Rapporto di Ricerca CS P. Baldan, A. Corradini, H. Ehrig, R. Heckel, B. König

Transcription

1 UNIVERSITÀ CA FOSCARI DI VENEZIA Dipartimento di Informatica Technica Report Series in Computer Science Rapporto di Ricerca CS Novembre 2006 P. Badan, A. Corradini, H. Ehrig, R. Hecke, B. König Bisimiarity and Behaviour-Preserving Reconfigurations of Open Petri Nets Dipartimento di Informatica, Università Ca Foscari di Venezia Via Torino 155, Mestre Venezia, Itay

2 Bisimiarity and Behaviour-Preserving Reconfigurations of Open Petri Nets Paoo Badan 1, Andrea Corradini 2, Hartmut Ehrig 3, Reiko Hecke 4, and Barbara König 5 1 Dipartimento di Informatica, Università Ca Foscari di Venezia, Itay 2 Dipartimento di Informatica, Università di Pisa, Itay 3 Institut für Softwaretechnik und Theoretische Informatik, Technische Universität Berin, Germany 4 Department of Computer Science, University of Leicester, UK 5 Institut für Informatik und Interaktive Systeme, Universität Duisburg-Essen, Germany Abstract. We propose a framework for the specification of behaviourpreserving reconfigurations of systems modeed as Petri nets. The framework is based on open nets, a mid generaisation of ordinary Pace/Transition nets suited to mode open systems which might interact with the surrounding environment and endowed with a coimit-based composition operation. We show that natura notions of (strong and weak) bisimiarity over open nets are congruences with respect to the composition operation. We aso provide an up-to technique for faciitating bisimiarity proofs. The theory is used to identify suitabe casses of reconfiguration rues (in the doube-pushout approach to rewriting) whose appication preserves the observationa semantics of the net. Introduction Petri nets are a we-known mode of concurrent and distributed systems, widey used both in theoretica and appicative areas [17]. In cassica approaches nets represent cosed, competey specified systems evoving autonomousy through the firing of transitions. However, ordinary Petri nets are not adequate to mode open systems, namey systems which can interact with the surrounding environment or, in a different view, systems which are ony partiay specified. Firsty, in this setting a arge (possiby sti open) system is typicay buit out of smaer open components. Syntacticay, an open system is characterised by the description of suitabe interfaces, over which the interaction with the externa environment can happen. Semanticay, openness can be represented by defining the behaviour of the component as if it were embedded in genera environments, determining any possibe interaction over the interfaces. Secondy, often the buiding components of an open system are not staticay determined, but they can change during the evoution of the system, according to predefined reconfiguration rues triggered by interna or externa soicitations. Research partiay supported by the EC RTN SegraVis, the EU IST SEnSOria the MIUR Project ART and the DFG project SANDS.

3 In this paper we provide a framework where open systems can be modeed as Petri nets and suitabe reconfigurations of such systems can be specified, which preserve the behaviour of the system. The framework is based on so-caed open nets, a mid generaisation of ordinary Petri nets introduced in [2, 3] to answer the first of the requirements above, i.e., the possibiity of interacting with the environment and of composing a arger net out of smaer open components. An open net is an ordinary net with a distinguished set of paces, designated as open, through which the net can interact with the surrounding environment. As a consequence of such interaction, tokens can be freey generated and removed in open paces. In the mentioned papers open nets are endowed with a composition operation, characterised as a pushout in the corresponding category, suitabe to mode both interaction through open paces and synchronisation of transitions. A deterministic process semantics à a Gotz-Reisig is shown to be compositiona with respect to such composition operation. In the foowing sections we first introduce marked open nets, i.e., open Petri nets with a distinguished initia marking. The existing theory of open nets, incuding the pushout-based composition operation, is extended to the marked case and a compositionaity resut for step sequences is proved in this context. Next we introduce bisimuation-based observationa equivaences for open Petri nets. Foowing the intuition about reactive systems discussed in [9], such equivaences are based on the observation of the interactions between the given net and the surrounding environment. The framework treats uniformy strong bisimiarity, where every transition firing is observed, and weak bisimiarity, where a subset of unobservabe transition abes is fixed and the firings of transitions carrying such abes (corresponding to τ transitions) are considered invisibe. Bisimiarity is shown to be a congruence with respect to the composition operation over open nets. Interestingy enough, this hods aso when the set of non-observabe events is not empty, i.e., for weak bisimiarity (some natura questions regarding the reation with weak bisimiarity in CCS are addressed in an Appendix). In addition, we aso define an up-to technique for faciitating bisimuation proofs. Expoiting the resuts in the first part of the paper we introduce next a framework for open Petri net reconfigurations. The fact that open net components are combined by means of categorica coimits, naturay suggests a mathematica framework for specifying net reconfigurations, based on doube-pushout (DPO) rewriting [6]. Using the congruence resut for bisimiarity we identify casses of transformation rues which ensure that reconfigurations of the system do not affect its observationa behaviour. A concuding section discusses some reated work, incuding the resuts on strong bisimiarity for Petri nets presented in [13,18] and the work on reconfigurations of ordinary Petri nets by means of rewriting rues in [1,10,16]. 1 Marked open nets An open net, as introduced in [2,3], is an ordinary P/T Petri net with a distinguished set of paces. These paces are intended to represent the interface of the net towards the environment, which can interact with the net by adding or removing some tokens in the open paces. Concretey, an open pace can be an input or an output pace 2

4 (or both), meaning that the environment can put or remove tokens from that pace. In this section we introduce the basic notions for open nets as presented in [3], generaising them to nets with initia marking. Given a set X we write 2 X for the powerset of X and X for the free commutative monoid over X. Moreover, given a function h : X Y we denote by the same symbo h : 2 X 2 Y its extension to sets and by h : X Y its monoida extension. Given a mutiset u X, with u = x X u x x, for x X we wi write u(x) to denote the coefficient u x. The symbo 0 denotes the empty mutiset. Definition 1 (mutiset projection). Given a function f : X Y and a mutiset u Y we denote by (u f) the projection of u aong f, which is the mutiset over X defined as (u f) = x X u f(x) x. In other words, ( f) : Y X is the monoida extension of the function ( f) : Y X defined by (y f) = x 1... x n when f 1 (y) = {x 1,..., x n }. In the foowing we wi mainy work with injective functions, for which the projection operation satisfies some expected properties, such as f ((u f)) u and (f ((u f)) f) = (u f). We consider nets where transitions are abeed over a fixed set of abes Λ. Definition 2 (P/T Petri net). A P/T Petri net is a tupe N = (S, T, σ, τ, λ) where S is the set of paces, T is the set of transitions (with S T = ), σ, τ : T S are functions mapping each transition to its pre- and post-set and λ : T Λ is a abeing function for transitions. In the foowing we wi denote by ( ) and ( ) the monoida extensions of the functions σ and τ to functions from T to S. Furthermore, given a pace s S, the pre- and post-set of s are defined by s = {t T : s t } and s = {t T : s t}. Definition 3 (Petri net category). Let N 0 and N 1 be Petri nets. A Petri net morphism f : N 0 N 1 is a pair of tota functions f = f T, f S with f T : T 0 T 1 and f S : S 0 S 1, such that for a t 0 T 0, f T (t 0 ) = f S ( t 0 ), f T (t 0 ) = f S (t 0 ) and λ 1 (f T (t 0 )) = λ 0 (t 0 ). The category of P/T Petri nets and Petri net morphisms is denoted by Net. Reca that category Net is a subcategory of the category Petri of [11], which has the same objects, but more genera morphisms which can map a pace to a mutiset of paces. We next introduce the notion of open net. As anticipated above, differenty from [2, 3], we work here with marked nets. This wi be used in the treatment of bisimiarity. Definition 4 (open net). An open net is a pair Z = (N Z, O Z ), where N Z = (S Z, T Z, σ Z, τ Z, λ Z ) is a P/T Petri net and O Z = (O + Z, O Z ) 2SZ 2 SZ are the sets of input and output open paces of the net. A marked open net is a pair (Z, û) where Z is an open net and û S Z is the initia marking. Hereafter, uness stated otherwise, a open nets wi be impicity assumed to be marked. An open net wi be denoted simpy by Z and the corresponding initia 3

5 marking by û. Subscripts carry over to the net components. The graphica representation for open nets is simiar to that for standard nets. In addition, the fact that a pace is input or output open is represented by an ingoing or outgoing danging arc, respectivey. For instance, in net Z 1 of Fig. 3, pace s is both input and output open, whie s is ony output open. The notion of enabedness for transitions is the usua one, but, besides the changes produced by the firing of the transitions of the net, we consider aso the interaction with the environment which is modeed by actions, denoted by + s and s, which produce or consume a token in an open pace s. Definition 5 (set of extended events). Let Z be an open net. The set of extended events of Z, denoted by T Z and ranged over by ǫ is defined as T Z = T Z {+ s : s O + Z } { s : s O Z }. Defining + s = 0 and + s = s, and symmetricay, s = s and s = 0, the notion of pre- and post-set extends in the obvious way to mutisets of extended events. We are now abe to define firing and steps in an open net. Given a marking u S we wi denote by + u the mutiset s S u(s) + s (provided that u contains ony input open paces). Simiary u = s S u(s) s. Definition 6 (firings and steps). Let Z be an open net. A step in Z consists of the execution of a mutiset of (extended) events A T Z, i.e., u A [A u A. A step is caed a firing when it consists of a singe event, i.e., A = ǫ T Z. Note that a firing can be (i) the execution of a transition u t [t u t, with u S Z, t T Z ; (ii) the creation of a token by the environment u [+ s u s, with s O + Z, u S Z ; (iii) the deetion of a token by the environment u s [ s u, with u S Z, s O Z. A step consists of the firing of a mutiset of transitions and of interactions with the environment, i.e., it is of the kind A w + v for w, v S Z. Definition 7 (open nets category). An open net morphism f : Z 1 Z 2 is a Petri net morphism f : N Z1 N Z2 such that, if we define in(f) = {s S 1 : f S (s) f T ( s) } and out(f) = {s S 1 : f S (s) f T (s ) }, then 1. (i) f 1 S (O+ 2 ) in(f) O+ 1 1 and (ii) fs (O 2 ) out(f) O û 1 = (û 2 f) (refection of initia marking). The morphism f is caed an open net embedding if both f T and f S are injective. We wi denote by ONet the category of open nets and open net morphisms. Intuitivey, an embedding f : Z 1 Z 2 inserts net Z 1 into a arger net Z 2, which constrains the behaviour of Z 1. Conditions 1.(i) and 1.(ii) first require that open paces are refected and hence that paces which are interna in Z 1 cannot be promoted to open paces in Z 2. Furthermore, the context in which Z 1 is inserted can interact with Z 1 ony through the open paces. To understand how this is formaised, 4

6 observe that in(f) incudes a the paces of Z 1 whose image f S (s) is in the post-set of a new transition, where new means not in the image of Z 1. Intuitivey we can think that in Z 2 new transitions are attached to s and can produce tokens in such pace. This is the reason why condition 1.(i) aso asks any pace in in(f) to be an input open pace of Z 1. Condition 1.(ii) is anaogous for output paces. Consistenty with the intuition that f inserts Z 1 into a arger context represented by Z 2, condition 2 requires that the marking of Z 1 is the projection of the marking of Z 2 : any pace s 1 S 1 must carry the same number of tokens of its image f(s 1 ) S 2, i.e., u 1 (s 1 ) = u 2 (f(s 1 )) for any s 1 S 1. A morphisms f 1, f 2, α 1 and α 2 in Fig. 3 are exampes of open net embeddings (the mappings on paces and transitions are those suggested by the shape and abeing of the nets). Consider, for instance, morphism f 1 : Z 0 Z 1. Note that in Z 1 a new c-abeed transition is attached to the paces s and s. This is ega since the corresponding paces in Z 0 are output open and input open, respectivey, in Z 1. Note aso that the number of tokens in paces in Z 0 and in their image through f 1 is the same. Instead, the number of tokens in the pace s in Z 1 is not constrained since it is not in the image of f 1 : the pace is marked, but f 1 woud have been a ega morphism aso if s were not marked. It is worth observing that most of the constructions in the paper wi be defined for open net embeddings, hence readers can imit their attention to embeddings if this heps the intuition. Sti, on the forma side, working in a arger host category with more genera morphisms is essentia to obtain a characterisation of the composition operation in terms of pushouts. Specificay, non-injective open net morphisms are needed as mediating morphisms (reca that the category of sets with injective functions does not have pushouts). Observe that the constraints characterising open nets morphisms have an intuitive graphica interpretation: The connections of transitions to their pre-set and post-set have to be preserved. New connections cannot be added. In the arger net, a new arc may be attached to a pace ony if the corresponding pace of the subnet has a danging arc in the same direction. Danging arcs may be removed, but cannot be added in the arger net. The number of tokens in each pace in the source net must be preserved in the target. Instead, there are no restrictions on the marking of paces of the target net which are not in the image of the source net. In the seque, given an open net morphism f = f S, f T : Z 1 Z 2, to ighten the notation we wi omit the subscripts S and T in its pace and transition components, writing f(s) for f S (s) and f(t) for f T (t). Moreover we wi write f : T Z 1 T Z 2 to denote the monoida function defined on the generators by f (t 1 ) = f(t 1 ) and f (x s1 ) = x f(s1) for x {+, }. The next proposition expicity shows that category ONet, as introduced in Definition 7, is we defined. To prove this fact we wi use the we-definedness of the category of non marked open nets, introduced in [3]. This category, denoted here by ONet u has (unmarked) open nets as objects and mappings satisfying ony condition 1 in Definition 7 as morphisms. These wi be referred to as unmarked open net morphisms. 5

7 Fig.1. Open net morphisms are not simuations. Proposition 1. Open net morphisms are cosed under composition. Proof. Let f 1 : Z 1 Z 2 and f 2 : Z 2 Z 3 be open net morphisms. Then f 1 and f 2 are unmarked open net morphisms and thus, since ONet u is a we-defined category, aso f 2 f 1 is an unmarked open net morphism. In order to prove that f 2 f 1 is a we defined open net morphism it remains to show that it satisfies aso condition 2 in Definition 7, i.e., that it refects the initia marking. But this fact foows easiy from the definition. In fact, for any s 1 S 1, Therefore û 3 (f 2 (f 1 (s 1 ))) = [since f 2 is an open net morphism] = û 2 (f 1 (s 1 )) [since f 1 is an open net morphism] = û 1 (s 1 ) Note that, different from most of the morphisms considered over Petri nets in the iterature, open net morphisms are not simuations. Instead, since open net embeddings are designed to capture the idea of inserting a net into a arger one, they are expected to refect the behaviour, in the sense that given an embedding f : Z 0 Z 1, the behaviour of Z 1 can be projected aong f to the behaviour of Z 0. This is consistent with the fact that we perform system composition by coimits: any component wi have a morphism into the fu system and we can not expect that the arger system is abe to simuate a (ess specified) component of itsef. The target net of a morphism is in genera more instantiated and thus more constrained than the source net (e.g., a pace which is open in the source net can be cosed in the target). As an exampe, consider the open net embedding in Fig. 1. Whie the transition abeed c in the net N 1 can fire infinitey many times, its image in the second net N 2 can fire ony once. Another possibiity woud be to perform system composition via imits and to use morphisms that are simuations (simiar to the morphisms considered in [19]). But doing composition via coimits is usuay more straightforward and gives us more contro over how paces shoud be attached. To formaise refection of the behaviour aong open nets embeddings, we define the projection operation aso over steps. 6

8 Definition 8 (projecting extended events). Given an open net embedding f : Z Z and an extended event ǫ T Z we define the projection of ǫ aong f as foows: if ǫ = t T Z { is a transition then t if t (t TZ and f(t) = t f) = ( t f) + (t f) if t f(t Z ) if ǫ = x s, with x {+, }, is an interaction with the environment, then (x s f) = x (s f). The projection operation over mutisets of extended events ( f) : T Z defined as the monoida extension of the projection of firings. T Z, is In words, if we think of the embedding as an incusion, given a transition t, the projection (t f) is the transition itsef if t is in Z. Otherwise, if t is not in Z but it consumes or produces tokens in paces of Z, the projection of t contains the corresponding extended events, expressing the interactions over open paces. The projection of an extended event + s is the empty mutiset, if s f(s Z ) and it is the same extended event if, instead, s f(s Z ). It is worth noticing that the step produced by the projection operation is wedefined, in the sense that, e.g., if + s (ǫ f) then s O + Z. In fact, if + s (t f) then s in(f) O + Z. If, + s (+ s f) this means that s O + Z and thus, since f(s) = s, by definition of open net morphism, s O + Z. The next emma express some properties of the projection operator over mutisets and over steps. Lemma 1 (properties of projection). Let f : Z Z be an open net embedding. Then 1. for x 1, x 2 S Z we have and for x S Z 2. for x 1, x 2 T Z we have and for x S Z 3. given A T Z 4. for u S Z we have ((x 1 x 2 ) f) = (x 1 f) (x 2 f) and (0 f) = 0 (f (x) f) = x ((x 1 x 2 ) f) = (x 1 f) (x 2 f) and (0 f) = 0 (f (x) f) = x (( A ) f) = (A f) and ((A ) f) = (A f) f ((u f)) u 7

9 Proof. We prove ony the third point. Since ( ) and ( ) are monoida functions it is sufficient to prove the resut ony on the generators. We concentrate on ( ), since the proof for ( ) is competey anaogous. We distinguish various cases: A = t T Z If there exists t T Z such that f(t) = t, then (t f) = t. Since f is an open net morphism f ( t) = t and thus, as desired (t f) = t = (f ( t) f) = (( t ) f) where the second equaity is justified by point (2). If, instead, t f(t Z ) we have that (t f) = ( t f) + (t f). Hence, in this case the resut is obvious since (t f) = ( ( t f) + (t f)) = (( t ) f) = (( t ) f) A = + s or A = s = s. In this case (A f) = (s f) and the re- Suppose, e.g., that A sut triviay hods. Lemma 2 (refection of behaviour). Let f : Z Z be an open net embedding. For every step u [A v in Z there is a step (u f) [(A f) (v f) in Z, caed the projection of the step u [A v over Z. Proof. Let f : Z Z be an open net embedding and assume that u [A v is a step in Z. Therefore u = u A and v = u A Now, we have (u f) = [by Lemma 1.(1)] = (u f) (( A ) f) [by Lemma 1.(3)] = (u f) (A f) and simiary (v f) = (u f) (A f) Therefore, as desired, there is the step (u f) = (u f) (A f) [(A f) (u f) (A f) = (v f). Observe that there is an obvious forgetfu functor F : ONet Net, defined by F(Z) = N Z and F(f : Z 0 Z 1 ) = f : N Z0 N Z1. Since functor F acts on arrows as identity, with abuse of notation, given an open net morphism f : Z 0 Z 1 we wi often write f : F(Z 1 ) F(Z 2 ) instead of F(f) : F(Z 1 ) F(Z 2 ). 8

10 2 Composing open nets We introduce next a basic mechanism for composing open nets which is characterised as a pushout construction in the category of open nets. The case of unmarked nets was aready discussed in [3]. Here we extend the theory to dea with marked open nets. This wi aow ater to define reconfigurations of open nets, where the appicabiity of a reconfiguration rue can depend on the marking. Intuitivey, two open nets Z 1 and Z 2 are composed by specifying a common subnet Z 0, and then by joining the two nets aong Z 0. The categorica characterisation wi be usefu for proving the resuts about behaviour preserving transformations in Section 4, but the concrete characterisation in terms of a join aong a common subnet can hep the intuition. Let us start with a technica definition which wi be usefu beow. Proposition 2 (composition of mutisets). Consider a pushout diagram in the category of sets as beow, where a morphisms are injective. Given u 1 S 1 and u 2 S 2 such that (u 1 f 1 ) = (u 2 f 2 ) = u 0, then there is a (unique) mutiset u 3 S 3 such that (u 3 α i ) = u i, for i {1, 2}. Such a marking u 3 wi be denoted by u 3 = u 1 u0 u 2. f 1 S 0 f 2 S 1 S 2 α 1 α S 2 3 Additionay, if u 3 = u 1 u0 u 2 and u 3 = u 1 u 0 u 2, then u 3 u 3 = (u 1 u 1 ) (u 0 u 0 ) (u 2 u 2 ). Proof. Define u 3 = s 3 S 3 max{u 1 (α 1 1 (s 3)), u 2 (α 1 2 (s 3))} s 3. 6 In order to prove that (u 3 α 1 ) = u 1, notice that, since f 1 is an embedding, this amount to show that for any s 1 S 1 we have u 1 (s 1 ) = u 3 (α 1 (s 1 )). Now that for s 1 S 1, we have have two cases: s 1 = f 1 (s 0 ) Then, since by hypothesis (u 1 f 1 ) = u 0 = (u 2 f 2 ) we have u 3 (α 1 (s 1 )) = max{u 1 (f 1 (s 0 )), u 2 (f 2 (s 0 ))} = max{u 0 (s 0 ), u 2 (f 2 (s 0 ))} = u 0 (s 0 ) = u 1 (s 1 ) s 1 not in the image of f 1 (s 0 ) In this case we have: u 3 (α 1 (s 1 )) = max{u 1 (s 1 ), u 2 (α 1 2 (α 1(s 1 )))} = max{u 1 (s 1 ), u 2 ( )} = u 1 (s 1 ) The uniqueness of u 3 is immediate, since α 1 and α 2 are jointy epi. Concerning the second part of the statement, et u 3 = u 1 u0 u 2 and u 3 = u 1 u u 0 2. Then just observe that by Lemma 1.(1), we have for i {1, 2} (u 3 u 3 α i ) = (u 3 α i ) (u 3 α i ) = u i 6 Reca that the α i are injective and note that we are abusing the notation considering u i( ) = 0 and u i({s i}) = u i(s i). 9

11 f 1 Z 0 f 2 Z 1 α 1 Z 2 α 2 Z 3 Fig.2. Pushout in ONet. hence the resut u 3 u 3 = (u 1 u 1) (u0 u 0 ) (u 2 u 2), foows by the defining property of the sum of markings. Intuitivey, the mutiset u 1 u0 u 2 can be seen as the east upper bound of the images of the mutisets over S 1 and S 2. As in [2,3], we say that two embeddings f 1 : Z 0 Z 1 and f 2 : Z 0 Z 2 are composabe if the paces which are used as interface by f 1, namey the paces in(f 1 ) and out(f 1 ), are mapped by f 2 to input and output open paces of Z 2, respectivey, and aso the symmetric condition hods. If, and ony if, these conditions hod the pushout of f 1 and f 2 can be computed in Net and then ifted to ONet. Definition 9 (composabiity). Let f 1 : Z 0 Z 1, f 2 : Z 0 Z 2 be embeddings in ONet (see Fig. 2). We say that f 1 and f 2 are composabe if 1. f 2 (in(f 1 )) O + Z 2 and f 2 (out(f 1 )) O Z 2 ; 2. f 1 (in(f 2 )) O + Z 1 and f 1 (out(f 2 )) O Z 1. Proposition 3 (pushouts in ONet). Let f 1 : Z 0 Z 1 and f 2 : Z 0 Z 2 be embeddings in ONet (see Fig. 2). Compute the pushout of the corresponding diagram in category Net (componentwise on paces and transitions) obtaining the net N Z3 and the morphisms α 1 and α 2, and then take as open paces, for x {+, }, O x Z 3 = {s 3 S 3 : α 1 1 (s 3) O x Z 1 α 1 2 (s 3) O x Z 2 } and as marking u 3 = u 1 u0 u 2, defined according to Proposition 2. Then (α 1, Z 3, α 2 ) is the pushout in ONet of f 1 and f 2 if and ony if f 1 and f 2 are composabe. In this case we write Z 3 = Z 1 + f1,f 2 Z 2. Proof. We know, that the above resut hods for unmarked nets, i.e., in the category ONet u. Here we must additionay show that (i) α i are marked morphisms and that (ii) if we take any other net Z 3, with α i : Z i Z 3 making the diagram commute, then the mediating morphism γ : Z 3 Z 3 (which exists uniquey as an unmarked net morphism by the resut in [3]) respects the condition on the marking. Now, (i) is immediate since Proposition 2 tes us that (û 3 α i ) = û i for i {1, 2}. Property (ii) can be proved aong the same ines. As an exampe, the open net embeddings f 1 and f 2 in Fig. 3 are composabe and Z 3 is the resuting pushout object. 10

12 Fig.3. An exampe of a pushout in ONet. 3 Composing steps In this section we anayse the behaviour of an open net Z 3 arising as the composition of two open nets Z 1 and Z 2 aong an interface net Z 0. More specificay, concentrating on steps, we show that steps of the component nets Z 1 and Z 2 can be composed to give rise to a step of Z 3 when they agree on the interface and satisfy suitabe compatibiity conditions. We start with a technica emma which wi be pivota in the paper. Lemma 3. Let f 1 : Z 0 Z 1 and f 2 : Z 0 Z 2 be composabe embeddings in ONet and et Z 3 = Z 1 + f1,f 2 Z 2 (see Fig. 2). Let u 1 [A 1 v 1 and u 2 [A 2 v 2 be steps in Z 1 and Z 2, respectivey, such that (u 1 f 1 ) = (u 2 f 2 ) = u 0 and A 2 = f 2 ((A 1 f 1 )). Then, (v 1 f 1 ) = v 0 = (v 2 f 2 ) and, if we define A 3 = α 1 (A 1 ), u 1 u0 u 2 [A 3 v 1 v0 v 2. Proof. First observe that, since A 2 = f 2 ((A 1 f 1 )), by Lemma 1.(2), (A 2 f 2 ) = (A 1 f 1 ). Let A 0 = (A i f i ) be the common projection. By the above, obviousy, we have (A 2 f 2 ) = (A 1 f 1 ) and thus, by Lemma 1.(3) (( A 1 ) f 1 ) = (( A 2 ) f 2 ) 11

13 so that we can consider the sum of markings A 1 A 0 A 2. We caim that and simiary, ((A 1 ) f 1 ) = ((A 2 ) f 2 ) and A 3 = A 1 A 0 A 2 (1) A 3 = A 1 A0 A 2 In fact, et us concentrate on ( ). We have that (( A 3 ) α 1 ) = [by Lemma 1.(3)] = (A 3 α 1 ) = [by definition of A 3 ] = (α 1 (A 1 ) α 1 ) = [by Lemma 1.(2)] = A 1 Therefore, to concude the vaidity of (1) we ony need to show that aso ( A 3 α 2 ) = A 2 (see Proposition 2). Now, we have (( A 3 ) α 2 ) = [by Lemma 1.(3)] = (A 3 α 2 ) = [by definition of A 3 ] = (α 1 (A 1 ) α 2 ) Therefore we must get (α 1 (A 1 ) α 2 ) = A 2 and this is proved by showing (α 1 (A 1 ) α 2 ) = A 2 (2) To this aim, we proceed by induction on the cardinaity of A 1 : A 1 = 0 In this case A 2 = f 2 ((A 1 f 1 )) = 0 = (α 1 (A 1 ) α 2 ), as desired. A 1 = t 1 We distinguish two subcases. If (t 1 f 1 ) = t 0 T Z0 then A 2 = f 2 (t 0 ) = (α 1 (t 1 ) α 2 ), as desired, by construction of the pushout. If, instead, (t 1 f 1 ) = ( t 1 f 1) + (t1 f 1), then by hypothesis, A 2 = f 2 ((A 1 f 1 )) and thus A 2 = f2 (( t 1 f 1)) + f2 ((t 1 f 1)) On the other hand, we have (α 1 (A 1 ) α 2 ) = (α 1 (t 1 ) α 2 ) = (( α 1(t 1)) α 2) + (((α1(t 1)) α 2) Now, by expoiting the fact that Z 3 is a pushout, it is easy to see that f 2 (( t 1 f 1 )) = (( α 1 (t 1 )) α 2 ) and simiary f 2 ((t 1 f 1 )) = ((α 1 (t 1 )) α 2 ). Hence we concude that A 2 = (α 1 (A 1 ) α 2 ), as desired. A 1 = + s1 or A 1 = s1 Assume, for instance, that A 1 = + s1 (the other case is competey anaogous). Therefore A 2 = f 2 ((A 1 f 1 )) = + f2 ((s 1 f 1)) On the other hand (α 1 (A 1 ) α 2 ) = (+ α1(s 1) α 2 ) = + (α1(s 1) α 2) and, again, by the fact that Z 3 is a pushout, we deduce easiy that f 2 ((s 1 f 1 )) = (α 1 (s 1 ) α 2 ), hence the desired equaity. 12

14 A 1 = A 1 A 1, with A 1, A 1 0. In this case we have (α 1 (A 1 ) α 2 ) = = ((α 1 (A 1 ) α 1 (A 1 )) α 2) = [by Lemma 1.(1)] = (α 1 (A 1) α 2 ) (α 1 (A 1) α 2 ) On the other hand A 2 = f 2 ((A 1 f 1 )) = [by Lemma 1.(1)] = f 2 ((A 1 f 1 ) (A 1 f 1 )) = = f 2 ((A 1 f 1 )) f 2 ((A 1 f 1 )) = [by inductive hypothesis] = (α 1 (A 1 ) α 2) (α 1 (A 1 ) α 2) Hence, the desired equaity foows. This concudes the proof of (2), from which (1) foows. Now, by expoiting (1) we can easiy concude. In fact, the steps in Z 1 and Z 2 are of the kind u i = u i A i [A i u i A i for i {1, 2}. First observe that, since (u 1 f 1 ) = (u 2 f 2 ) and ( A 1 f 1 ) = ( A 2 f 2 ), we immediatey get: (u 1 f 1 ) = (u 2 f 2 ) Let u 0 = (u i f i) be the common projection. Since v i = u i A i, for i {1, 2}, by the fact that (A 1 f 1 ) = (A 2 f 2 ), we deduce that, as desired (v 1 f 1 ) = (v 2 f 2 ) Hence, if v 0 = (v i f i ) is the common projection, we can define v 3 = v 1 v0 v 2. Now, if we set u 3 = u 1 u u 0 2 we have u 3 = u 1 u0 u 2 = = (u 1 A 1 ) u 0 A 0 (u 2 A 2 ) = [by Proposition 2] = (u 1 u 0 u 2 ) ( A 1 ) A 0 A 2 ) = [by (1)] = u 3 A 3 Therefore we have the step u 3 [A 3 u 3 A 3. By a sequence of passages anaogous to those used above, we can show that u 3 A 3 = v 3 and thus, as desired u 3 [A 3 v 3. The fact that such step projects to u i [A i v i for i {1, 2} immediatey foows by construction. We are then abe to show how steps of the components net can be joined to ead a step of their composition. This can be done when the steps satisfy a suitabe compatibiity condition, defined beow. Definition 10 (compatibe steps). Let Z 3 = Z 1 + f1,f 2 Z 2 be a pushout in ONet (see Fig. 2). We say that two steps u i [A i v i (i {1, 2}) are compatibe if (u 1 f 1 ) = (u 2 f 2 ) and we can decompose the steps as A i = A i A i (i {1, 2}) such that 13

15 A 2 = f 2 ((A 1 f 1)) and A 1 = f 1 ((A 2 f 2)) It is immediate to see that if A 1 and A 2 are compatibe, then (A 1 f 1 ) = (A 2 f 2 ). Lemma 4 (composing steps). Let f 1 : Z 0 Z 1 and f 2 : Z 0 Z 2 be composabe embeddings in ONet and et Z 3 = Z 1 + f1,f 2 Z 2. Let u 1 [A 1 v 1 and u 2 [A 2 v 2 be compatibe steps Then there exists a unique step u 3 [A 3 v 3 which is projected to u i [A i v i aong α i for i {1, 2}. Proof. By definition of compatibiity, we know that A 1 and A 2 can be decomposed as A i = A i A i (i {1, 2}) such that A 2 = f 2 ((A 1 f 1 )) and A 1 = f 1 ((A 2 f 2 )). Moreover, (u 1 f 1 ) = u 0 = (u 2 f 2 ). Now, since u i [A i A i v i, we can find markings u i, u i, v i, v i such that u i [A i v i and (u 1 f 1) = (u 2 f 2) = u 0 u i [A i v i and (u 1 f 1) = (u 2 f 2) = u 0 In fact, just observe that, since u i [A i v i, the marking u i must be of the kind w i A i A i and thus we coud choose u i = A 1, v i = A i, u i = A i w i and v i = A i w i. Therefore, we can use Lemma 3 and, defining u 3 = u 1 u 0 u 2, u 3 = u 1 u 0 u 2, v 3 = v 1 v 0 v 2, v 3 = v 1 v 2, we concude 0 v u 3 [α 1 (A 1) v 3 and u 3 [α 2 (A 2) v 3 Therefore u 3 u 3 [α 1 (A 1 ) α 2 (A 1 ) v 3 v 3 By expoiting Proposition 2, we easiy see that u 3 u 3 = (u 1 u 1) u 0 u (u 0 2 u 2 ) = u 1 u0 u 2, where u 0 denotes the common projection of u 1 and u 2 over Z 0. Simiary, v 3 v 3 = v 1 v0 v 1 and thus u 1 u0 u 1 [α 1 (A 1 ) α 2 (A 1 ) v 1 v0 v 1 is the desired step. The fact that it projects over the steps we started from in Z 1 and Z 2 foows by construction. 4 Bisimiarity of open nets In this section we study (strong and weak) bisimiarity for open nets. Then we prove that bisimiarity is a congruence with respect to the coimit-based composition of open nets. First, we define a abeed transition system associated to an open net. Net transitions carry a abe which is observed when they fire. Additionay, in the abeed transition system we aso observe what happens at the open paces. As 14

16 discussed in the concusions, this resembes the abeed transition system arising from the view of Petri nets as reactive systems in [12,18]. Given an open net Z, the corresponding abeed transition system has the markings of the net as states. Transitions are generated by the firings of Z and abeed over the set Λ Z = Λ {+ s : s O + Z } { s : s O Z }. For notationa convenience we extend the abeing function λ Z to the set of extended events T Z, by defining λ Z (x) = x for x T Z T Z (i.e., when x = + s or x = s with s S Z ). Definition 11 (ts for an open net). The abeed transition system associated to an open net Z, denoted by ts(z), is the pair S Z, Z, where states are markings u Z S Z and the transition reation Z S Z Λ Z S Z incudes a transitions u Z λ Z(x) Z u Z such that there is a firing u Z [x u Z in Z. It commony happens that, when observing the behaviour of a system, ony a subset of events is considered observabe or important. In our setting this is formaised by seecting a subset of abes representing interna/not observabe firings (paying the roe of τ-transitions) and then considering a corresponding notion of weak bisimiarity. Let Λ τ Λ be a subset of unobservabe abes, which is fixed for the rest of the paper. Given a Λ-abeed open Petri net Z, for markings v, v S Z et us write v Z v if v Z v with Λ τ, and v Z v if v Z v with Λ Z Λ τ. Then we define v = Z v when v( Z ) v. v = Z v when v( Z ) Z ( Z ) v. 4.1 Bisimiarity The notion of weak bisimiarity is now defined in a standard way (but note that when the set of unobservabe abes is empty, this actuay corresponds to strong bisimiarity). Ony, we need to specify for each open pace of one net which is the corresponding open pace in the other net, i.e., bisimuations between two nets are parametrised by a bijection between their open paces. Given two open nets Z 1 and Z 2 we write η : O 1 O 2 and say that η is a correspondence between Z 1 and Z 2 if η : O + 1 O 1 O+ 2 O 2 is a bijection such that for s 1 O 1, x {+, } we have s 1 O x 1 iff η(s 1) O x 2. Definition 12 ((weak) bisimiarity). Let Z 1, Z 2 be open nets and η : O 1 O 2 be a correspondence between Z 1 and Z 2. A (weak) η-bisimuation over Z 1 and Z 2 is a reation over their markings R S 1 S 2 such that if (u 1, u 2 ) R then if u 1 Z1 u 1, then there exists u 2 such that u 2 = Z2 u 2 and (u 1, u 2 ) R; 15

17 if u 1 Z1 u 1, then there exists u 2 such that u η() 2 = Z2 u 2 and (u 1, u 2 ) R; the symmetric conditions hods; where η(+ s ) = + η(s), η( s ) = η(s), and η() = for any Λ. Two open nets Z 1 and Z 2 are (weaky) η-bisimiar, denoted Z 1 η Z 2, if η : O 1 O 2 is a correspondence and there exists a (weak) η-bisimuation R over Z 1 and Z 2 such that (û 1, û 2 ) R. Sometimes we wi simpy say that Z 1 and Z 2 are (weaky) bisimiar, omitting the correspondence η. Notice that weak bisimuation bois down to the notion of strong bisimuation when a abes are observabe, i.e., when Λ τ =. For convenience of the reader we make expicit the notion of strong bisimiarity. Definition 13 (strong bisimiarity). When Z 1 and Z 2 are weaky η-bisimiar open nets, with Λ τ = we say that Z 1 and Z 2 are strongy η-bisimiar. Expicity, a strong η-bisimuation over Z 1 and Z 2 is a reation over their markings R S 1 S 2 such that if (u 1, u 2 ) R then if u 1 Z1 u 1, then there exists u 2 such that u η() 2 Z2 u 2 and (u 1, u 2 ) R; if u 2 Z1 u 2, then there exists u η 1 such that u 1 () 1 Z2 u 1 and (u 1, u 2) R; As aready mentioned, open net morphisms are not simuations, since the target net can be more instantiated than the source net. However, according to the foowing emma, which is a coroary of Lemma 3 and Lemma 4, given composabe embeddings f 1 : Z 0 Z 1 and f 2 : Z 0 Z 2, the firing of a transition in Z 2, projected aong f 2 to Z 0 can then be simuated in Z 1. We first prove a preiminary emma on unabeed steps, which wi have the desired resut on abeed transitions as an easy coroary. Lemma 5. Let f 1 : Z 0 Z 1 and f 2 : Z 0 Z 2 be composabe embeddings in ONet and et Z 3 = Z 1 + f1,f 2 Z 2. Let u 2 [A 2 v 2 be a step consisting of transitions ony, i.e., A 2 T Z2. Then u 1 [f 1 ((A 2 f 2 )) v 2 and u 3 [α 2 (A 2 ) v 3 = v 1 v 0 v 2. Proof. Let A 1 = f 1 ((A 2 f 2 )). First note that A 1 is we-defined, i.e., A 1 T Z 1. In fact, et + s1 A 1 and et us show that s 1 is input open. From the assumption we deduce that there is + s0 (A 2 f 2 ) with f 1 (s 0 ) = s 1. Examining the definition of projection for steps, since A 2 consists ony of transitions, this impies that f(s 0 ) t 2, with t 2 f 2 (T Z0 ) and thus s 0 in(f 2 ). Since f 1 and f 2 are composabe, we have that s 1 = f 1 (s 0 ) f 1 (in(f 2 )) O + Z 1, as desired. Now observe that A 1 = (f 1 ((A 2 f 2 ))) [by def. of open net morphism] = f 1 ((A 2 f 2 )) [by Lemma 1.(3)] = f 1 (( A 2 f 2 )) Now, since the step u 2 [A 2 v 2 is enabed, we know that A 2 u 2, and thus A 1 = f 1 (( A 2 f 2 )) f 1 ((u 2 f 2 )) [since (u 2 f 2 ) = (u 1 f 1 )] = f 1 ((u 1 f 1 )) [by Lemma 1.(4)] u 1 16

18 f 1 (Z 0, û 0) f 2 f 1 (Z 0, û 0) g 2 (Z 1, û 1) α 1 (Z 2, û 2) α 2 (Z3, û 3) (a) (Z 1, û 1) β 1 (W 2, ˆv 2) β 2 (W 3, ˆv 3) (b) Fig.4. Pushouts in ONet. Hence, the step u 1 [A 1 v 1 can be performed. Ceary, the two steps in Z 1 and Z 2 are compatibe, and thus we concude by Lemma 4. Lemma 6. Let Z 0, Z 1, Z 2 be open nets and et f i : Z 0 Z i (i {1, 2}) be composabe embeddings, as in Fig. 2. Furthermore, et Z 3 = Z 1 + f1,f 2 Z 2. Assume u 2 Z2 u 2 where Λ and et t T 2 such that λ 2 (t) = and u 2 [t u 2, et u 0 [A 0 u 0 be its projection over Z 0 (hence A 0 = (t f 2 )) and et u 1 0 Z0... n Z0 u 0 be any sequence of transitions in ts(z 0 ) arising as a inearisation of such step in Z 0. Then we have, for any u 1 S 1 such that (u 1 f 1 ) = u 0 : u 1 1 Z1... n Z1 u 1 and u 1 u0 u 2 Z3 u 1 u u 0 2. Proof. Immediate by Lemma 5. Note that above, when transition t is in the image of Z 0 the sequences of transitions in ts(z 0 ) and ts(z 1 ) are actuay singe firings. Instead, they are sequences of interactions over open paces, possiby of ength greater than one, otherwise. By expoiting this emma we can prove that bisimiarity is a congruence with respect to the composition operation on open nets. Theorem 1 (bisimiarity is a congruence). Let Z 0, Z 1, Z 2, W 2 be open nets. Let Z 2 η W 2, for some η. Consider the nets Z 3 = Z 1 + f1,f 2 Z 2 and W 3 = Z 1 + f1,g 2 W 2, as in Fig. 4 where f 1, f 2, g 2 are embeddings and f 1, f 2 and f 1, g 2 are composabe. If g 2 O0 = η (f 2 O0 ) (i.e., f 2 and g 2 are consistent with η on open paces) then Z 3 η W 3, where η is defined as foows: η (α 1 (s 1 )) = β 1 (s 1 ), whenever α 1 (s 1 ) is open, and η (α 2 (s 2 )) = η(s 2 ), whenever α 2 (s 2 ) is open. Proof. Let us compose the nets Z 0, Z 1, Z 2 and W 2 as shown in Fig. 4, where (η f 2 ) O0 = g 2 O0. To simpify the notation, assume, without oss of generaity, that, a the morphisms in the diagrams of Fig. 4 are incusions and η = id. Hence f 2 O0 = g 2 O0. Now et R be a weak η-bisimuation over Z 2 and W 2 such that (û 2, ˆv 2 ) R, which exists by hypothesis. Consider the reation R over Z 3 and W 3 defined as R = {(u 1 u0 u 2, v 1 v0 v 2 ) : (u 2, v 2 ) R u, v S Z0. u 1 u = v 1 v} The condition on u 1 and v 1 means that the two markings can differ for the number of tokens in paces of the interface net Z 0. Notice that the marking of Z 0 is competey determined by the marking of components Z 2 and W 2. 17

19 We caim that R is a weak η -bisimuation over Z 3 and W 3, where η is again identity on open paces. Since, by the construction of the pushout, (û 3, ˆv 3 ) = (û 1 û 0 û 2, û 1 û 0ˆv 2 ) R, this provides the desired resut. In fact, assume that u 3 Z3 u 3. We distinguish various possibiities: 1. = + s (token put by the environment in an open input pace) There are severa possibiities: (a) s S 0. In this case, since s O + Z 3, necessariy s O + Z i for i {0, 1, 2} and thus we can have the same firing in a the other components Z 0, Z 1 and Z 2, i.e., for i {1, 2, 3} and notice that u 3 = u 1 u 0 u 2. Since (u 2, v 2 ) R, we have that u i + s Zi u i. v 2 + s = W2 v 2 The above sequence of firings projects over Z 0 to h Z0 + s Z0 1 k Z0 v 0 v 1 0 Z0... Z0... where any i Λ can be a τ-transition or an interaction with the environment. Hence, by Lemma 6 (appied severa times) and recaing that s O 1 + we deduce that there wi be a firing sequence in Z 1 v 1 1 Z1... h + s 1 Z1 Z1 Z1... k Z1 v 1 (3) + and v s 3 = W3 v 3 for v 3 = v 1 v 0 v 2. Since (u 3, v 3 ) R, by construction u 1 and v 1 differ ony for the number of tokens in the (the image of) Z 0. Hence, by construction, aso u 1 and v 1 can differ ony for the number of tokens in the paces in Z 0 (these are the ony paces affected by the transitions abeed + s, 1,..., h, 1,..., k ). Hence (u 3, v 3 ) R, as desired. (b) s S 1 S 0 Since s O + Z 3, necessariy s O + Z 1. The projections of the step in Z 3 over Z 1 is + u s 1 Z1 u 1 s. whie Z 0 and Z 2 stay ide. Ceary an anaogous step can be performed starting from v 1 : + v s 1 Z1 v 1 s. and, since s S 0, pace s is open input aso in W 3 The empty step in Z 2, by Lemma 4, can be composed with the above step eading to: + v 3 = v 1 v0 v s 2 Z3 (v 1 s) v0 v s = v 3 and, as desired, (u 3, v 3 ) R. (c) s S 2 S 0 Anaogous to first case, but for the fact that firing + s is not in the firing sequence (3) for S 1. 18

20 2. = s (token taken by the environment from an open output pace) Anaogous to the previous case, by repacing + s with s. We ony need the additiona observation that in the firing sequence (3), the firing s is possibe, because it is possibe in Z 0 and the marking of s is the same in Z 0 and Z = a Λ Λ τ In this case, the associated transition in Z 3 is of the kind u 3 [t u 3 where λ 3 (t) = a. We distinguish various subcases. (a) t in Z 1 and not in Z 2 The firing of t in Z 3 projects to the firing of a transition in Z 1, and to a (possiby empty) step consisting ony of interactions with the externa environment in Z 0 and Z 2. Let us consider a inearisation of such steps where a s firings precede the + s firings, i.e., u 1 i Zi... n Zi u i i {0, 2}. where i = si for i {1,...,h} and i = + si for i {h + 1,...,n}. By the fact that (Z 2, u 2 ) and (W 2, v 2 ) are weaky bisimiar we deduce that there is a sequence v 1 2 = Z2... n = Z2 v 2 (4) such that (u 2, v 2) R. Again, we can rearrange the sequence (4) in order to have first a the τ firings occurring in transitions i = Z2 (i {1,..., h}), which produce a marking aowing for the firings si to be executed, then 1,..., o n and finay by the τ firings in = i Z2 (i {h + 1,...,n}). This wi project to a sequence v 1 0 Z0... k Z0 v 0 1 Z0... n 1 Z0 Z0... m Z0 v 0 (5) in Z 0. By Lemma 6, we deduce that the first part of the the sequence 1,..., k can be performed aso in Z 1, thus producing a marking v 1 where 1,..., h are enabed. Now, since t is enabed in u 1 and the markings u 1 and v 1 differs ony for tokens in the paces of the interface and, on such tokens there are enough tokes, as witnessed by the enabing of 1,..., h, we have that the a-abeed transition t is enabed in Z 1 and can be performed. From the produced marking, again by Lemma 6 the firings 1,..., m can be performed, i.e., we have: v 1 1 Z1... k Z1 v 1 a 1 Z1 Z0... m Z0 v 1 which projects over (5) in Z 0. Therefore we can construct a marking v 3 = v 1 v 0 v 2 a and v 3 = W3 v 3. By definition of R we thus finay have (u 3, v 3 ) R, as desired. 19

21 (b) t in Z 0 (hence t is both in Z 1 and Z 2 ) In this case the firing projects to the firing of a transition in Z i (i {0, 1, 2}) a u i Zi u i i {0, 1, 2}. By the fact that (Z 2, u 2 ) and (W 2, v 2 ) are weaky bisimiar we deduce that v 2 a = Z2 v 2 (6) such that (u 2, v 2) R. Expicity, the sequence (6) wi be of the kind v 1 2 Z2... k a Z0 1 Z2 Z2... m Z2 v 2. and thus it projects in Z 0 to v 0 1 Z0... h Z0 v 0 a Z0... v 0 1 Z0... m Z0 v 0 (7) where 1,..., h, 1,..., m can be either τ s (projection of τ-transitions of Z 1 which are aso in the interface Z 0 ) or interaction with the environment of the kind + s, s. By Lemma 6, we deduce that the same sequence can be performed aso in Z 1 v 1 1 Z1... h Z1 v 1 a Z1... v 1 1 Z1... m Z1 v 0 (8) and we can construct a marking v 3 = v 1 v 0 v 2 a and v 3 = W3 v 3. By definition of R we thus finay have (u 3, v 3 ) R, as desired. (c) t in Z 2, but not in Z 0 In this case the firing projects to the firing of a transition in Z 2, and to (possiby empty) sequences of interactions with the externa environment in Z 0 and Z 1 u 1 1 Z1... n Z1 u 1 ( ) and u 1 0 Z0... n Z0 u 0. By the fact that (Z 2, u 2 ) and (W 2, v 2 ) are weaky bisimiar we deduce that there exists a = W2 v 2 (9) v 2 such that (u 2, v 2) R. The firing sequence (9) projects over Z 0 to a sequence of interactions with the environment v 1 0 Z0... k Z0 v 0. and, as in the previous cases, by using Lemma 6, we deduce that there wi be a firing sequence in Z 1 v 1 1 Z1... k Z1 v 1. ( ) a = W3 v 3. As in the previous and we can define v 3 = v 1 v 0 v 2 such that v 3 cases we deduce that (u 3, v 3) R. If instead, u 3 Z3 u 3, then we must prove that v 3 = W3 v 3, with (u 3, v 3 ) R. This shoud be competey anaogous to item 3 above. 20

22 4.2 Some properties and proof techniques We next present some properties of (strong and weak) bisimiarity, which can be used in bisimuation proof. The next resut shows that given two bisimiar nets, if we cose the same open paces in both nets we sti get two bisimiar nets. Given an open net Z and an open pace s OZ x, et us denote by Z (s, x) the open net obtained from Z by cosing pace s, i.e., Z = (N, O Z ), where OZ x = Ox Z {s}. The initia marking remains the same. Proposition 4. Let Z 1 η Z 2. Let s O x 1 (x {, +}) be an open pace in Z 1. Then the nets Z 1 (s, x) and Z 2 (η(s), x) are strongy bisimiar. Proof. Let Z 1 = Z 1 (s, x) and Z 2 = Z 2 (η(s), x). Let R S 1 S 2 be an η-bisimuation such that (û 1, û 2 ) R. Then R is a bisimuation between Z 1 and Z 2. In fact, if (u 1, u 2 ) R and u 1 Z 1 u 1 then ceary u 1 Z1 u 1. Since R is a η() bisimuation for Z 1 and Z 2 this impies that u 2 = Z2 u 2 with (u 1, u 2 ) R. Since is a abe in Z 1 where pace s has been cosed, we are sure that x s, and η() thus u 2 = Z2 u η() 2 impies u 2 = Z 2 u 2. The case in which u 1 Z 1 u 1 is treated anaogousy. Hence we get the desired resut. We next provide a kind of up-to technique for open net bisimiarity. Given an open net Z, et us define the out-degree of a pace s S as deg(s) = max ( {( t)(s) : t T Z } {1 : s O Z }) The idea, formaised in the notion of up-to bisimuation, is to aow tokens to be removed from open input paces, when they exceed the out-degree of the pace. More precisey, given a net Z and a marking u S, et us say that a marking v (O + Z ) is subtractabe from u if s O + Z. deg(s) u(s) v(s). Note that this impies that a transitions enabed in u are aso enabed in u v. Definition 14 (up-to bisimuation). Let Z 1 and Z 2 be open nets, and et η : O 1 O 2 be a correspondence between Z 1 and Z 2. A reation R S 1 S 2 between markings is caed an up-to η-bisimuation if whenever (u 1, u 2 ) R then if u 1 Z1 u 1, then there exists u 2 such that u 2 = Z2 u 2 and v 1 (O 1 + ) subtractabe from u 1 with (u 1 v 1, u 2 η (v 1 )) R. if u 1 Z1 u 1, then there exists u η() 2 such that u 2 = Z2 u 2 and v 1 (O 1 + ) subtractabe from u 1 with (u 1 v 1, u 2 η (v 1 )) R; the symmetric conditions hod; A first technica emma shows an invariance property of up-to bisimuations, with respect to adding tokens in open paces. Lemma 7. Let Z 1 and Z 2 be open nets, et η : O 1 O 2 be a correspondence between Z 1 and Z 2, and et R be an up-to η-bisimuation between Z 1 and Z 2. Then, given any s O + 1, the reation R = R {(u 1 s, u 2 η(s)) : (u 1, u 2 ) R} is an up-to η-bisimuation. 21

23 Proof. In order to simpify the notation, et us assume, without oss of generaity, that η is the identity (i.e., O + 1 = O+ 2 and O 1 = O 2 ). Let (u 1 s, u 2 s) R. Let us show that if u 1 s Z1 u 1 then there exists u 2 s = Z2 u 2 and v O 1 + subtractabe from u 1 with (u 1 v, u 2 v) R. The other cases are competey anaogous. Observe that, since s O 1 +, we have u 1 + s Z1 u 1 s. By definition of R, we have (u 1, u 2 ) R and thus u 2 + s = Z2 u 2 and (u 1 s v, u 2 v ) R (10) for a suitabe v O + 1 subtractabe from u 1 s. Aso notice that, since a + s can aways be performed, we can assume that the firing sequence (10) is of the kind u 2 + s Z2 u 2 s = Z2 u 2 (11) Now, if u 1 s Z1 u 1, then, since v is subtractabe from u 1 s, aso u 1 s v Z1 u 1 v. Thus, by (10) u 2 v = Z2 u 2 and (u 1 v v, u 2 v ) R (12) for a suitabe v O + 1, subtractabe from u1 v. Putting the above together with (11), we have that u 2 s = Z2 u 2 = Z2 u 2 v i.e., u 2 s = Z2 u 2 v and, if we denote u 2 = u 2 v, (u 1 v v, u 2 v v ) R. It is immediate to see that v v is subtractabe from u 1, and thus we concude. Coroary 1. Let Z 1 and Z 2 be open nets, et η : O 1 O 2 be a correspondence between Z 1 and Z 2, and et R be an up-to η-bisimuation between Z 1 and Z 2. Then the reation R = R {(u 1 v 1, u 2 η (v 1 )) : (u 1, u 2 ) R v 1 O + 1 } is an up-to η-bisimuation. Proof. By an inductive reasoning, expoiting the previous resut, we can show that R n = R {(u 1 v 1, u 2 η (v 1 )) : (u 1, u 2 ) R v 1 O 1 + v1 n} is a weak bisimuation up-to for any n. Then we expoit the fact that the union of weak bisimuations up-to is again a weak-bisimuation up-to. Proposition 5. Let Z 1 and Z 2 be open nets, and et η : O 1 O 2 be a correspondence between Z 1 and Z 2. Let R be an up-to η-bisimuation. Then for any (u 1, u 2 ) R we have that (Z 1, u 1 ) η (Z 2, u 2 ). Proof. In order to simpify the notation, et us assume, without oss of generaity, that η is the identity (i.e., O + 1 = O+ 2 and O 1 = O 2 ). Let show that 22

24 R = {(u 1 v, u 2 v) : (u 1, u 2 ) R v (O + 1 ) } is an η-bisimuation. Let (u 1 v, u 2 v) R, with (u 1, u 2 ) R and v O + 1, and assume that u 1 v Z1 u 1 (the case in which u 1 v Z1 u 1 is treated anaogousy). By Coroary 1 we know that R is an up-to bisimuation, and thus there exists a transition u 2 v = Z2 u 2 and v O + 1, subtractabe from u 1, such that (u 1 v, u 2 v ) R. However, by construction of R, this impies that (u 1, u 2 ) R as desired. Note that, as it often happens with up-to techniques, the above resut aows to show that two nets are bisimiar by exhibiting finite reations (whie bisimuations are typicay infinite). E.g., consider the open nets beow, where abe a is observabe: Then a bisimuation woud incude at east the pairs {(k s, k s) : k N}, where s is the ony pace. Instead, according to the definition above {(0, 0), (s, s)} is an up-to bisimuation. 5 Behaviour preserving transformations The resuts in the previous sections are used here to design a framework where a system specified as a (possiby open) Petri net can be dynamicay reconfigured by transformation rues, triggered by the state/shape of the system. The congruence resut aows to characterise casses of reconfigurations which preserve the observationa behaviour of the system. 5.1 Transforming open nets The fact that the composition operation over open nets is defined in terms of a pushout construction suggests naturay a way of reconfiguring open nets by using the doube-pushout approach to rewriting [6]. A rewriting rue over open nets consists of a pair of morphisms in ONet: p = L p p Kp r p Rp 23