On the construction of pullbacks for safe Petri nets

Transcription

1 On th construction of pullbacks for saf Ptri nts Eric Fabr Irisa/Inria Campus d Bauliu Rnns cdx, Franc Eric.Fabr@irisa.fr 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 papr, w considr a mor gnral way of combining nts, calld a pullback. Th pullback opration gnralizs th product to nts which intract both by synchronizd transitions and/or by a shard sub-nt (i.. shard placs and transitions). To obtain all pullbacks, w actually show that all qualizrs can b dfind in th catgory of saf nts. Combind to th known xistnc of products in this catgory, this givs mor than what w nd : w actually obtain that all small limits xist, i.. that saf nts form a complt catgory. 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 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. It thrfor offrs a vry natural way to build larg concurrnt systms from lmntary componnts. As a nic proprty, 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. Pullbacks can b usd, for xampl, to combin two concurrnt systms that synchroniz through common vnts and at th sam tim shar som rsourcs (.g. locks to accss data). Th notion of pullback has bn xtnsivly xplord for othr modls of concurrncy (transition graphs, graph grammars, tc.) [7], 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 consquntly unfoldings!), and alrady has a product.

2 Unfoldings hav bcom an important tool for th vrification of concurrnt systms [10,11,12,13,14,15]. Thy hav also bn advocatd for th monitoring of concurrnt systms [16]. In particular, this scond application domain rlis intnsivly on factorization proprtis of unfoldings : th fact that th unfolding of a product systm can b xprssd as a product of unfoldings of its componnts [17]. This proprty is actually th ky to distributd or modular monitoring algorithms (surprisingly, this approach has not bn xplord in modl chcking applications, to th knowldg of th author). Th drivation of th factorization proprty on unfoldings (or on othr structurs lik trlliss [18,19]) rlis on catgorical argumnts, and in particular on th fact that th unfolding opration prsrvs limits, lik th product for xampl. In ordr to obtain a similar proprty for othr ways of combining componnts, it is thrfor crucial to charactriz thm as catgorical limits. This is th main motivation of th prsnt work. Lt us mntion som contributions to th topic. B. Konig provids in [9] a dfinition for spcific pullback diagrams. M. Bdnarczyk t al. prov in [8] 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. Finally, lt us strss that [8] mntions in its introduction (p.3) that th xistnc of a pullback construction for saf Ptri nts has bn rportd... although th authors hav not bn abl to locat any rfrnc! It is thrfor usful to provid a simpl and dirct dfinition for this construction. 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 (sction 4). Scondly, w rcall (sction 2) that a pullback opration can b drivd from a product and an qualizr (s [1], chap. V-2, thm. 1, and [7], sc. 5). Sinc all products xist in Nts, w simplify th construction (and proofs) by building qualizrs, which is th hart of th contribution (sction 3). W finally gathr all pics to giv a comprhnsiv dfinition of th pullback of labld Ptri nts (sction 4), first in th gnral cas, thn in th spcific cas whr morphisms ar partial functions. Th conclusion undrlins som important consquncs of this construction. 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 [2] 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

3 C1. φ T : T 1 T 2 is a partial function, and φ P a rlation btwn P 1 and P 2, C2. P2 0 = φ P (P1 0 ) and p 2 P2 0, a uniqu 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, 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. Obsrv that condition C3 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 op (P 2 T 2 ). Notic that condition C3 ntails that th pair (φ P, φ T ) prsrvs th flow rlation (on its domain of dfinition). Togthr with C4 and C2, this guarants that a run of N 1 is mappd into a run of N 2 by φ T (s [2]), which is th last on should rquir from nt morphisms. Simplr dfinitions of nt morphisms would nsur this proprty, but C1-C4 ar actually ncssary to provid xtra catgorical proprtis, as w shall s in th squl. Rmark. Notic that condition C2 bcoms a consqunc of C3 and C4 whn on assums th xistnc of a fak initial transition t i,0 in ach N i, fd with a fak initial plac p i,0 i t i,0, such that t i,0 = Pi 0 and t 2,0 = φ(t 1,0 ), φ p 2,0 p 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 [2,4]. For labld nts, w naturally considr labl-prsrving morphisms to dfin th catgory λnts. Sction 4 will dtail th dfinition of this catgory. N 3 h 1 ψ h 2 N π 1 π 2 N 1 N 2 Fig. 1. Commutativ diagram of th product N = N 1 N 2. Product. Lt N 1, N 2 b nts, thir catgorical product N 1 N 2 in Nts is a nt N associatd to morphisms π i : N N i, i = 1, 2, satisfying th so-calld univrsal proprty of th product (fig. 1) : for vry othr candidat tripl (N 3, h 1, h 2 ) with h i : N 3 N i, thr xists a uniqu morphism ψ : N 3 N such that h i = π i ψ. This nt N = (P, T,, P 0 ) and th π i ar givn by [4,6]

4 1. P = {(p 1, ) : 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. T = (T 1 { }) ({ } T 2 ) (T 1 T 2 ), π i (t 1, t 2 ) = t i if t i and is undfind othrwis, 4. th flow is dfind as follows : for t T, t = π 1 1 ( π 1 (t)) π 1 2 ( π 2 (t)) and symm. for t, assuming π i (t) = π i (t) = if π i is undfind at t. At first sight, this catgorical product may look uslss sinc vry transition is fr to fir alon or jointly with any transition of th othr nt. Again, th intrst of this construction appars whn it is applid to labld nts, in association with a synchronization algbra [4]. Its practical intrst thn bcoms obvious to build larg systms starting from lmntary componnts. Sinc labls bring no tchnical difficulty othr than notational, w put thm asid until sction 4. N 3 h 1 ψ N g g 1 2 h 2 N 1 N 2 f 1 f 2 N 0 Fig. 2. Commutativ diagram of th pullback N = N 1 N 2. 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. 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. 2) : 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 mdiating morphism ψ : N 3 N such that h i = g i ψ. W dnot th pullback by N 1 N0 N 2, or by N 1 N 2 for short. 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 2, and th associatd canonical projctions π i : N 1 N 2 N i, i = 1, 2. In gnral, N 1 N 2 and th π i do not satisfy th pullback condition, i.. f 1 π 1 f 2 π 2. Howvr,

5 N 3 ψ N h N x 1 N 2 f 1 f 2 o π1 o π2 N 0 Fig. 3. Equalizing f 1 π 1 and f 2 π 2. 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. 3). It is straightforward to chck that (N, π 1, π 2 ) thn yilds th dsird pullback. For dtails, w rfr th radr to [1], chap. V-2, thm. 1, or to [7], sc. 5 whr this construction is also usd. 3 Equalizr in Nts Considr two nts N i = (P i, T i, i, Pi 0 ), 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. 4). N 3 ψ N h N 1 f g N 2 Fig. 4. 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 sts 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)

6 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. 4 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. S 2 F G S 1 a b c d E S _ a=b=c _ d Fig. 5. 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 ]. S fig. 5 for an xampl. 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.

7 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 P1 givn by P = {[p 1 ] t 1 : t 1 T, p 1 t 1 } {[p 1 ] t1 : 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) 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 1 p 2 t 2 : p 1 f p 2 g p 1. Lt t 2 = f(t 1 ) = g(t 1 ), by C3 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. t 1 t 2 p 1 p 1 f g p 2 t 1 t 2 Fig. 6. Idntity of quivalnc classs. 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. 1 g f Th othr possibility is p 1 p 2 p 1, but this dosn t affct th proof.

8 Initial placs. In q. (5), w assum th xistnc of (fak) transitions t i,0 with t i,0 = Pi 0 and f(t 1,0 ) = g(t 1,0 ) = t 2,0. So initial placs in P ar givn by P 0 = {[p 1 ] t1,0 : 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 ] t1 ) 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. 7. Convrsly, assum an quivalnc class [p 1 ] t 1 (for x.) 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. 7 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). t p p t t 1 t 1 p 1 p 1 p" 1 f f g f g g g t 2 p 2 t 2 Fig. 7. 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.

9 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. C4 obviously holds by construction of placs of P : lt t 1 = (t), thn op : t 1 t dfind by op (p 1 ) = [p 1 ] t 1 is a total function. And 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 [2] : 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 (s fig. 4). Dfinition of ψ. On transitions, ψ is uniquly givn by ψ = 1 h, as it was sn in sction 3.1. 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 th construction of co-qualizrs in sction 3.1, 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) Spcifically, h op (p 1 ) t 3 xists and is formd by a singl plac p 3 bcaus h is a nt morphism and thus satisfis C4. Morovr, this valu p 3 dosn t dpnd on th choic of p 1 in [p1] t 1 bcaus, as a co-qualizr h op is ncssarily class invariant on t 1 (s 3.1). W procd similarly to dfin ψ op : t t 3. 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 h p 1, connctd to at last on transition t 3 in N 3. If h is dfind at p 3 and p 3 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 ).

10 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 Fig. 8. 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 2 t 3 3 p 3, t 1 1 p 1, t p and p = [p 1 ] t 1 (s Fig. 8). 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 (t 3, t, t 1 ), s (10), ψ 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 h 1. W know that p 3 p 1, bcaus h op : t 1 t 3 is class invariant (a consqunc of f h = g h). From h 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 Application to pullbacks of labld nts W now rassmbl all lmnts to provid a dfinition for pullbacks of saf labld nts. Th first task is to dfin th catgory λnts. Considr labld 2 Equivalntly, w could hav assumd that th rlatd placs ar in th prsts (instad of post-sts) of a transition triangl.

11 nts N i = (P i, T i, i, P 0 i, λ i, Λ i ), φ : N 1 N 2 is a morphism in λnts iff φ is a nt morphism (as dfind in sction 2 by C1-C4), with th xtra rquirmnts : C5. φ T prsrvs labls, C6. Λ 1 Λ 2, C7. Dom(φ T ) = λ 1 1 (Λ 2). Th nxt sction rcalls th dfinition of th product in this catgory, that w combin to th qualizr to obtain th pullback. 4.1 Product Lt N i = (P i, T i, i, P 0 i, Λ i, λ i ), i = 1, 2 b two labld nts. To build nt products, w assum a simpl synchronization algbra [5] : 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 2 and th associatd projctions π i : N Ni ar dfind as follows 3 : 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 = π1 1 (P 1 0 ) π2 1 (P 2 0 ), 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 = π1 1 ( π 1 (t)) π2 1 ( π 2 (t)) and symm. for t, assuming π i (t) = π i (t) = if π i is undfind at t, 5. Λ = Λ1 Λ 2 and λ is th uniqu labling prsrvd by th π i. Lt us rcall that th product of labld nts can also b obtaind by taking th product of non-labld nts, and thn discarding transition pairs that violat th ruls of th synchronization algbra. For our choic of morphisms, it is straightforward to chck that th abov dfinition actually yilds th catgorical product in λnts : Th π i ar nt morphisms that obviously satisfy C5-C7. And for th univrsal proprty, with notations of fig. 1, th ψ computd in Nts (ignoring labls) is dfind by 4 t 3 T 3, ψ(t 3 ) = (h 1 (t 3 ), h 2 (t 3 )), so it clarly satisfis C5-C7 whn h 1, h 2 do. 3 Rmark : if ons wishs to us th trick of fak initial transitions t 0 i to dfin initial markings P 0 i by P 0 i = t 0 i, on has to assum that ach Λ i contains a spcial labl ɛ 0 rsrvd to th transition t 0 i. 4 with th convntion that ψ(t 3) = (, ) mans undfind.

12 4.2 Equalizr Similarly, th construction of qualizrs drivd in Nts naturally xtnds to qualizrs of labld nts. With notations of fig. 4, w tak Λ = Λ 1 for th labl st of N, and dfin th labling function by λ = λ 1. Th morphism : N N 1 thn clarly satisfis C5-C7. For th univrsal proprty, th morphism ψ : N 3 N is dfind on transitions by ψ T = 1 T h T. So Dom(ψ T ) = Dom(h T ), and ψ clarly satisfis C5-C 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 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) T = T s T p (13) 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. 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 symmtrically 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 t 2 such that p 2 f 2 p0, so (p 1, ) R (t 1,t 2) (, p 2 ) in th product N 1 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.

13 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 ), (t 1, t 2 ) T s, Q 1 Q 2 quiv. class of (t 1,t 2) or of (t1,t2) } (15) P = P p P s (16) In (14), th dot in (t 1, ) stands for ithr t 2 or, and symmtrically for th scond lin. 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. 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 undfind othrwis. Lt (Q 1, Q 2 ) b a gnral plac in P, on has g i (Q 1, Q 2 ) pi iff p i Q i. 4.4 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 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 [9] (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. An xampl is givn in fig. 9.

14 N a c g γ t 4 t 3 γ α β δ δ t 1 t 2 t 5 t 6 b d h i N 1 a c c g N 2 γ γ α β α t 4 t 3 t 1 t 2 t 1 α β δ δ t 1 t 2 t 5 t 6 b d d h i f c N 0 β α β α t 2 t 1 t 2 t 1 d Fig. 9. Exampl of a pullback : N = N 1 N 0 N 2, in th simpl cas of injctiv morphisms. Morphisms ar rprsntd by common nams on transitions and placs. Transition labls ar indicatd by Grk lttrs. Obsrv that transition t 1 of N 1 disappars in N sinc it finds no partnr in N 2 with th sam imag in th intrfac nt N 0. This xampl dosn t rflct th full gnrality of th pullback construction sinc outsid th domains of f 1 and f 2, transitions of N 1 and N 2 don t synchroniz : (Λ 1 Λ 2) \ Λ 0 =.

15 5 Conclusion Th original motivation for this work was th drivation of a simpl construction for pullbacks of saf nts, thus providing a way to xprss in a catgorical framwork th combination of nts that intract by sharing placs and transitions. W actually obtaind mor : w provd th xistnc of all qualizrs in Nts, which, in conjunction with th xistnc of all products, provs th xistnc of all (small) limits in Nts. Exprssing th combination of nts as a catgorical limit has som advantags. Considr for xampl th unfolding opration [4], that associats th unfolding U(N ) to a saf nt N. U is actually a functor from Nts to th subcatgory Occ of occurrnc nts, and w know that U : Nts Occ has a lft adjoint, and so prsrvs limits. As a consqunc, whn N = N 1 N0 N 2, on immdiatly obtains U(N ) = U(N 1 ) U(N0) O U(N 2 ) whr O dnots th pullback in Occ. This rsult xprsss that th factorizd form of a nt immdiatly givs ris to a factorizd form on runs of this nt. Morovr, on obtains for fr th xistnc of pullbacks in Occ, with a formal xprssion for O : lt O 0, O 1, O 2 b occurrnc nts, on has O 1 O0 O O 2 U(O 1 O0 O 2 ), whr th last pullback is computd in Nts, and whr mans isomorphic to. Th rsults abov naturally xtnd to gnral limits : whatvr th way on combins lmntary nts to build a largr systm (by products, pullbacks, tc.), a similar dcomposition holds on th unfolding (or on th trllis [19]) of th global systm. W bliv this is an important ky to study larg systms by parts (s [17,18] for xampls of modular diagnosis basd on ths idas). Acknowldgmnt : th author would lik to thank Mark Bdnarczyk for fruitful discussions, and Philipp Darondau for his usful commnts. Rfrncs 1. S. Mac Lan : Catgoris for th Working Mathmatician, Springr-Vrlag, G. Winskl : A nw Dfinition of Morphism on Ptri Nts, LNCS 166, pp , M. Nilsn, G. Winskl : Ptri Nts and Bisimulation, BRICS rport no. RS-95-4, Jan G. Winskl : Catgoris of modls for concurrncy, Sminar on Concurrncy, Carngi-Mllon Univ. (July 1984), LNCS 197, pp , G. Winskl : Evnt structur smantics of CCS and rlatd languags, LNCS 140, 1982, also as rport PB-159, Aarhus Univ., Dnmark, April G. Winskl : Ptri Nts, Algbras, Morphisms, and Compositionality, Information and Computation, vol. 72, pp , M.A. Bdnarczyk, L. Brnardinllo, B. Caillaud, W. Pawlowski, L. Pomllo : Modular Systm Dvlopmnt with Pullbacks, ICATPN 2003, LNCS 2679, pp , M.A. Bdnarczyk, A. Borzyszkowski, R. Somla : Finit Compltnss of Catgoris of Ptri Nts, Fundamnta Informatica, vol. 43, no. 1-4, pp , 2000.

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