfis [p 1 ] t 1 P1 0. By lmma 1, [p 1 ] t 1 = [p 1 ] t0 1 which corrsponds to an initial plac of N. W could thus tak as an altrnat dfinition : P 0 = {p P : (p) P 0 1 } (8) Flow rlation. It is obviously dfind by p t whn (t) = t 1 and p = [p 1 ] t 1 for som p 1 t 1. But, using lmma 1, w can driv th simplr critrion : W procd symmtrically for t p. p t (p) (t) in N 1 (9) Exampl. Fig. 6 illustrats this construction. Obsrv that p 1 R t 1 p 1 and p 1 R t 1 p 1, which rsults in two classs/placs in N, both rlatd to p 1 by. Ths placs must indd b distinguishd : by mrging placs p and p in N, i.. by aggrgating classs sharing on or mor placs of P 1, th rsulting wouldn t b a morphism (C3 violatd). 3.3 Cohrnc of th dfinition : N N 1 is a nt morphism. C1 holds by dfinition, and with th trick of fak initial transitions, C2 is a consqunc of C3 and C4, which w only nd to xamin. 2 g f Th othr possibility is p 1 p 2 p 1, but this dosn t affct th proof. Irisa

11 Pullbacks of saf Ptri nts 9 t p p t t 1 t 1 p 1 p 1 f p"1 f g f g g g t 2 p 2 t 2 Figur 6: Th qualizr (N, ) (lft) for nts N 1 (cntr) and N 2 (right) rlatd by two morphisms f, g. Notic that t, t 1, t 2 could b th fak initial transitions, providd thir input placs would b rmovd. C4 obviously holds by construction of placs of P : if t 1 = (t), thn op : t 1 t dfind by op (p 1 ) = [p 1 ] t 1 is a total function. Similarly for op : t 1 t. For C3, considr p t in N, such that p p 1 and (t) = t 1. W want to chck that p 1 1 t 1 in N 1. By dfinition of th flow in N, on has p t iff (p) (t) = t 1, and p p 1 iff p 1 p, so p 1 1 t 1 holds. Th sam rasoning provs that : p p 1 is also a total function. N is a saf nt. By a standard argumnt : sinc : N N 1 is a nt morphism, it maps runs of N to runs of N 1. So if N is not saf, on of its run fills som plac with mor than on tokn, which rvals by a non saf run in N 1, bcaus is a total function on T. (N, ) satisfis th commutativ diagram. This is tru by construction for th partial functions on transitions. It also holds locally for rlations on placs, i.. around tripls of transitions (t, t 1, t 2 ) with t 1 = (t), t 2 = f(t 1 ) = g(t 1 )). This allows to rach compltly th plac rlations, f, g. 3.4 Univrsal proprty Assum th pair (N 3, h) satisfis f h = g h, with N 3 = (P 3, T 3, 3, P 0 3 ) and h : N 3 N 1. W look for a (uniqu) ψ : N 3 N satisfying h = ψ. Dfinition of ψ. On transitions, ψ is uniquly givn by ψ = 1 h, as it was sn in sction For placs, considr a tripl (t 3, t, t 1 ) T 3 T T 1 of rlatd transitions : ψ(t 3 ) = t and h(t 3 ) = t 1 = (t). W say that such a tripl (t 3, t, t 1 ) forms a triangl. From sction 3.1.2, w know that ψ op : t t 3 is uniquly dfind from h op : t 1 t 3 by p 1 t 1, ψ op ([p1] t 1 ) = h op (p 1 ) t 3 (10) (rcall that h op is ncssarily class invariant on t 1 ). W procd similarly to dfin ψ op : t t 3. PI n 1750

12 10 E. Fabr ψ satisfis th commutativ diagram. By construction of ψ, h = ψ is obvious on transitions, and locally on placs (i.. around triangls of transitions). To show that th rlation holds globally on placs, considr p 3 P 3. By assumption, p 3 is connctd to at last h on transition t 3 in N 3. If h is dfind at p 3 and p 3 p 1, thn h is also dfind at t 3 (by C3), h(t 3 ) = t 1 T and p 1 is connctd to t 1. W thn us h = ψ around th triangl (t 3, t, t 1 ), whr t = ψ(t 3 ). ψ is a nt morphism. It obviously satisfis C1, and C4 is imposd by th construction of ψ on placs. So only C3 has to b chckd, which is th difficult part of th proof. t 3 t 3 p 3 h t 1 p 1 ψ t p p t 1 t Figur 7: Proof that ψ satisfis C3. ψ For C3, considr a pair of placs (p 3, p) P 3 P rlatd by ψ (i.. p 3 p) and assum p 3 t 3 in N 3. W want to show that ψ is dfind at t 3, and ψ(t 3 ) p in N. By dfinition of ψ on placs, thr xists a triangl (t 3, t, t 1) T 3 T T 1 such that for xampl 3 t 3 3 p 3, t 1 1 p 1, t p and p = [p 1 ] t 1 (s Fig. 7). h is dfind at p 3, thus also at t 3 by C3. Sinc f h = g h, on has t 1 = h(t 3 ) T. So thr xists t T with (t) = t 1 and thus w alrady know that ψ is dfind at t 3 : ψ(t 3 ) = t. In othr words, (t 3, t, t 1 ) T 3 T T 1 forms anothr triangl. Sinc is a morphism, lt p b th imag of p 1 by op : t 1 t, so p = [p 1 ] t 1. By dfinition of ψ in th prsts of th triangl ψ (t 3, t, t 1 ), s (10), on has p 3 p. To conclud th proof, w thus hav to show that p = p. W ssntially us th fact that h is a morphism satisfying f h = g h. Lt p 1 b a plac of t 1 such that p 1 t 1 p 1. W know that p h 3 p 1, bcaus hop : t 1 t 3 h is class invariant (a consqunc of f h = g h). From p 3 3 t 3 in N 3 and p 3 p 1, w driv by C3 that p 1 1 t 1 = h(t ). W ar now xactly in th situation of lmma 1, so p 1 t 1 p 1. W hav thus provd that [p 1 ] t 1 and [p 1 ] t 1 ar idntical, or in othr words p = p. 4 Synthsis W now rassmbl all lmnts to provid a dfinition for pullbacks of saf labld nts. Th nxt sction rcalls th product dfinition (assuming a simpl synchronization algbra), that w combin to th qualizr to obtain th pullback. 3 Equivalntly, w could hav assumd that th rlatd placs li in th prsts (instad of post-sts) of a transition triangl. Irisa

13 Pullbacks of saf Ptri nts Product Lt N i = (P i, T i, i, P 0 i, Λ i, λ i ), i = 1, 2 b two labld nts. For nt products, w assum a simpl synchronization algbra : two transitions carrying th sam labl hav to synchroniz, whil transitions carrying a privat labl rmain privat. Privat labls ar thos in (Λ 1 \ Λ 2 ) (Λ 2 \ Λ 1 ). Th product N = N 1 N N 2 and th associatd projctions π i : N Ni ar dfind as follows 4 : 1. P = {(p1, ) : p 1 P 1 } {(, p 2 ) : p 2 P 2 } : disjoint union of placs, π i (p 1, p 2 ) = p i if p i and is undfind othrwis, 2. P 0 = π 1 1 (P 0 1 ) π 1 2 (P 0 2 ), 3. th transition st T is givn by T = {(t 1, ) : t 1 T 1, λ 1 (t 1 ) Λ 1 \ Λ 2 } {(, t 2 ) : t 2 T 2, λ 2 (t 2 ) Λ 2 \ Λ 1 } {(t 1, t 2 ) T 1 T 2 : λ 1 (t 1 ) = λ 2 (t 2 ) Λ 1 Λ 2 } π i (t 1, t 2 ) = t i if t i and is undfind othrwis, 4. th flow is dfind by t = π 1 (t) π 2 (t) and t = π 1 (t) π 2 (t), assuming π i (t) = π i (t) = if π i is undfind on t, 5. Λ = Λ1 Λ 2 and λ is th uniqu labling prsrvd by th π i. 4.2 Pullback Assum th f i : N i N 0 ar morphisms of labld nts. Th pullback N = N 1 N 2 is dfind as follows, by combining th dfinitions of product and qualizr (sction 2). Transitions. W distinguish shard transitions in N 1 and N 2, i.. thos having an imag in N 0, from privat ons, th othrs. For privat transitions, th dfinition of th pullback mimics th dfinition of th product. For shard transitions, only pairs that match through th f i ar prsrvd. T s = {(t 1, t 2 ) T 1 T 2 : t i Dom(f i ), f 1 (t 1 ) = f 2 (t 2 )} (11) T p = {(t 1, t 2 ) T 1 T 2 : t i Dom(f i ), λ 1 (t 1 ) = λ 2 (t 2 )} T = T s T p {(t 1, ) : t 1 T 1, t 1 Dom(f 1 ), λ 1 (t 1 ) Λ 1 \ Λ 2 } {(, t 2 ) : t 2 T 2, t 2 Dom(f 2 ), λ 2 (t 2 ) Λ 2 \ Λ 1 } (12) Notic that th labl condition dosn t appar in (11) : it coms as a consqunc of f 1 (t 1 ) = f 2 (t 2 ), sinc morphisms prsrv labls. 4 Rmark : if ons wishs to us th trick of fak initial transitions t 0 0 i to dfin initial markings Pi by Pi 0 = t 0 i, on has to assum that ach Λ i contains a spcial labl ɛ 0 rsrvd to th transition t 0 i. (13) PI n 1750

14 12 E. Fabr Placs. Placs ar obtaind by inspcting transitions slctd in T. Considr first a privat transition (t 1, t 2 ) T p, whr on (at most) of th t i can b. Assum p i i t i (or quivalntly t i i p i ) in N i, with t i. Obsrv that ncssarily p i Dom(f i ), othrwis f i would b dfind at t i. Such a plac p i inducs a singlton quivalnc class in P, ithr (p 1, ), or (, p 2 ). W dnot by P p all such privat placs. Considr now a pair of shard transitions (t 1, t 2 ) T s, whr f 1 (t 1 ) = t 0 = f 2 (t 2 ). Considr for xampl a plac p 1 t 1 (or quivalntly p 1 t 1, and symm. for a plac p 2 t 2 ). a. If p 1 Dom(f 1 ), thn [(p 1, )] (t 1,t 2 ) is rducd to (p 1, ), which yilds anothr privat plac in P p. b. If p 1 Dom(f 1 ), lt p 0 P 0 f 1 t 0 satisfy p 1 p0. By C4 applid to f 2, thr xists p 2 f 2 t 2 such that p 2 p0, so (p 1, ) R (t 1,t 2 ) (, p 2 ) in th product N 1 N N 2. Th rsulting quivalnc class [(p 1, )] (t 1,t 2 ), taks th form (Q 1, Q 2 ), with Q i P i, and yilds a shard plac in th pullback. In summary : P p = { (p 1, ) : p 1 P 1, p 1 Dom(f 1 ), (t 1, ) T, p 1 t 1 } { (, p 2 ) : p 2 P 2, p 2 Dom(f 2 ), (, t 2 ) T, p 2 t 2 } (14) P s = { (Q 1, Q 2 ) : Q i P i, Q i Dom(f i ), P = P p P s (t 1, t 2 ) T s, Q 1 Q 2 quiv. class of (t 1,t 2 ) In (14), th dot in (t 1, ) stands for ithr t 2 or, and symm. for th scond lin. or of (t 1,t 2 ) } (15) Initial placs. By abus of notation, lt us idntify a privat plac lik (p 1, ) to (Q 1, Q 2 ) = ({p 1 }, ), and (, p 2 ) to (Q 1, Q 2 ) = (, {p 2 }). So (Q 1, Q 2 ) dnots a gnral plac in P. (16) P 0 = {(Q 1, Q 2 ) P : Q 1 P 0 1, Q 2 P 0 2 } (17) Flow. Lt (Q 1, Q 2 ) P and (t 1, t 2 ) T (whr on of th t i can b ). Thn (Q 1, Q 2 ) (t 1, t 2 ) Q 1 t 1 in N 1, Q 2 t 2 in N 2 (18) (t 1, t 2 ) (Q 1, Q 2 ) Q 1 t 1 in N 1, Q 2 t 2 in N 2 (19) with th convntion that and hold. Morphisms g i. Lt (t 1, t 2 ) b a transition of T, on has g i (t 1, t 2 ) = t i if t i, and is g i undfind othrwis. Lt (Q 1, Q 2 ) b a gnral plac in P, on has (Q 1, Q 2 ) pi iff p i Q i. 4.3 Spcial cas W xamin hr th spcial cas whr morphisms f i : N i N 0 ar partial functions not only on transitions, but also on placs (instad of bing rlations on placs). Th dfinition changs Irisa

15 Pullbacks of saf Ptri nts 13 only for P s in (15) : whn plac duplications ar forbiddn, quivalnc classs of shard placs ar rducd to two lmnts only. P s = { (p 1, p 2 ) : p i P i Dom(f i ), f 1 (p 1 ) = f 2 (p 2 ) = p 0, (t 1, t 2 ) T s, f 1 (t 1 ) = f 2 (t 2 ) = t 0, p 0 t 0 } (20) This dfinition coincids with th proposition of [8] (and also to an arly vrsion of th prsnt nots), apart from th xtra condition that placs cratd in (14) and (20) b connctd to at last on transition of th pullback. Acknowldgmnt : spcial thanks to Mark Bdnarczyk for fruitful discussions. Rfrncs [1] S. Mac Lan, Catgoris for th Working Mathmatician, Springr-Vrlag, [2] G. Winskl, A nw Dfinition of Morphism on Ptri Nts, LNCS 166, pp , [3] M. Nilsn, G. Winskl, Ptri Nts and Bisimulation, BRICS rport no. RS-95-4, Jan [4] G. Winskl, Catgoris of modls for concurrncy, Sminar on Concurrncy, Carngi-Mllon Univ. (July 1984), LNCS 197, pp , [5] G. Winskl, Evnt structur smantics of CCS and rlatd languags, LNCS 140, 1982, also as rport PB-159, Aarhus Univ., Dnmark, April [6] M.A. Brnarczyk, L. Brnardinllo, B. Caillaud, Modular Systm Dvlopmnt with Pullbacks, ICATPN 2003, LNCS 2679, pp , [7] M.A. Brnarczyk, A. Borzyszkowski, R. Somla, Finit Compltnss of Catgoris of Ptri Nts, Fundamnta Informatica, vol. 43, no. 1-4, pp , [8] B. Konig, Paralll Composition and Unfolding of Ptri Nts (Including Som Exampls), privat communication, PI n 1750

On the construction of pullbacks for safe Petri nets

On the construction of pullbacks for safe Petri nets On th construction of pullbacks for saf Ptri nts Eric Fabr Irisa/Inria Campus d Bauliu 35042 Rnns cdx, Franc Eric.Fabr@irisa.fr Abstract. Th product of saf Ptri nts is a wll known opration : it gnralizs