On an abstract characterization of bisimulation Mila Majster-Cederbaum Universitat Mannheim
Markus Roggenbachy Universitat Mannheim
Abstract
[AM89] and [JNW94] present abstract concepts of bisimulation in terms of category theory. This paper deals with the question how to relate these approaches. Futheron it shows how di erent types of bisimulations on event structures can be modelled in terms of the abstract concepts.
Bisimulation was introduced by [Mil80] and [Par81] in order to identify processes that cannot be distinguished by an external agent. Since then various notions of \bisimulation" have been studied, e.g. on labelled transition systems by [DNMV90], [MS92], on event structures by [GG89], [GKP92], on petri nets by [GV87], [ABS91]. Recently attempts have been made to develop an abstract characterization of the various notions of bisimulation, see for example [DDNM93] and [Mal95]. We focus here on the work of [AM89] and [JNW94]. [AM89] characterize bisimulation (AM-bisimulation) as a coalgebra relative to a functor on the category Class. [JNW94] work with a general category M of models with a distinguished subcategory P of path objects. Two objects X1 and X2 are called P-bisimular i there exists an object X in M together with so-called P-open morphisms fi : X ! Xi ; i = 1; 2: We study here how AM-bisimulation and P-bisimulation are related. To connect these concepts we use the formalism of path-P-bisimulation of [JNW94]. Starting in a setting where one may speak about path-P-bisimulation we prove the equivalence of (strong) path-P-bisimulation and (strong) AM-bisimulation. As every P-bisimulation induces a strong path-P-bisimulation we get the result: If one can introduce the concept of P-bisimulation in a category of models M this bisimulation induces a strong AM-bisimulation. For the reverse direction { i.e. to characterize a given AM-bisimulation on a category M of models in terms of P-bisimulation { numerous assumptions have to be made. In a rst step we switch from AM-bisimulation to path-P-bisimulation. Therefore we have to construct a suitable category P of path objects. Using a theorem of [JNW94] which characterizes the situations where strong path-P-bisimulation coincides with P-bisimulation one may conclude that AM-bisimulation is a more general concept than P-bisimulation. As an application we study AM-bisimulation and P-bisimulation on labelled event structures where we consider the concepts of interleaving, step, pomset, history-preserving and strong-historypreserving bisimulation. This paper is a short version of [Rog97], which provides the proofs of all theorems.
1 AM-Bisimulation A coalgebra for an endofunctor F on a category C is a pair (A; ) where A is an object of C and : A ! FA a morphism. A morphism : A ! B in C is called a homomorphism between Lehrstuhl f ur Praktische Informatik II, Universitat Mannheim, D-68131 Mannheim;
[email protected] y Lehrstuhl f ur Praktische Informatik II, Universitat Mannheim, D-68131 Mannheim;
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1
A
1
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FA
2
R
F1
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-B
F2
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- FB FR Figure 1: AM-Bisimulation
coalgebras (A; ) and (B; ) i = (F) holds. The coalgebras and homomorphisms form itself a category denoted by CF : Let F be an endofunctor on Set. We call a coalgebra (R; ) an AM-bisimulation between two coalgebras (A; ) and (B; ) i R A B and the projections 1 : (R; ) ! (A; ) and 2 : (R; ) ! (B; ) of R on A resp. B are homomorphisms, i.e. they make the diagram in gure 1 commute. In the view of [AM89] a labelled transition system over a xed set of labels L is an object in SetF ; where F := P (L ) and P stands for the powerset operator. In the rest of this paper SetF denotes the category of coalgebras for this special functor. Each coalgebra (A; ) in SetF encodes a labelled transition system T(A; ) = (A; ?!) and vice versa: A is the set of states of a y with label a 2 L between two states x; y 2 A in T T(A; ) and there is a transition x ?! (A; ) i (a; y) 2 (x) L A: For a given coalgebra (A; ) in SetF let (A; ? ) denote its \inverse coalgebra" which consists of the same set A and the \inverse" of as morphism: (a; x) 2 ? (y) i (a; y) 2 (x): A coalgebra (R; ) is a strong AM-bisimulation between coalgebras (A; ) and (B; ) i it is an AM-bisimulation and (R; ? ) is an AM-bisimulation between (A; ? ) and (B; ? ): In order to relate two transition systems by an AM-bisimulation one should not only require that there exists a coalgebra (R; ) which makes the diagram of gure 1 commute. (R; ) = (;; ;) is an AM-bisimulation between any two coalgebras. To model a special kind of bisimulation as AMbisimulation it is therefore necessary that the set R of the coalgebra (R; ) includes a distinguished pair of states, e.g. the initial states of the transition systems. The concept of strong AM-bisimulation is new. One shoud note that strong AM-bisimulation is not an \abstraction" of the concept of back and forth bisimulation which [DNMV90] introduce on transition systems with initial states. An instance of strong AM-bisimulation can be found in [GKP92]. Among other kinds of bisimulations they introduce forward bisimulation and backward-forward-bisimulation on prime event structures and characterize both by temporal logics: Two event structures are forward bisimilar i their related models cannot be distinguished by formulas of the logic S4N; backward-forward bisimulation can be characterized by the logic POL { an extension of S4N by two modalities. In section 4 we show how one can model interleaving bisimulation (this is just another term for forward bisimulation) as AM-bisimulation. Backward-forward-bisimulation arises as the strong case of this AM-bisimulation.
2
P-Bisimulation and Path-P-Bisimulation
To give an abstract characterization of bisimulation [JNW94] choose a category M of models and a subcategory P of M of \path objects". A path is a morphism p : P ! X from an object P in P to an object X in M: In M a morphism f : X ! Y is called P-open, i whenever there are objects P; Q and a morphism m : P ! Q in P and paths p : P ! X; q : Q ! Y; then there exists a path r : Q ! X with r m = p and f r = q: P-open morphisms include all the identity morphisms 2
and are closed under composition. Two objects X1 and X2 of M are called P-bisimilar, i there exists an object X in M and P-open morphisms f1 : X ! X1 and f2 : X ! X2 : To introduce the concept of path-P-bisimulation [JNW94] assume that P is a small subcategory of M and that P and M have a common initial object I: Two objects X1 and X2 of M are called pathP-bisimilar i there is a set R of pairs of paths (p1 ; p2 ) with common domain P; so p1 : P ! X1 is a path in X1 and p2 : P ! X2 is a path in X2 ; such that (o) (1 ; 2 ) 2 R; where 1 : I ! X1 and 2 : I ! X2 are the unique paths starting in the initial object, and for all (p1 ; p2 ) 2 R and for all m : P ! Q; where m is in P; holds (i) if there exists q1 : Q ! X1 with q1 m = p1 then there exists q2 : Q ! X2 with q2 m = p2 and (q1 ; q2 ) 2 R and (ii) if there exists q2 : Q ! X2 with q2 m = p2 then there exists q1 : Q ! X1 with q1 m = p1 and (q1 ; q2 ) 2 R: We call two objects X1 and X2 strong path-P-bisimilar i they are path-P-bisimilar and the set R further satis es: (iii) If (q1 ; q2 ) 2 R; with q1 : Q ! X1 and q2 : Q ! X2 and m : P ! Q; then (q1 m; q2 m) 2 R: Sometimes we call the set R a (strong) path-P-bisimulation between the objects X1 and X2 :
3 Relating the concepts [JNW94] give the following relation between P-bisimulation and strong path-P-bisimulation:
Theorem 3.1
1. Let M be a category of models, let P be a small subcategory of M of path objects, such that P and M have a common initial object I: If two objects X1 and X2 of M are P-bisimilar then they are strong path-P-bisimilar. 2. Let M be the subcategory of rooted presheaves in [Pop ; Set]: Rooted presheaves X1 and X2 are strong path-P-bisimilar i they are P-bisimilar.
We show how the concept of (strong) AM-bisimulation ts into this picture. First we start with a given path-P-bisimulation: Let M be a category of models, let P be a small subcategory of M of path objects, such that P and M have a common initial object I: De ne for each object M a labelled transition system Tpath?P (X ) = (S; ) in SetF over the set of labels S X off(m; P; Q) j m 2 Mor(P; Q)g : P;Q 2 P S := fp : P ! X j P 2 P; p 2 Mor(P; X )g: (m; P; Q; q ) 2 (p) :() q m = p; i.e. the states of Tpath?P (X ) are all morphism from path objects to X; the labels are morphisms between path objects.
Theorem 3.2 Let M be a category of models, let P be a small subcategory of M of path objects, such that P and M have a common initial object I: Two objects X1 and X2 of M are (strong) path-P-bisimilar i there exists a (strong) AM-bisimulation (R; ) between (A; ) := Tpath?P(X1 ) and (B; ) := Tpath?P (X2 ) with (1 ; 2 ) 2 R; where 1 : I ! X1 and 2 : I ! X2 are the unique pathes from I to X1 resp. X2 : 3
In order to translate the concept of AM-bisimulation into path-P-bisimulation we have to introduce an abstract formulation of AM-bisimulation: Let M be a category of models. Let mt be an operator which makes from an object X of M a coalgebra mt X in SetF { i.e. a transition system over a suitable set of labels L: Call two objects X1 and X2 of M (strong) AM-bisimilar i mt X1 and mt X2 are (strong) AM-bisimilar in SetF : Let T be the category of transition systems which consists of all objects (A; ) of SetF which have a unique initial state iA 2 A and all states s 2 A are reachable from iA : Take as morphisms between two objects (A; ) and (B; ) of T the mappings f : A ! B with Ff f and f (iA ) = iB ; where iA and iB are the initial states of (A; ) resp. (B; ): We use the category T as link between SetF and M : For a given path-P-bisimulation in M we construct an AM-bisimulation in T and vice versa. An AM-bisimulation in T is an AMbisimulation in SetF as the projections are morphism in both categories. If there is an AMbisimulation (R; ) between (A; ) and (B; ) in SetF where (R; ); (A; ) and (B; ) are objects in T and (iA ; iB ) 2 R then (R; ) is an AM-bisimulation between (A; ) and (B; ) in T. Call an AM-bisimulation (R; ) between (A; ) and (B; ) strong in T i (R; ?) is an AM-bisimulation between (A; ? ) and (B; ? ) in SetF : Assume the following conditions: 1. The operator mt evolves into a functor from M to T, i.e. mt; which in the setting of AMbisimulation needs only to map objects from M into transition systems, has now to map furtheron morphism of M into morphisms of T. The choice of T instead of SetF is due to this condition: In T we nd \more" morphisms. To illustrate the problem take the category T as model M, let mt be the inclusion operator to SetF : Then mtacannot evolve into a functor, because there is a morphims in T from (fxg; ;) to (fx; yg; fx ?! yg); but there is no morphisms between these two objects in SetF : 2. There is a small subcategory P in M; such that P and M have a common initial object I: 3. For objects P in P holds: The transition system mt P has a unique nal state, which is reachable from all other states in mt P: l ?1 l2 : : : ?! l1 s ?! sn of a transition system in T exists an object P of 4. For any derivation s1 ?! 2 l ?1 l2 : : : ?! l1 t ?! tn ; where t1 is the initial state of P such that in mt P exists a derivation t1 ?! 2 mt P; tn is the nal state of mt P: For this object P holds furtheron: Whenever there exists l ?1 l2 : : : ?! l1 u ?! un in mt X; where u1 is the inital state of mt X; an object X of M with u1 ?! 2 then there exists a morphism p : P ! X in M; such that (mt p)(ti ) = ui ; i = 1; 2; : : : ; n: 5. For the transition systems in T which constists of just one state and no transition the initial object I of P is one of the objects condition 4 speaks about. 6. Let P and Q be objects of P; X be an object of M; p : P ! X; q : Q ! X; m : P ! Q l ?1 l2 : : : ?! l1 t ?! tn be a derivation in mt P; where t1 is the morphims in M resp. P: Let t1 ?! 2 initial and tn the nal state of mt P: Then holds: q m = p () 81 i n : (mt q mt m)(ti ) = mt p(ti ): n
n
n
n
Theorem 3.3 Assume the above conditions, let X and X be objects of M: In T exists an AM1
2
bisimulation (R; ) between mt X1 and mt X1 i there exists a path-P-bisimulation R0 between X1 and X2 : Remark 3.4 Example 3.5 of [Rog97] shows: In the setting of theorem 3.3 exist AM-bisimilar objects X1 and X2 which are strong path-P-bisimilar but there exists no strong AM-bisimulation between mt X1 and mt X2 : Thus it is not possible to prove a general equivalence between strong AM-bisimulation and strong path-P-bisimulation.
4
4 Bisimulations on event structures Let Act be a set of actions. A (prime) event structure E = (E; ; ]; l) over the set of actions Act consists of a set of events E; a causal dependency relation E E , which is a partial order, an irre exive and symmetric con ict relation ] E E and a labelling function l : E ! Act ; which together satisfy: 1. For all e 2 E the set # (e) := fe0 2 E j e0 eg is nite and 2. for all d; e; f 2 E holds: if d e and d]f then e]f: We call a set X E a con guration of E i X is a nite set, leftclosed in E and for all e; f 2 X holds: : e]f: Sometimes we look on a con guration X not just as a set but as an lposet. In this case X inherits the causal dependency relation and the labelling function from E : X = (X; \ (X X ); ;; ljX ): Conf (E ) denotes the set of all con gurations of an event structure E : We call two events e1 ; e2 2 E concurrent, e1 co e2 ; i they are not related by or ]: The category EAct has as objects the prime event structures E = (E; ; ]; l) over a xed set of actions Act ; where E Ev for some \universal" set of events Ev: This condition ensures that EAct is small and therefore all subcategories P of EAct which we introduce to de ne some kind of path-P-bisimulaton are small either. Let E = (E; E ;]E ; lE ) and F = (F; F ;]F ; lF ) be objects of EAct: A total map : E ! F is a morphism from E to F i for all e 2 E : lE (e) = lF ( (e)); 8X 2 Conf (E1 ) : (X ) 2 Conf (E2 ) and 8X 2 Conf (E1 ) 8e; e0 2 X : (e) = (e0 ) ) e = e0 . Lin denotes the full subcategory of EAct which consists of con ict free event structures (E; ;;; l) where E is a nite set and the dependency relation is a total order. Let E = (E; E ; ;; lE ); M = (M; M ; ;; lM ) be event structures with E \ M = ; and M = f(m; m) j m 2 M g: Then F := E ; M denotes the event structere (E [ M; F ;;; lE [ lM ); where e F f i e = f or (e 2 E; f 2 M ) or e E f: Call an event structure S
:= M1 ; M2; : : : ; Mn ; n 0;
a step, where Mi = (Mi ; M ; ;; li ) are event structures, Mi are nite sets, Mi are pairwise disjoint and M = f(m; m) j m 2 Mi g: The representation of a step by nonempty event structures Mi is uniquely determined. Step denotes the full subcategory of EAct which consists of steps as objects. Call Pom the full subcategory of EAct which has as objects those con ict free event structures (E; ; ;; l) where E is a nite set. A pomset [E ] is the equivalence class of an event structure E from Pom where we take isomorphy as equivalence relation. P denotes the set of all pomsets which we can derive from EAct : For an event structure E = (E; ; ]; l) of EAct we construct di erent coalgebras in SetF ; where the functor F := P (L ): These transition systems Tint (E ); Tstep (E ) and Tpom(E ) consist all of the same set Conf (E ) in the rst component but are de ned over di erent sets of labels L. In case of Tint (E ) we choose L := Act and de ne (a; X 0 ) 2 int (X ) i X X 0; X 0nX = feg and l(e) = a: Tstep (E ) we choose L := NAct and de ne (M; X 0) 2 step (X ) i 0 0 0 X X ; 8e; f 2 X nX : e 6= f ) e co f and 8a 2 Act : M (a) = jfe 2 X 0 nX j l(e) = agj: i
i
5
Tpom (E ) we choose L := P and de ne (p; X 0 ) 2 pom (X ) i X X 0 and p = [X 0nX ]: De nition 4.1 Let E1; E2 be event structures. A relation R Conf (E1 )Conf (E2) with (;; ;) 2 R is called
interleaving bisimulation i for all (X; Y ) 2 R; a 2 Act holds:
a X 0 in Conf (E ) then Y ?! a Y 0 in Conf (E ) for some Y 0 2 Conf (E ) with if X ?! 1 1 2 2 2 0 0 (X ; Y ) 2 R and a a if Y ?!2 Y 0 in Conf (E2 ) then X ?!1 X 0 in Conf (E1 ) for some X 0 2 Conf (E1 ) with (X 0 ; Y 0 ) 2 R: bf-bisimulation (backward-forward-bisimulation) [GKP92] i it is an interleaving bisimulation and for all (X 0 ; Y 0 ) 2 R; a 2 Act holds:
a X 0 in Conf (E ) then Y ?! a Y 0 in Conf (E ) for some Y 0 2 Conf (E ) with if X ?! 1 1 2 2 2 (X; Y ) 2 R and a a if Y ?!2 Y 0 in Conf (E2 ) then X ?!1 X 0 in Conf (E1 ) for some X 0 2 Conf (E1 ) with (X; Y ) 2 R: step bisimulation i for all (X; Y ) 2 R; M 2 NAct holds: 0
M X 0 in Conf (E ) then Y ?! M Y 0 in Conf (E ) for some Y 0 2 Conf (E ) with if X ?! 1 1 2 2 2 0 0 (X ; Y ) 2 R and M M if Y ?!2 Y 0 in Conf (E2 ) then X ?!1 X 0 in Conf (E1 ) for some X 0 2 Conf (E1 ) with (X 0 ; Y 0 ) 2 R: pomset bisimulation i for all (X; Y ) 2 R; p 2 P holds:
p p 0 0 0 if X ?! 1 X in Conf (E1 ) then Y ?!2 Y in Conf (E2 ) for some Y 2 Conf (E2 ) with 0 0 (X ; Y ) 2 R and p p 0 0 0 if Y ?! 2 Y in Conf (E2 ) then X ?!1 X in Conf (E1 ) for some X 2 Conf (E1 ) with 0 0 (X ; Y ) 2 R:
Theorem 4.2 Let E ; E be event structures in EAct : 1
2
are interleaving (step, pomset) bisimular i there exists a an AM-bisimulation (R; ) between Tint (E1 ) and Tint (E2 ) (Tstep (E1 ) and Tstep (E2 ) resp. Tpom(E1 ) and Tpom(E2 )) with (;; ;) 2 R: 2. E1 and E2 are bf-bisimular i there exists a strong AM-bisimulation (R; ) between Tint (E1 ) and Tint (E2 ) with (;; ;) 2 R: 3. E1 and E2 are interleaving bisimilar i they are Lin-bisimilar. 1.
E1 ; E2
Event structures, which are interleaving, step or pomset bisimilar, are in general not strong AMbisimilar, for a counter example see [Rog97]. Corollary 4.3 Let E1; E2 be event structures in EAct : The following are equivalent: 1. E1 and E2 are interleaving-bisimilar. 2. There exists an AM-bisimulation (R; ) between Tint (E1 ) and Tint (E2 ) with (;; ;) 2 R: 6
3. There exists a (strong) AM-bisimulation (R; )between Tpath?Lin (E1 ) and Tpath?Lin(E2 ) with (1 ; 2 ) 2 R: 4. E1 and E2 are (strong) path-Lin-bisimilar. 5. E1 and E2 are Lin-bisimilar. Applying theorem 3.3 on the characterization of step bisimulation in theorem 4.2 we get:
Corollary 4.4 Two event structures of EAct are step bisimular i they are path-Step-bisimilar.
Concerning pomset bisimulation it is not possible to apply theorem 3.3 on the equivalent characterization as AM-bisimulation from theorem 4.2: The operator Tpom(E ) fails to evolve into a functor. This coincides with a result of [JNW94]: path-Pom-bisimulation is equivalent with history preserving bisimulation.
De nition 4.5 Let E1; E2 be event structures. A set R of triples (X; Y; f ); where X 2 Conf (E1 ); Y 2 Conf (E2 ) and f : X ! Y is an isomorphism in Pom, is called history preserving bisimulation i for all (X; Y; f ) 2 R; p 2 P holds: p p if X ?!1 X 0 in Conf (E1 ) then Y ?!2 Y 0 in Conf (E2 ) for some Y 0 2 Conf (E2 ); f 0 : E1 ! E2 with (X 0; Y 0 ; f 0 ) 2 R; fj0X = f and p p if Y ?!2 Y 0 in Conf (E2 ) then X ?!1 X 0 in Conf (E1 ) for some X 0 2 Conf (E1 ); f 0 : E1 ! E2 with (X 0; Y 0 ; f 0 ) 2 R; fj0X = f: strong history preserving bisimulation i R is a history preserving bisimulation and satis es further
(X 0 ; Y 0 ; f 0 ) 2 R and X X 0 for some con guration X 2 Conf (E1 ) implies (X; Y; f ) 2 R for some Y Y 0 and f = fj0X and (X 0 ; Y 0 ; f 0 ) 2 R and Y Y 0 for some con guration Y 2 Conf (E1 ) implies (X; Y; f ) 2 R for some X X 0 and f = fj0X :
[JNW94] give the following characterizations of (strong) history preserving bisimulation on event structures with consistency relation:
Theorem 4.6 Two event structeres E and E are Pom-bisimilar i they are strong history preserving bisimular. (strong) history preserving bisimilar i they are (strong) path-Pom-bisimilar. 1
2
Applying theorem 3.2 on this result we get a characterization of (strong) history preserving bisimulation in terms of AM-bisimulation on event structures with consistency relation as well as on prime event structures.
Corollary 4.7 Event structures E and E are (strong) history preserving bisimilar i there exists a (strong) AM-bisimulation (R; ) between Tpath?Pom(E ) and Tpath?Pom(E ) with ( ; ) 2 R: 1
2
1
2
1
2
Concerning the exibility of P-bisimulation on event structures [JNW94] write: It might be thought that strong history-preserving bisimulation, presented as Pom-bisimilarity, is a ected by restricting the category Pom to a smaller class of objects. However, no matter how much the objects in the path category Pom are restricted, provided they include all pomsets of the \stick" and \lollipop" forms in the proof of Proposition 7, then the relation of bisimulation that results 7
P-bisimulation
(theorem 3.1) strong path-P-bisim. m (theorem 3.2) strong AM-bisim. on Tpath?P 6+ (remark 3.4) strong AM-bisim. on T 6= Tpath?P m
) ) )
path-P-bisim. (theorem 3.2) AM-bisim. on Tpath?P m (theorem 3.3) AM-bisim. on T 6= Tpath?P m
Figure 2: Relations between the di erent bisimulation concepts. will coincide with strong history-preserving bisimulation. Thus one does not expect that step, pomset or history preserving bisimulation can be modelled as P-bisimulation. Only interleaving and strong history preserving bisimulation t in the concept of P-bisimulation. For AM-bisimulation holds that all mentioned types of bisimulation on event structures can be modelled in a unifying way: Two event structures E and F are -bisimilar, i there exists a (strong) AM-bisimulation (R; ) with (i1 ; i2 ) 2 R between T (E ) and T (F ); where 2 finterleaving, bf, step, pomset, history-preserving, strong-history-preservingg; (i1 ; i2 ) is a distinguished pair of states and T is an operator which maps an event structure on a suitable coalgebra. In case of interleaving, step and pomset bisimulation we choose (i1 ; i2) = (;; ;) and one of the operators Tint ; Tstep resp. Tpom : bf-bisimulation is modelled as strong AM-bisimulation with (i1 ; i2 ) = (;; ;) and T = Tint : For history preserving bisimulation we take (i1 ; i2 ) = (1 ; 2 ) and the operator Tpath?Pom; for strong history preserving bisimulation we make the same choice but take this time the strong version of AM-bisimulation.
5 Conclusion Figure 2 summarizes the general relations between (strong) AM-bisimulation, (strong) path-Pbisimulation and P-bisimulation. For simplicity we do not mention the conditions which are (sometimes) necessary to establish an equivalence. Combining theorem 3.1 and theorem 3.2 we can conclude: If a bisimulation between objects of a category of models M can be modelled as P-bisimulation for a suitable subcategory P of M and the assumptions of theorem 3.1 are full lled, then this bisimulation can also be modelled as strong AM-bisimulation where we choose the operator Tpath?P to get a transition system. Applications of this combination are interleaving and strong history preserving bisimulation on event structures, see section 4, and Bran-bisimulation on transition systems, see [MCR96], [Rog97]. For the converse direction one obtains: If a bisimulation can be modelled as AM-bisimulation, the assumptions of theorem 3.3 are full lled, the AM-bisimulations on Tpath?P are always strong, the assumptions of theorem 3.1 are full lled, then this bisimulation can also be modelled as Pbisimulation. Applications of this equivalence are interleaving bisimulation on event structures, see section 4, and AM-bisimulation on transition systems, see [MCR96], [Rog97]. Applying these results to concrete models we showed: For transition systems the concepts of AM-bisimulation, Bran-bisimulation and (strong) path-Bran-bisimulation coincide [Rog97]. Di erences arise for the more complex model of event structures: Looking for an approach which is able to model various types of bisimulations on event structures AM-bisimulation turned out to be the most exible of the three concepts. It is left as open question how strong AM-bisimulation and strong path-P-bisimlation are related. In order to get more insight into the \nature" of bisimulation other types of bisimulation on event strucures, bisimulations on other models of concurrency and other abstract characterizations of bisimulation should be studied. 8
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