Transition systems from event structures revisited

Inform$ion Riaesmg Information Processing Letters 67 (1998) 119-124

Transition systems from event structures revisited Mila Majster-Cederbaum Univemitiit Mannheim, Lehrstuhlflr

*, Markus Roggenbach

’

Praktische Infornmtik II, D-68131 Mannheim, Germany

Received 4 December 1997; received in revised form 25 June 1998 Communicated by W.M. Turski

Abstract For event structures various approaches to the definition of an operational semantics can be distinguished. how the obtained transition systems are related. 0 1998 Elsevier Science B.V. All rights reserved. Keywords: Bisimulation;

Transition

systems

Concurrency;

Event structures; Formal semantics; Transition

play an important

role in con-

currency theory. Associating a transition system with, e.g., a true concurrency model has proved to be a suitable technique for studying various problems related to reactive systems including consistency, bisimulation, implementation and verification. In the case of event structures various types of transition systems T(E) were associated with an event structure I [24,6,7]. The operators T differ in the choice of the set of states and/or the set of labels and/or the transition relation. The first question we address here concerns the influence of the choice of the set of states. Next we deal with the modelling of bisimulation on event structures by a suitable transition system. The classical notions of interleaving, step and pomset bisimulation on event structures can be viewed as bisimulations on suitable transition systems. We show that the more complex notions of weak, strong and normal history preserving bisimulation can be captured in the same way. Finally majstex-cederbaum@pi2. author. Email: * Corresponding informatik.uni-mannheim.de. 1Email: roba@pi2,informa&uni-mannheim.de. 0020-0190/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PII: SOO20-0190(98)00105-7

We investigate

systems

we consider the question which of the operators T yields a functor from the category of event structures to the category of transition systems. This question is motivated by the work of Winskel and Nielsen [ 1 l] who take a rigorous view stipulating that a semantic model M should form a category, the semantic operations should posess a categorical characterization and that bisimulation can be expressed via a subcategory of paths [5]. It would support this categorical view if each of the above operators T, that maps a category M (semantic model) into the category T of transition systems where bisimulation in M is mapped on bisimulation in T. extends to a functor from M to T.

1. Associating transition systems with event structures Let Act be a set of actions. A (prime) event structure E = (E, <, #, 1) over the set of actions Act consists of a set of events E, a causal dependency relation < C E x E, which is a partial order, an irreflexive and symmetric conflict relation fl C E x E and a labelling function 1: E -+ Act, which together satisfy:

hf. Majster-Cederbaum,

120

M. Roggenbach /Information

For all e E E the set J(e) := (e’ E E I e’ 6 e) is finite and for all d, e, f E E holds: if d < e and dflf then euf. An event structure is called finite if its set of events is finite. An event structure is called conjictfree if its conflict relation is the empty set. Call a set X 2 E a conjiguration of & iff X is a finite set, leftclosed in E (i.e., Ve E X: {e’ E E 1 e’ < e) 5 X) and for all e, f E X holds: letif. Sometimes we consider a configuration X itself as event structure (X, < II (X x X), 0, Zlx). Conf(E) denotes the set of all configurations of an event structure E. Two events et, e2 E E are called concurrent, et coez, iff they are not related by < or #. For X E Conf(E) let O&(X):= {f E E (3e E X: e$f}, define E’ := E\(X

Processing Letters 67 (1998) 119-124

Let L be a set of labels. A transition system 7 = (S, +, is) over L consists of a set of states S, a transition relation -+ E S x L x S and an initial state is E S. The category TL has as objects the transition systems over L. Let 70 = (So, -+, io) and I1 = (S1,-+,it)beobjectsofTL.Atotalmapo:.Sa+St is a morphism from 70 to z iff o(i0) = il and for all s, s’ E S, Z E L holds: if s & s’ in 70 then a(s) & a(~‘) in 71. R G SO x S1 is a bisimulution [8,9] between 70 and 7tiff(iu,it)~Randforall(s,t)~R,Z~L:ifs_fts’ in lo then t -$ t’ in ;’ and (s’, t’) E R for some t’, and if t 5 t’ in z then s 5 s’ in 70 and (s’, t’) E R for some s’. A bisimulation R is a backward-forward iff for all (s’, t’) E R, 1 E L: ifs 5 s’ in

bisimulation U #E(X))

70 then there exists some t such that t & t’ in 7r and

and

(s, t) E R, and if t 5 t’ in 7t then there exists some s

E\X := (E’, < n (E’ x E’), g rl (E’ x E’), Zl,,).

such that s & s’ in 70 and (s, t) E R.

The category EAct has as objects the prime event structures E = (E, 6, g, 1) over Act, where E E Ev for some “universal” set Ev of events. Let E = (E, denotes the set of all equivalence classes [ele2 . . . e,] of derivations of a configuration X, DerA,r :=

U

Der(X). XECOflf(E) &EEacr

1.1. The operators TC,: conjigurations

as states

Let Lint := Act, Ls, := @“’ and Lpom := PomAcr be sets of labels. Relative to these sets of labels various notions of bisimulations on the configurations of the event structures have been defined, e.g., in [3, 41. It is easy to see that each of these definitions gives rise to a transition system in a natural way such that event structures & and F are interleaving, step respectively pomset bisimilar iff the transition systems are bisimilar. In addition we define two new operators which are capable of modelling more complex bisimulations on event structures. Let & be an event structure. TCint(E) := (Conf (E), Lint, 0) is a transition

SYS-

tern over Lint, where X :int X’

iff

X C X’, X’\X = (e) and Z(e) = a. := (Conf(&), IsteP, 0) isatransitionsys-

TC,,,(E)

tern over L,,,

where

M X-

step X’ XCX’, Vu E

iff Ve,f

EX’\X:

e#

Act: M(a) = I( e E X’\X

f =+ecof I Z(e) =a)/.

and

M. Majster-Cederbaum,

K,,,(E) := (C&(E), system over Lpom, where x-

P

M. Roggenbach /Information

Aporn, 0) is a transition

Y := (Y, < fl (Y x Y), 0,lJ~) TCwhp(E) := (Conf(E), system over Lt,Om, where P

121

Let I’ := E\X for some conjguration X E Conf (E), E” := E’\X’for some configuration X’ E Conf (E’). Then X U X’ is a configuration

iff

Porn X’

(2)

X C_X’ and p = [Y], where

x-

(1)

Processing Letters 67 (1998) 119-124

iff

whP X’

0) is a transition

(3) For all X E Conf (E) holds: E\X E Reachi,,( (4) Reachi,t (8) = ReachStep (E) = Reachp,, (E) . exist configurations

) x E Conf(&)), ‘hp, system over DerA,t , where

Ielq...+e,+ll ele2 . . . enen+ -hp X’\X = {e,+l), where

such

(5) Let E’, E” E Reachi,!(

X 2 X’ and p = [X’].

etez...e,

holds:

X E Conf(E)

that E’ = E\X.

TChp(&) := ({Der(X>

transition

of E and E” =

E’ E Reachi,,

There exists a configuration

and Y := X’\X.

-++,

E\(X U X’). For all event structures

If E’ L7* E” then there X’, X” E Conf (E) such that

E) is a I’ = E\X’,

E” = E\X”,

X’ L*

X”.

(6) Let X’, X” E Conf (E) with X’ C X”. Define E’ :=

iff

E\X’, E” := E\X” and X := X”\X’. configuration

X=Iel,e2,...,enl,

Then X is a

of E’ and E” = E’\X.

(7) Let X’, X” E Conf (E). IfX’ L,* X” then E\X’ -fT* E\X”.

1.2. The operators TE,: event structures as states Transition and labels in over to other F = (F,
4jnl3

systems with event structures as states Act occur, e.g., in [2,6]. The idea carries labellings. Let E = (E, GE, tf,y, 1~) and OF, 1~) be prime event structures.

iff

3e E E: J,(e) = {e}, l(e) =a, & Zsfep 3 iff there exists Conf(&) such that

3=

E\{e).

a configuration

X E

Ve, f E X: eco f, Va EAct: M(a)=({eEX)lE(e)=a}\and3=E\X. & A,,, 3 iff there exists a configuration Conf(&) such that p = [Xl and 3 = E\X. Let

X E

Proof. (l), (4), (6) Straightforward. (2) Let E’ E Reach&E). The existence of a configuration X E Conf (E) with E’ = E\X is shown by induction on the length n of a shortest derivation from & to E’. (3) Let X be a configuration of E. If X = 0 then E\X = E\0 = E E Reachi,t(&). If X # 0 any topological sorting of X induces a path in TEjnt(E) from E to E\X. (5)LetE’=(E’,&,ti~~,1E/)andE”=(E”,<~~, #al, 1~“) E Reachint(E). If E’ :jnt E” e E E with l;(e) a configuration Let X” := X’ U know that X” is E”=

(E\X’)\[e]

we know that E” = E’\(e) for some = a. AS E’ E Reachi,t(E) there exists X’ E Conf(E) such that I’ = E\X’. (e}. From part (1) of this lemma we a configuration of E and we get: = E\(X’U

(e}) = E\X”

and

a II X’ ‘in? X .

Reach,(&) :=(3EEActI3k~O,3Eo,&l,...,EkEEAct: EO = E, Ek = 3,

Ei LT* Ei+l

for some label I where i <

k} .

Lemma 1. Let E = (E,
of Act,

Rom,&t and * E { int, step, porn}.

IV;“,

respectively

Let E’ ES, E”, respectively E’ &pom E”. In both cases we know from part (2) of this lemma that there exists a configuration X E Conf(&) such that E’ = E\X. By definition of both transition relations there exists a configuration X’ E Conf (E’) with E” = E’\X’. Let X” := X U X’. X” is a configuration of E, E\X” = E”. If we start with a multistep the elements of X’ =

M. Majster-Cederbaum,

122

E:

..

l

.

e?

M. Roggenbach /Information

‘b

e2

Processing Letters 67 (1998) 119-124

fTg

G: Jy-+;; 0:

.

l

P: . . . . . . . . . . . . . P; PT Pi

. . .

4

4

Fig. 1.

X”\X are concurrent in E, if we start with a pomset X’ is an Iposet. (7) X’ C X := &‘f:=

Let X’, X” E Conf(&). If X’ L, X” we get X” and conclude with part (6) of this lemma: X”\X’ is a configuration of &’ := E\X’ and &\X” =&‘\X. 0

With Lemma 1 we can define the operators TE, on an event structure E as follows: TE,(E)

:= (Reach,(&),

-*,

E),

where * E {int, step, porn). The counterparts of TC,,+,, respectively TCh, are given by T&+(E)

:= (Reach&&),

where for all 3,3’

-whp,

Remark 2. The map u : TC, (E) + TE, (E), where for X E Con@) we define a(X) := &\X, is a morphism in TL for all * E {int, step, porn, whp, hp]. There exist event structures & such that for all * E {int, step,pom, whp, hp} the sets HomT,(TE,(E), TC, (E)) are empty, for example, the event structure E of Fig. 1 where the dotted lines represent the relation d. Theorem3. Let & be an event structure.

For * E (int, step,pom) holds: T&(E) and TE,(&) are bisimilul; but in general not backward-forward bisimilal: In particular; T&(3) and TE*(3) are not isomorphic. For * E {whp, hp) holds: T&(E) and TE,(E) are not bisimilal:

I),

Proof (Sketch). The relation

E &a&,,(E),

R := ((X, E’) I X E Conf(E), p c pOmAct : F &hp 3x, x

x’

E Conf(&):

2 x’,

F’

:+

.F = &\X,

he2

= E\X’,

p = [X’].

TEhp(E) := (Reuchj,l(f),

3,3’

F’

7hp, &), where for all

E Rf?UChint(E), . ..e.+l]EDerAct:3

h~2-4+11

7

&,3‘I:+

E’ = E\X}

is a bisimulation between TC, (E) and TE, (0, where * E (int, step,pom]. To establish this result use Lemma 1. For the event structure 3 in Fig. l-the dotted lines represent the relation fl, while the arrow stands for <-TC,(3) and TE,(3) are not backward-forward bisimilar. For the event structure E in Fig. 1, TCwhP(E) and TEwhp(&), respectively TCh, (&) and T.hP (&) are not bisimilar.

3x, x’ E Conf (&): 3 = &\X, 3’ = &\X’, X 5 X’, ele2 . . . e, f Der(X), ele2..

Another distinction between TC, (E) and TE, (E) is given in Lemma 6.

. en+1 E Der(X’>.

1.3. Relating TC,(&) and TE,(&)

2. Using the operators to model bisimulations

The transition systems TC,(Q and TE,(&) differ basically in their set of states. Thus the question arises what consequences this choice has. As T&(Z) and TE, (E) are both objects in TL for some set of labels L one first attempt to clarify their relationship is to look for morphisms in TL .

Event structures & and 3 are interleaving, step, respectively pomset bisimilar, iff TC,(E) and T&(3) are bisimilar for a suitable choice of * E (int, step, porn}. Moreover E and 3 are backward-forward bisimilar [4] iff TC&E) and T&&F) are backwardforward bisimilar. We consider here the question if

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M. Majster-Cederbaum, M. Roggenbach /Information Processing Letters 67 (1998) 115124

weak, strong and normal history preserving bisimulation on event structures may also be captured by bisimulation on a suitable transition system. Let & = (R,
F

=

(F,


$F, 1~)

x Conf(3)

is called weak history preserving bisimulution [3] iff (PI,O)ERandif(X,Y)ERthen (1) there is an isomorphism between (X, , 0, kIY), (2) if there exists a configuration X’ E Conf(&) with X C_ X’ then there exists a configuration Y’ E Conf(3) with Y E Y’ and (X’, Y’) E R, and (3) if there exists a configuration Y’ E Conf(3) with Y G Y’ then there exists a configuration X’ E Conf(E) with X C_X’ and (X’, Y’) E R. A set R of triples (X, Y, r]) where X E Conf(E), Y E Conf(3) and n : X -+ Y is an isomorphism in EA,-r is called history preserving bisimulution iff (0,0,0) E R and if (X, Y, r]) E R then (1) if there exists a configuration X’ E Conf(E) with X C X’ then there exists a configuration Y’ E &n.(3) with Y 2 Y’ and a map n’ : X’ -+ Y’ such that (X’, Y’, n’) E R and r]‘]x = 9, and (2) if there exists a configuration Y’ E Conf(3) with Y C Y’ then there exists a configuration X’ E Conf(E) with X G X’ and a map 0’ : X’ + Y’ such that (X’, Y’, r]‘) E R and $1~ = n. A history preserving bisimulation R is called strong [5] iff it satisfies further (1) (X’, Y’, t,~‘)E R and X C_X’ for some configuration X E Co@(&) implies (X, Y, n) E R for some Y 2 Y’ and n = q’(x, and (2) (X’, Y’, 8’) E R and Y G Y’ for some configuration Y E Conf(E) implies (X, Y, n) E R for some X c X’ and r] = r]‘lx.

Theorem 4. Event structures & and 3 are weak his-

Let R be a bisimulation between TC,h,,(&) and T&+(3). Let for all (X, Y), (X’, Y’) E R and all p E Rom,Jct (X, Y) 3 (X’, Y’) :+ X 4: X’, Y A Y’. ConsideGn the transition system R := (R, +, (0,0)) the set R of all states (X, Y), which are reachable from (0,0). We claim that R^ is a weak history preserving bisimulction between E yd 3. Obviously we have (0,0) E R. Let (X, Y) E R. We prove first [X] = [Y]. If (X, Y) = (0,0) we obtain [X] = [Y]. If (X, Y) # (0,0) there exists some element (U, V) E R^ with (U, V) 5 (X, Y). This implies (I 3 X. By definition of T&h, we obtain p = [Xl. In the same way we obtain p = [Y] and thzefore [X] = [Y]. To prove the gosure properties of R consider an element (X, Y) E R and let X C X’ for some configuration X’ E Conf(&). By definition of TC,h,, this implies lx’1 X - X’. As R is a bisimulation there exists some Y’ E Conf(3) such that Y ‘%I Y’ and (X’, Y’) E R. Thus by definition of TC,h, we obtain Y G Y’. As (X, Y) is reachable in R and (X, Y) !F, (X’, Y’) we conclude (X’, Y’) E R^. Let R be a history preserving bisimulation between & and 3. Then Z:={(e1e2...e,,fif2...fJ

I

3(X, Y, q) E R: ele2.. .flf2...fn vhe2

.e, E Der(X),

EDer(Y), . ..d=fif2...fn}

is a bisimulation between TChp(E) and TChP(3). If R is strong then R^is a backward-forward bisimulation. Now let R^be a bisimulation between TChp(&) and T+,(3). Then R:={(X,Y,q)I3(ele2...e,,fif2...f,)~z: X=(el,ez,..., 17: X

e,),

Y=Ifiyf2,...,fnJ,

-+ Y is an isomorphism

dele2..

with

.en>=fif2...fn}

tory preserving bisimilar iff T&h,(&) and TCwh,,(3) are bisimilal: & and 3 are (strong) history preserving bisimilar iff TChp(&) and TChP(3) are (backwardforward) bisimilal:

is a history preserving bisimulation between & and 3. If R^ is a backward-forward bisimulation then R is strong. q

Proof. A weak history

Remark 5. It is an open problem whether for the op-

and T&h, (3).

erator TE,,+ and weak history preserving bisimulation, respectively for the operator TEh, and history

preserving bisimulation R between E and 3 is a bisimulation between TC,,,h,,(&)

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M. Majster-Cederbaum,

M. Roggenbach /Information

preserving bisimulation a result similar to Theorem 4 holds. 3. Considering the operators as functors As we explained in the introduction the question if an operator T yields a functor is of some importance for the categorical view of [ 1I]. In addition this question is relevant in connection with another view of bisimulation, i.e., the work of [I] where transition systems and bisimulation are seen as co-algebras of a suitable functor in the category Class. One step in relating these two different views is to establish a functor from a category M (semantic model) to the co-algebras. We showed in [7] that given the category M with a bisimulation in the sense of [5] then there is always a functor F from M to the category of transition systems that respects bisimulation. However, the obtained transition systems are very abstract. Thus in the case of event structures one may ask if the “natural” transition systems obtained via one of the previously defined operators TC,, respectively TE, also fit in the categorical framework. Lemma 6. The operators TCint and TCStep yieldjknctors from EARN to TACO,respectively TNA~~. The operators TCpom, TC,+, and TCh, do not yield functors from EARN to TPomAct , respectively TarA,, . The operators TEinr, TEStep, TEpoms, TE,h,, and TEhp fail to yieldfunctors from EARN to TL, where L is choosen from {Act, Np, POm.&r, DerAct ) . Proof. Let & and 3 be prime event structures, n : E -+ 3 a morphism in EARN. Defining TCint (r])(X) := v(X) and TC,,(q)(X) := q(X) for configurations X E Conf(8) yield functors. Let B and % be the event structuresfromFig.l.~:&+‘Hwithr](gl)=htand I = h2 is a morphism between 6 and Z in EARN. In TL exists no morphism from TC,(G) to TC,(li) for * E born, whp, hp}. For the event structures 0 and P of Fig. 1 there is no morphism in TL from TE,(O) to TE,(P) for * E [int, step,pom, whp, hp], while r] : 0 + P with r](ol) = p1 and ~(02) = p2 is a morphism between 0 and P in EAct . 0

Processing Letters 67 (1998) 119-124

This result is interesting for two reasons: first, it reveals a difference between the TC-operators and the TE-operators. Secondly, it shows that the categorical framework of [l I] is only partly adequate to include “natural” notions of bisimulation on event structures. On the one hand Joyal et al. [5] give categorical characterizations for (strong) history preserving bisimulation within the category EAct. On the other hand these bisimulations have “natural” operational interpretations in terms of transition systems TC@(&). As TCh, does not evolve into a functor there is no categorical connection between both views-the abstract and the natural one.

References [ 11 P Aczel, N. Mendler, A final coalgebra theorem, Lecture Notes in Comput. Sci., Vol. 389, Springer, Berlin, 1989. [2] C. Baier, M.E. Majster-Cederbaum. The connection between an event structure semantics and an operational semantics for TCSP, Acta Inform. 3 1 (1994). [3] R. van Glaabeek, U. Goltz, Equivalences and refinement, Lecture Notes in Comput. Sci., Vol. 469, Springer, Berlin, 1990. [4] U. Gohz, R. Kuiper, W. Penczek, Propositional temporal logics and equivalences, Lecture Notes in Comput. Sci., Vol. 630, Springer, Berlin, 1992. [5] A. Joyal, M. Nielsen, G. Winskel, Bisimulation maps, Technical Report RS-94-7, BRIGS, 1994.

from open

[6] R. Loogen, U. Goltz, Modelling nondeterministic concurrent processes with event structures, Fund. Inform. XIV (1991). [7] M. Majster-Cederbaum, M. Roggenbach, On an abstract characterization of bisimulation, in: Selected Papers 8th Nordic Workshop on Programming Theory, Research Report No. 248, Department of Informatics, University of Oslo, 1997. [S] R. Milner, A calculus of communicating systems, Lecture Notes in Comput. Sci., Vol. 92, Springer, Berlin, 1980. [9] D. Park, Concurrency and automata on infinite sequences, Lecture Notes in Comput. Sci., Vol. 104, Springer, Berlin, 1981. [lo] M. Roggenbach, Categorical characterization of bisimulation, Technical Report l/97, Fakultat fur Mathematik und Informatik, Universit;it Mannheim, 1997. [ll]

G. Winskel, M. Nielsen, Models for Concurrency, Handbook of Logic in Computer Science, Oxford University Press, 1995.