Event Structures for Arbitrary Disruption

Transcription

1 Fundment Informtice XX (2005) IOS Press Event Structures for Arbitrry Disruption Hrld Fecher Christin Albrecht Universität zu Kiel Technische Fkultät, Institut für Informtik, Hermnn-Rodewldstr. 3, Kiel, Germny hf@informtik.uni-kiel.de Mil Mjster-Cederbum Universität Mnnheim Fkultät für Mthemtik und Informtik, D7, 27, Mnnheim, Germny mcb@pi2.informtik.uni-mnnheim.de Abstrct. In process lgebrs tht llow for some form of disruption, it is importnt to stte when process termintes. One option is to include termintion ction. Another pproch is tht the finl executed ction of process termintes the process. The semntics of the former pproch hs been investigted in the literture in detil, e.g. by providing consistent true-concurrency nd opertionl descriptions. The finl executed ction termintion pproch, which is more dequte for modelling, still lcks the existence of detiled true concurrent description. In order to give true concurrency model of such termintion view, we introduce new clss of event structures. This type of event structures models disbling by indicting sets of precursor events. We show tht the introduced clss of event structures hs more expressive power with respect to event trces thn the common event structures. We lso give clssifiction of the expressive power in terms of sets of event trces. A consistency result of n opertionl nd denottionl semntics is shown. Keywords: event structures, disruption, termintion, expressive power, process lgebr Address for correspondence: Hrld Fecher, Christin Albrecht Universität zu Kiel, Technische Fkultät, Institut für Informtik, Hermnn-Rodewldstr. 3, Kiel, Germny phone: , fx: , hf@informtik.uni-kiel.de

2 2 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption 1. Introduction Disrupt mechnisms re importnt to model mny relistic systems nd hve hence found their wy into vrious process lgebrs [2, 3, 14, 15]. The disrupt opertor of LOTOS [11], clled disbling opertor, is denoted by B 1 [>B 2. Here, ny ction executed by B 2 disbles B 1 s long s B 1 hs not terminted. It is importnt to understnd disruption in order to hndle timeout dequtely. Timeouts re n importnt concept in mny pplictions. For the description of the disrupt opertor, in the definition of n opertionl semntics it is necessry to specify when process termintes. This cn be chieved in two wys: By introducing n dditionl internl ction, which will be executed in order to terminte the process. An dditionl syntcticl expression 1 is lso provided to indicte the process tht my terminte immeditely. In prticulr the process consisting of ction results in 1 by executing nd therefter ction cn be executed. This pproch is for exmple tken in the cse of LOTOS [11]. An lterntive pproch to del with termintion is to specify tht process termintes when it executes its finl ction [8, 2]. For exmple the process b, which denotes the prllel composition of ctions nd b, termintes by executing if b ws executed before or it termintes by executing b if ws executed before. In this pproch, which is clled f-pproch in the following, there is no need to extend the syntcticl expressions by further expressions, s 1, to hndle termintion. The -pproch cn be esily imbedded in the f-pproch by using n dditionl ction. For exmple, process B of the -pproch corresponds to the process B;, where ; denotes the sequentil composition. On the other hnd, the f-pproch llows us to specify broder clss of models s cn be seen in the following exmple: Exmple 1.1. We wnt to model nucler power plnt. Let P sd be process tht controls the shut down of the rector nd let P nr be the process tht describes the norml running behvior of the rector. The process P nr is controlled utomticlly nd only informs the environment bout its ctivity. In other words, in our model the ctions of process P nr re observble but not ccessible for the environment nd in prticulr they must not be used for synchroniztion outside P nr. In process P nr n ction stop indictes tht the norml running is terminted nd the control is given immeditely to the process tht controls the shut down of the rector. Then simple specifiction of nucler power plnt is given by P nr ;P sd where ; denotes the sequentil composition. A more relistic specifiction of nucler power plnt my invoke the shut down process P sd for vrious other resons, e.g. if the temperture reches criticl point. In ddition, the shutting down my be combined with other ctivities s, e.g., setting n lrm off. Let us consider nucler power plnt with the ction stop nd temperture triggered shut down tht lso invokes n lrm. Then in ny system run either norml termintion of P nr by stop or disruption of P nr by n externl criticl temperture messge my hppen, but it should be prevented tht both occur. In prticulr, once the stop ction is executed no lrm should be set off. Let ction t denote tht the temperture of the rector reches criticl point nd let ction denote tht n lrm is set off. The nturl representtion of the rector control in LOTOS is given by ((P nr [>t);p sd ) {t} (t;)

3 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption 3 where B 1 A B 2 denotes the prllel execution of B 1 nd B 2 with synchroniztion on the ctions in A. However, ccording to the semntics of LOTOS, t cn hppen fter stop. This origintes in the fct tht the ction nd not the stop ction termintes P nr. Hence, n lrm with expensive consequences my be unnecessrily set off. Furthermore, the f-pproch is more pproprite to model timeout expressions in time extensions. For exmple, we wnt to model timeout opertor where timeout t time t shll occur during the execution of process B s long s B is not terminted. In the -pproch the environment hs no mens to prevent the timeout, since the -ction is n internl ction nd therefore the system cn dely the termintion until the timeout tkes plce. The only possibility to circumvent this sitution is to use n urgency pproch, i.e. time my not proceed if for exmple -ction is enbled. But urgency increses the complexity nd should be voided if possible. On the other hnd, it is esily seen tht the f-pproch is suitble to model the mentioned timeout opertor. Event structures re suitble model for the denottionl semntics of, e.g., process lgebrs. In prticulr they re useful to give deeper insight to the cuslity of the event-execution of processes. For exmple, this ws helpful to estblish n opertionl semntics of process lgebrs with respect to ction refinement, e.g. [13] nd the cittion within. For the -ction termintion pproch, [22] provided n event bsed denottionl semntics tht is consistent with the opertionl one. But it is not cler how to obtin n event bsed denottionl semntics for process lgebr tht contins disrupt opertor nd tht follows the f-philosophy: Prime event structures [26], flow event structures [12] nd stble event structures [29] require symmetricl conflict reltion which mkes it hrd to model disruption. Extended bundle event structures [22], which re used to give denottionl semntics to LOTOS, dul event structures [21] nd symmetric event structures [7] llow the modelling of disruption, since the symmetry condition for the conflict reltion is dropped. Inhibitor event structures [6] cn lso model disruption. If (nd how) these types of event structures could be used to define denottionl semntics tht lso incorportes the f-philosophy is highly questionble. E.g. it is not possible to model the process ( b)[> c with only three events. Also configurtions [17] do not provide smooth wy to model disruption. Consider for exmple the process B, which consists of the disruption of ;b ( sequentilly followed by b) by the ction c ( i.e. B = (;b)[>c). An intuitive pproch is to ssume tht B hs three events denoted by,b,c. Then the sets, {}, {c}, {,b} cn be considered s configurtions, but wht bout {,c}? Assuming it is not configurtion contrdicts the existing execution. c On the other hnd, ssuming tht {,c} is lso configurtion leds to the interprettion tht the execution c is legl 1, which contrdicts the brnching structure of B, since fter the disruption (by c) no further ctions from the left process my be executed. Event utomt [28], which consist of set of configurtion with n explicit event execution reltion between the configurtions, nd locl event structures [20], which re event utomt where the execution reltion lso considers step execution, cn model disruption with respect to the f-pproch. But they do not describe the dependency between the events intensionlly, i.e. they do not describe cuslities or conflicts between events by using reltions on events. An intentionl representtion hs usully the dvntge of mking the extension with time spects less expensive. In [19] logicl pproch to cuslities of events is presented, where ech event is ssigned to formul, which indictes when this event is enbled. In this pproch, it is not esy to model disbling. 1 by the definition of [17]

4 4 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption Another drwbck is tht it is hrd to find suitble grphicl nottion for this pproch. On the other hnd most clsses of event structures hve suitble grphicl nottion, which increses redbility. In this pper, we present new clss of event structures tht is suitble to model the f-philosophy, introduce grphicl representtion for these event structures nd study their properties. This new clss of event structures, clled precursor event structures, is generliztion of Winskel s event structures [29] in the sense tht reltion between sets of events nd events, i.e. P(E) E, is used to model disbling. The mening of this reltion is nlogous to the mening of the cuslity reltion of Winskel s event structures [29], i.e. condition (Z e) indictes tht event e is disbled in system run if every event of Z ppers in the system run. The result of this pper re: We present n f-bsed denottionl semntics of process lgebr tht contins disruption in terms of precursor event structures nd we show its consistency with n opertionl semntics. We show tht the clss of precursor event structures hs more expressive power with respect to event trces thn the clss of prime [26], flow [12], stble [29], bundle [23], extended bundle [22] nd dul event structures [21]. We give clssifiction of the expressive power of the clss of precursor event structures in terms of sets of event trces. Prts of the results of this pper ppered in [16]. The pper is orgnized s follows. Section 2 presents the syntx nd Section 3 presents the opertionl semntics of our process lgebr. The denottionl semntics is given in Section 4, where lso the clss of precursor event structures is introduced. This section lso exmines the expressive power of this clss of event structures nd contins the consistency result tht the trnsition system derived from the denottionl semntics is bisimilr to the opertionl semntics. 2. Syntx of the Process Algebr Let τ denote the internl ction. Furthermore, let Obs be set such tht τ / Obs. We cll Obs the set of observble ctions. The set of ll ctions Act is defined by Act = {τ} Obs. A relbelling function f is function from Act to Act such tht f(τ) = τ. We denote the set of ll lbelling functions by F L. Furthermore, ssume fixed countble set of process vribles Vr which is disjoint from Act. The process lgebr expressions EXP re defined by the following BNF-grmmr. B ::= 0 B + B B;B B [>B B A B B[f] B\\A x where f F L, x Vr, Act nd A Obs. A process with respect to EXP is pir decl,b consisting of declrtion decl : Vr EXP nd n expression B EXP. Let PA denote the set of ll processes. We sometimes cll n expression B EXP process if decl is cler from the context. The expressions hve the following intuitive mening: 0 is the inctive process, i.e. it cnnot execute ny ction. is the process tht executes nd termintes. B 1 + B 2 is choice between the behviors described by B 1 nd B 2. B 1 ;B 2 is the sequentil composition, i.e. B 1 proceeds until it termintes, fter which B 2 tkes over. B 1 [>B 2 is the disruption of B 1 by B 2, i.e. ny ction from B 2 disbles B 1 s long

5 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption 5 s B 1 hs not terminted. B 1 A B 2 describes the prllel execution of B 1 nd B 2 where both processes hve to synchronize on ctions from A. The process termintes if both sides terminte in the cse of synchroniztion or if one termintes nd the other one hs lredy terminted. The relbelling process B[f] executes ction f() if B executes ction. The restriction process B\\A executes ction if B executes ction provided is not contined in A. The behvior of x is given by the declrtion. Process lgebrs, like [18, 11], tht re bsed on the presented synchroniztion nd contin n expression for termintion process (denoted by 1) cn model restriction opertor in terms of the prllel opertor (B A 1). Since we do not introduce n expression for termintion process, we include the restriction opertor, s in [9, 25]. As it turns out, the restriction opertor plys lso crucil role for the opertionl definition of the prllel opertor in our setting. 3. Opertionl Semntics for PA The opertionl semntics of process is given by trnsition system. Definition 3.1. (Trnsition System) A lbelled trnsition system (with predictes) is tuple (S,L,, {P i i I}, s) with S, non-empty set of sttes L, set of lbels S L S, trnsition reltion i I : P i S, collection of predictes over the predicte index set I s S, the initil stte. We will write p q rther thn (p,,q). As stted in the introduction we dopt the philosophy tht the finl ction termintes the process. Therefore, we hve to distinguish between finl ctions nd non-finl ctions. In trnsition systems, the f-philosophy is usully modelled by using predicte with respect to ction [8, 2], where B indictes tht B termintes by executing. Insted of using trnsition system of the form (S,L,, { Act}, s), we encode the predicte directly in the trnsition reltion by S L (S { }), where B mens B. In the following, trnsition system will be qudruple (S,L,, s) where S L (S { }). The trnsition rules of decl EXP Act (EXP { }) with respect to decl : Vr EXP re presented in Figure 1. In the following, we explin the rules which devite from the stndrd ones: The process cn execute nd termintes by executing this ction. The process tht cn only execute ction nd evolves into non-terminted process tht is unble to proceed cn be modelled, for exmple, by ;0. The trnsition rule for the choice opertor is the stndrd CCS-rule [25]. If the first process of the sequentil composition termintes by executing (rule S 2 ), then B 1 ;B 2 evolves to B 2 by executing. The rules for the disrupt opertor re s expected, in prticulr if B 1 termintes by executing then so does B 1 [>B 2.

6 6 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption Here is bbrevition of decl A 1 : C 1 : B 1 B B 1 + B 2 B B 2 + B 1 B C 2 : B 1 B 1 + B 2 B 2 + B 1 S 1 : B 1 B 1 B 1 ;B 2 B 1;B S 2 : 2 B 1 B 1 ;B 2 B 2 D 1 : B 1 B 1 B 1 [>B 2 B 1 [>B 2 B 2 [>B 1 B 1 D 2 : B 1 B 1 [>B 2 B 2 [>B 1 P 1 : B 1 B 1 / A B 1 A B 2 B 1 AB 2 B 2 A B 1 B 2 A B 1 P 2 : B 1 / A B 1 A B 2 B 2 \\A B 2 A B 1 B 2 \\A P 3 : B 1 B 1 B 2 B 2 A B 1 A B 2 B 1 A B 2 P 4 : B 1 B 2 B 2 A B 1 A B 2 B 2 \\A B 2 A B 1 B 2 \\A P 5 : B 1 B 2 B 1 A B 2 A Lb : B B B[f] f() B [f] Res : B B / A B\\A B \\A Rec : decl(x) B x B B B[f] f() B / A B\\A decl(x) x Figure 1. Trnsition Rules for decl

7 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption 7 In the cse of the prllel opertor, we distinguish the cses whether the subprocesses terminte by executing the ctions or not. The second rule sttes tht if subprocess termintes by executing nonsynchronizing ction, then this process hs to be removed nd ll ctions in the synchroniztion set hve to be forbidden for the remining process. In the cse tht both processes terminte by executing n ction from the synchroniztion set, the whole process termintes by executing this ction. The rules for the relbelling opertor nd the restriction opertor only depend on the ction nme nd preserve termintion, s expected. 4. Denottionl Semntics for PA Clssicl event structures re not pproprite to define n f-bsed denottionl semntics for our process lgebr. To mke this cler, we consider dul event structures 2 [21], which re strictly more expressive thn prime, flow, stble nd extended bundle event structures [21]. Before we present the definition of dul event structures we introduce the following nottions: P(M) denotes the powerset of M, P f (M) denotes the set of ll finite subsets of M nd M 1 M 2 denotes the set of ll prtil functions from M 1 to M 2. Furthermore, the domin of prtil function f is the set {m f(m) is defined}, nd is denoted by dom(f). The function f M 1, where M 1 M 1, denotes the restriction of function f to the domin M 1. We write f(m 1) f (m 1 ) to denote tht f(m 1) is defined f (m 1 ) is defined nd f(m 1 ) is defined f(m 1 ) = f (m 1 ). Function π i denotes the projection to the i th component. For ny binry reltion we write p q if nd only if (p,q). The set M \ M denotes the set where ll elements of M re removed from M. We ssume fixed countble set of events U such tht e,e U : (e,e ),(,e),(e, ) U nd U nd / U. The constrints on set U result from technicl resons, i.e. they gurntee the well-definedness of the presented opertors (Subsection 4.2). Definition 4.1. (Dul event structure) A dul event structure E d = (E d, d, d,l d ) is n element of P(U) P(U U) P(P(U) U) (U Act) such tht d (E d E d ) nd d is irreflexive d P(E d ) E d dom(l d ) = E d A sequence of distinct events (e 1,...,e n ) is n event trce of E d if nd only if - every event is well cused, i.e. i < n : X i : X i d e i+1 X i {e 1,...,e i } nd - every event is not disbled t its execution position, i.e. i < n : j < i : (e i+1 d e j+1 ). The intuitive mening of dul event structure is the following: If e is in conflict to e, i.e. e d e, then e disbles e forever, but not vice vers. X d e mens tht before e my be executed n event from X hs to be executed. The lbelling function indictes which ction is observble when the event is executed. In dul event structures, it is implicitly ssumed tht n event is disbled by its execution. 2 Dul event structures re obtined from extended bundle event structures [22] by dropping the bundle stbility constrint.

8 8 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption Usully, termintion is modelled in dul event structures by introducing events lbelled with, which corresponds to the opertionl semntics using n dditionl internl ction. In the cse of the fphilosophy, it is more nturl to encode termintion informtion in event structures by predictes on events in order to sty close to the opertionl semntics. However, event structures including informtion bout termintion re not powerful enough to hndle disruption in n f-setting, which is illustrted in the following: The process (( b)[> c);d cn be modelled by dul event structures if we bse our modelling on the opertionl semntics of LOTOS 3. This is done by using n event e lbelled by τ nd by requesting e e, where e is the event corresponding to (lbelled by) c. Dul event structures re not pproprite to model the bove expression in n fsetting 4 : If we put c in conflict with, then c cn be disbled before b hppens nd by symmetry the sme rgument holds for b. But c hs to be in conflict with some ction since otherwise c remins enbled fter the execution of nd b. Therefore, conflict reltion tht is bsed on binry reltion on events is not pproprite to model this kind of disrupt opertor in the context of the f-philosophy. In the following, we introduce clss of event structures, which llows for the modelling of (( b)[>c);d in n f-setting by using reltion between sets of events nd events Precursor event structures Precursor event structures re generliztion of Winskel s event structures [29]: They contin n dditionl set of event sets (T ) to model termintion. Furthermore, sets of events (rther thn single events) disble other events. Definition 4.2. (prees) A precursor event structure (prees) E = (E,,,T,l) is n element of P(U) P(P(U) U) P(P(U) U) P(P(U)) (U Act) such tht P f (E) E nd e E : ( e) nd e E : {e} e 5 P f (E) E T P f (E) nd / T dom(l) = E A sequence of distinct events (e 1,...,e n ) is n event trce of E if nd only if - every event is well cused, i.e. i < n : X i {e 1,...,e i } : X i e i+1, - every event is not disbled t its execution position, i.e. i < n : X i {e 1,...,e i } : (X i e i+1 ) nd - it is not terminted during its execution, i.e. X {e 1,...,e n 1 } : X / T. 3 After the execution of nd b in (( b) [>c); d, τ-ction cn be executed, which disbles c. 4 Action c becomes immeditely disbled in (( b)[>c); d fter ctions nd b hve been executed. 5 Plese note tht we reversed the order of the conflict reltion used in dul event structures.

9 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption 9 An event trce (e 1,...,e n ) of E is terminted if nd only if X T : X {e 1,...,e n }. Let Tr e (E) denote the set of ll event trces of E, Tr e (E) denote the set of ll terminted event trces of E nd let PREES denote the set of ll precursor event structures. We cll E the set of events, the conflict reltion, the cuslity reltion, T the set of termintion sets nd l the ction-lbelling function. Set X is clled precursor (with respect to e) if X T or X e or X e. The intuitive mening of the conflict reltion is tht event e is disbled in system run (event trce) if there is conflict precursor Z of e (Z e) such tht the system run contins ll elements from Z. The intuitive mening of the cuslity reltion nd of the set of termintion sets T is similr to. For exmple, system run of prees is terminted if there is n element X of T where every element of X ppers in the system run. The constrints imposed on the conflict reltion re: no event is immeditely disbled ( ( e)), since otherwise the event cn be omitted. Furthermore, the execution of n event disbles itself ({e} e), i.e. every event cn hppen only once. The constrint / T on the set of termintion sets ensures tht precursor event structure my not terminte immeditely, i.e. it cn only terminte by executing n ction. It is lso resonble to consider only finite sets of events, since system run cn only contin finite sets. Note tht cuslity, disbling nd termintion re orthogonl concepts. In prticulr, termintion cnnot be modelled through disbling, since then no difference between dedlock nd termintion, which is essentil for the sequentil opertor, cn be mde. Exmple 4.1. Some precursor event structures re shown in Figure 2. Here, the events re depicted s dots, where their corresponding ction nmes re shown close to the dots (we do not nme the events explicitly nd identify them with the ction nmes if no confusion rises). The conflict reltion is illustrted by wvy lines. More precisely, conflict Z e is depicted by wvy rrow from the elements of Z to e. Furthermore, we do not drw the conflict precursors of the form Z e where e Z. Sometimes, the sme (wvy) lines re used in different precursors, for exmple the wvy line in E 3 corresponds to {} b nd {b}. The cuslity reltion is depicted similrly to the conflict reltion, except tht stright lines re used insted of wvy lines. A termintion set X is displyed by surrounding its events by closed line. Furthermore, we omit supersets, e.g. in E 1 we do not drw the cuslity {} b, since the constrint specified by this cuslity cn be derived from the constrint specified by the cuslity b. The set of event trces of E 4 is Tr e (E 4 ) = {(),(b),(c),(,b),(b,), (, c), (b, c)}. Herefter, E is considered to be (E,,,T,l) nd E i to be (E i, i, i,t i,l i ) Expressive Power We sy tht set V of finite sequences of events (set of event trces) is described by clss of event structures if there is n event structure of this clss which hs exctly the event trces of V s its event trces. The expressive power of clsses of event structures cn be mesured by compring the set of event trces described by them. This is more discriminting mesure thn tht of event utomt [27], which re bsed on configurtions. For simplicity, we neglect the termintion informtion when we compre

10 10 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption b b b ( b);c c ( b)[>c c b b c d (( b)[>c);d E 1 E 2 E 3 E 4 + b b E 5 b ( + b);c E 6 c ( + b)[>c c b E 7 Figure 2. Some Precursor Event Structures the expressive power. The termintion informtion cn be esily included by using predicte indicting which of the event trces re terminted. Theorem 4.1. Every set of event trces tht is described by prime [26], flow [12], stble [29], bundle [23], extended bundle [22] or dul event structure (Definition 4.1) is lso described by n precursor event structures, but not vice vers. The proof is given in Appendix A. We lso give clssifiction of the set of event trces tht corresponds to prees. In order to hve termintion sensitive clssifiction, we first introduce termintion sensitive nd lbelled sets of event trces. Definition 4.3. A termintion sensitive, lbelled set of event trces is triple (V, V, l) such tht V is set of event trces, i.e. set of finite sequences of distinct elements of U, V is the set of the terminted event trces with V V \ {ǫ} nd ll elements in V re mximl (cnnot be prober prefix of n event trces of V) nd l : U Act where every event ppers in V hve to be in the domin of l.

11 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption 11 The set of ll termintion sensitive, lbelled set of event trces re denoted by V. The termintion sensitive, lbelled set of event trces obtined from prees E is denoted by Tr(E), i.e. Tr(E) = (Tr e (E),Tr e (E),l). Definition 4.4. A termintion sensitive, lbelled set of event trces (V, V, l) is non-empty if V. is prefix closed if (e 1,..,e n ) V : (e 1,..,e n 1 ) V. is history-order independent if (e 1,...,e n+1 ),(e 1,...,e n ) V : {e 1,...,e n } = {e 1,...,e n } (e 1,...,e n,e n+1) V. is termintion-order independent if (e 1,...,e q ),(e 1,...,e n) V : {e 1,...,e q } {e 1,...,e n} (e 1,...,e q ) V (e 1,...,e n) V. is interrupt free if (e 1,...,e n+1 ),(e 1,...,e m,e) V,e 1,...,e q U : ( (e 1,...,e n+1,e) / V {e 1,...,e m} {e 1,...,e n } {e 1,...,e n+1 } {e 1,...,e q } ) (e 1,...,e q,e) / V. The prefix closedness mens tht ech event hs to be executed s singleton, i.e. there is no step execution enforced (this would be for exmple the cse if e cn only execute together with event e ). The non-emptiness together with the prefix closedness gurntees tht the empty sequence is contined. The history-order independence mens tht the future behvior only depends on the executed events nd not on the order of the executed events. The termintion-order independence sttes tht termintion is determined by sets of events, i.e. if set of events indicte termintion then every execution sequence tht contins the events of the set hs to be terminted execution sequence. The interrupt-freeness sttes tht n event tht ws enbled nd tht becomes disbled hs to sty disbled forever. Similrly to termintion the enbling nd the disbling is determined here by sets of events. Theorem 4.2. Let (V, V, l) be termintion sensitive, lbelled set of event trces. Then (V, V, l) is non-empty, prefix closed, history-order independent, termintion-order independent nd interrupt free if nd only if E PREES : V = Tr e (E) V = Tr e (E). The proof is given in Appendix A Trnsition Systems from PREES Here, we describe how to obtin n ction-lbelled trnsition system from prees. This construction will be used to estblish consistency result for the denottionl nd the opertionl semntics. First, we define the initil events of prees, i.e. those events which re redy to execute nd we define termintion predicte to indicte when prees termintes by executing n event e: Definition 4.5. Let E be prees. Then the set of initil events of E, denoted by init(e) nd the termintion predicte Υ P(P(U)) U re defined by init(e) = {e E e} Υ(T,e) {e} T.

12 12 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption In order to obtin trnsition system from prees, we define the reminder [4, 23, 24] of prees with respect to n initil event. The reminder with respect to event e describes the system fter the execution of e. All events which re disbled by e re removed. Plese remember tht {e} e, hence e disbles itself. After the execution of e, we remove lso those precursors tht contin disbled event, since these precursors cnnot be contined in further system runs nd hence hve no impct on the behvior. Definition 4.6. (Reminder of prees) Let E PREES nd e init(e). Then the reminder E [e] of E is given by (E,,,T,l ) where E = {e E ({e} e )} = {(Z,e ) e E Z E Z : Z e Z = Z \ {e}} T = {(X,e ) e E X E X : X e X = X \ {e}} = {X X E X T : X = X \ {e}} l = l E Plese note tht the reminder E [e] is prees whenever Υ(T,e). In the other cse, the empty set is n element of the set of termintion sets nd therefore the reminder is not prees. This is useful for our theory, since we forbid further event execution fter termintion. The definition of the reminder coincides with the definition of event trces in the following sense: Proposition 4.1. Suppose E PREES. Then for ll e 1,...,e n U we hve (e 1,..,e n ) Tr e (E) E [e1 ]...[e n] is defined. The proof is given in Appendix A. The reminders re used in the following definition to obtin n ction bsed interleving semntics for PREES. Definition 4.7. The trnsition reltion PREES Act (PREES { }) is defined by = {(E,l(e), E [e] ) e init(e) Υ(T,e)} {(E,l(e), ) e init(e) Υ(T,e)}. The trnsition system obtined from E 4 of Figure 2 is presented in Figure Complete Prtil Order ( ) on PREES In order to give denottionl semntics to PA we turn PREES into n ω-complete prtil order. The order on PREES is given by: Definition 4.8. Let E PREES nd E P f (U) such tht E E. Then the restriction of E to E is E E = (E, (P f (E ) E ), (P f (E ) E ),T P f (E ),l E ). A prees E 1 is smller thn prees E 2, written E 1 E 2, if E 1 E 2 nd E 1 = E 2 E 1.

13 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption 13 b E 4 c b b c c c c b (,,,, ) c Figure 3. Trnsition System Derived from PREES Plese note, tht the bove definition is n dption of the stndrd order on Winskel s event structures [29]. In prticulr, 1 = 2 (P f (E 1 ) E 1 ) holds whenever E 1 E 2. It is esily seen tht the restriction of prees is gin prees. Theorem 4.3. The set of ll prees ordered by is n ω-complete prtil order, where the lest upper bound of n ω-chin (E i ) i IN is given by i E i = ( i E i, i i, i i, i T i, i l i). The proof is given in Appendix A Opertors on PREES Here, we present the opertors on PREES tht re lter used to define the denottionl semntics. For simplicity, it is not explicitly defined in these opertors tht termintion set disbles every other event, since in the mening of event structures this is implicitly the cse. Definition 4.9. (Opertors on PREES) Let A Obs. Then define + : PREES PREES PREES with E 1 +E 2 (E 1 E 2,, 1 2,T 1 T 2,l 1 l 2 ) where = 1 2 {({e i },e j ) e i E i e j E j {i,j} = {1,2}} ; : PREES PREES PREES with E 1 ; E 2 (E 1 E 2,,,T 2,l 1 l 2 ) where = 1 2 (T 1 E 1 ) = 1 {(X 2,e 2 ) X 2 2 e 2 e 2 / init(e 2 )} (T 1 init(e 2 )) [> : PREES PREES PREES with E 1 [>E 2 (E 1 E 2,, 1 2,T 1 T 2,l 1 l 2 ) where = 1 2 {({e 2 },e 1 ) e 1 E 1 e 2 E 2 }

14 14 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption A : PREES PREES PREES with E 1 A E 2 = (Ẽ,,, T, l) where Ẽ = (E f 1 { }) ({ } Ef 2 ) Es E f i = {e E i l i (e) / A} E s = {(e 1,e 2 ) E 1 E 2 l 1 (e 1 ) = l 2 (e 2 ) A} = {( Z,(e 1,e 2 )) P f (Ẽ) Ẽ i : π i( Z) i e i } {( Z,(e 1,e 2 )) P f (Ẽ) Ẽ i : π i( Z) T i e i } = {( X,(e 1,e 2 )) P f (Ẽ) Ẽ i : e i = X i π i ( X) : X i i e i } T = { X { P f (Ẽ) i : X i π i ( X) : X i T i } l 1 (e 1 ) if e 2 = l((e1,e 2 )) = l 2 (e 2 ) otherwise Lb : (PREES F L ) PREES with Lb(E,f) = (E,,,T,f l). \\A : PREES PREES with E \\A = E {e E l(e) / A} To simplify our presenttion we introduce two uxiliry opertors: Shift 1 : PREES PREES with Shift 1 (E) = (Ẽ,,, T, l) where Ẽ = E { } = {(Z { },(e, )) Z e} = {(X { },(e, )) X e} T = {X { } X T } l(e, ) = l(e) Shift 2 : PREES PREES with Shift 2 (E) = (Ẽ,,, T, l) where Ẽ = { } E = {({ } Z,(,e)) Z e} = {({ } X,(,e)) X e} T = {{ } X X T } l(,e) = l(e). Plese note tht +, ; nd [> re defined if the involved E 1 nd E 2 hve disjoint set of events, i.e. E 1 E 2 =. We re only interested in such cses. The opertor symbols re overloded, which does not cuse ny problems, since they cn be uniquely determined from the context. We will give some comments on the definition of these opertors: For the sequentil composition, we hve to ensure tht ll events from the first event structure re disbled in the sequentil composition (which is done by T 1 E 1 ), since otherwise there exist trces tht contin events of the first event structure fter the execution of terminting set of events of the first event structures. For exmple, (,b,c,d) would become trce of (( b)[>c);d, which contrdicts the opertionl semntics. Plese note, tht the sequentil opertor is the only opertor tht genertes sets in the left hnd side of, since for the other opertors the disbling chrcter of the termintion is sufficient. Furthermore, in the sequentil composition the initil events of the second process cn only be executed if the first process hs terminted.

15 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption 15 The events of the prllel opertor re those which result from synchroniztion (E s ) together with the events tht cnnot synchronize (E f 1 { }) ({ } Ef 2 ). An event e = (e 1,e 2 ) of E 1 A E 2 is disbled if one of its component is disbled. Consequently, no event of component cn synchronize with more thn one component, since it disbles itself. Furthermore, if one component termintes, then ll its events nd the events tht results from synchroniztion re disbled. Such constrint is necessry, since otherwise the denottion of ([> b) c would be ble to perform ction b fter the execution of. An event obtined from synchroniztion is only enbled if its two components re enbled, hence ech projection of the precursor to component hs to be cuslity for this component. A process is terminted if nd only if both sides re terminted, which is defined nlogously to the cuslity reltion. The restriction opertor removes ll forbidden events, i.e. those lbelled with elements from A. Furthermore, only those precursors re considered tht do not contin events tht re lbelled with elements from A, since these precursors cn never be contined in system run. Lemm 4.1. All opertors of Definition 4.9 re well defined, i.e. they relly yield elements of PREES (if they re defined), nd they re continuous with respect to. Strightforwrd, where the continuity on events [29] technique cn be used to verify continuity. In order to rgue tht the bove opertors mtch the intuition, we dditionlly introduce opertors on termintion sensitive, lbelled set of event trces nd show tht they corresponds to the opertors on PREES. We write w 1 w 2 to denote the conctention of two finite sequences. The conctention of two sets of finite sequences V 1 nd V 2 is defined by V 1 V 2 = {w 1 w 2 w 1 V 1 w 2 V 2 }. Furthermore, the empty sequence is denoted by ǫ. Definition (Opertors on V) Let A Obs. Then define + : V V V with (V 1,V 1,l 1 )+(V 2,V 2,l 2 ) (V 1 V 2,V 1 V 2,l 1 l 2 ). ; : V V V with (V 1,V 1,l 1 ) ; (V 2,V 2,l 2 ) (V 1 (V 1 V 2 ),V 1 V 2,l 1 l 2 ). [> : V V V with (V 1,V 1,l 1 )[>(V 2,V 2,l 2 ) (V 1 ((V 1 \V 1 ) V 2 ),V 1 ((V 1 \V 1 ) V 2 ),l 1 l 2 ). A : V V V with (V 1,V 1,l 1 ) A (V 2,V 2,l 2 ) ( ˆV, ˆV,ˆl) where ˆV = ({ŵ (0) 1 ŵ(0) 2 ŵ(0) ŵ (1) 1 ŵ(1) 2 ŵ(1) ŵ (n) 1 ŵ(n) 2 ŵ(n) w (0) 1 w(0) 3 w (n) 1 w(n) 3 V 1,: w (0) 2 w(0) 4 w (n) 2 w(n) 4 V 2 : j n : ( (j) (w 1 = ǫ ŵ (j) 1 = ǫ) (w (j) 1 U ŵ (j) 1 = (w (j) 1, ) l 1(w (j) 1 ) / A)) ( (j) (w 2 = ǫ ŵ (j) 2 = ǫ) (w (j) 2 U ŵ (j) 2 = (,w (j) 2 ) l 2(w (j) 2 ) / A)) ( (j) (w 3 = ǫ w (j) 4 = ǫ ŵ (j) = ǫ) (w (j) 3 U w (j) 4 U ŵ (j) 2 = (w (j) 3,w(j) 4 ) l 1(w (j) 3 ) = l 2(w (j) 4 ) A)) } V ˆ is nlogously { defined s ˆV except tht V i is used insted of V i l 1 (e 1 ) if e 2 = l((e 1,e 2 )) l 2 (e 2 ) otherwise

16 16 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption Lb : V F L V with Lb((V,V,l),f) = (V,V,f l). \\A : V V with (V,V,l) \\A = ({e 0 e n V j n : l(e j ) / A}, {e 0 e n V j n : l(e j ) / A},l). We give some comments on the definition of these opertors. The choice opertor combines the possibilities of both rguments. An event trce of the sequentil opertor is either n event trce of the first process or conctention of terminted event trce of the first process with n event trce of the second process, where the resulting event trce is terminted if the event trce from the second process is terminted. Event trces of the disrupt opertor re obtined by tking ll event trces of the first process together with the conctention of event trces from the first nd the second process, where the first one must not be terminted (termintion disbles the disruption). The prllel opertor tkes those event trces tht re obtined by the interleving of event trces from the first nd the second process, where the synchroniztion condition hs to be stisfied. The prllel opertor lso renmes the events in order to be consistent with respect to the event denottion of the E-bsed prllel opertor. The relbelling opertor just relbels the events nd the restriction opertor just removes ll event trces tht contin n event tht is lbelled with n element from A. Proposition 4.2. Suppose E 1, E 2, E PREES such tht E 1 E 2 = then Tr(E 1 +E 2 ) = Tr(E 1 )+Tr(E 2 ) Tr(E 1 ; E 2 ) = Tr(E 1 ) ; Tr(E 2 ) Tr(E 1 [>E 2 ) = Tr(E 1 )[>Tr(E 2 ) Tr(E 1 A E 2 ) = Tr(E 1 ) A Tr(E 2 ) Tr(Lb(E,f) = Lb(Tr(E),f) Tr(E \\A) = Tr(E 1 ) \\A The proof is given in Appendix A Denottionl Mening First, we define the denottionl semntics of expressions (EXP) with respect to vrible ssignments. Then vrible ssignments re derived from declrtions, which re used to define the denottionl semntics of processes (PA). Definition Let [ ] : EXP (Vr PREES) PREES be defined s follows (where ρ : Vr PREES) [] ρ = ({ }, {({ }, )}, {(, )}, {{ }}, {(,)}) [0] ρ = (,,,, ) [B 1 + B 2 ] ρ = Shift 1 ([B 1 ] ρ )+Shift 2 ([B 2 ] ρ ) [B 1 ;B 2 ] ρ = Shift 1 ([B 1 ] ρ ) ; Shift 2 ([B 2 ] ρ ) [B 1 [>B 2 ] ρ = Shift 1 ([B 1 ] ρ )[>Shift 2 ([B 2 ] ρ ) [B 1 A B 2 ] ρ = [B 1 ] ρ A [B 2 ] ρ [B[f]] ρ = Lb([B] ρ,f) [B\\A] ρ = [B] ρ \\A [x] ρ = ρ(x) Remrk 4.1. [B] is continuous with respect to for every B EXP. Assume decl : Vr EXP. Then define F decl : (Vr PREES) (Vr PREES) with F decl (ρ) (x) = [decl(x)] ρ. From Remrk 4.1 it follows tht F decl is continuous. Therefore, from the

17 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption 17 complete prtil order theory [1] we get {[ ]} : (Vr EXP) (Vr PREES) with {[decl]} = fix(f decl ) = n Fn decl ( ) is well defined. Definition (Denottionl Semntics) Define [ ] : PA PREES by [ decl,b ] = [B] {[decl]}. Exmple 4.2. The denottionl semntics of some processes is illustrted in Figure Consistency of both Semntics Here, we show tht the trnsition system derived from the denottionl semntics nd the opertionl semntics yield bisimilr trnsition systems. Definition (Bisimilrity) Two trnsition systems (S,L,, s) nd (S,L,, s ) over the sme set of lbels re bisimilr if there is bisimultion, i.e. reltion R S S such tht ( s, s ) R nd for which for ll (s 1,s 1 ) R we hve: if s 1 s 2 then there is s 2 such tht (s 2,s 2 ) R nd s 1 s 2 nd s 2 = s 2 = if s 1 s 2 then there is s 2 such tht (s 2,s 2 ) R nd s 1 s 2 nd s 2 = s 2 =. Theorem 4.4. (Consistency) Suppose decl,b PA. Then the trnsition systems (EXP, Act, decl,b) nd (PREES, Act,, [ decl,b ]) re bisimilr. The proof is given in Appendix B. It uses new proof technique tht cn lso hndle ungurded recursion, which is not the cse for exmple in the consistency result of [4]. The two trnsition systems of Theorem 4.4 re not isomorphic in generl. Consider for exmple decl with decl(x) = ;x. Then the expression x yields finite trnsition system with respect to decl, wheres [ decl, x ] yields n infinite trnsition system with respect to. A. Proofs Theorem 4.1. Every set of event trces tht is described by prime [26], flow [12], stble [29], bundle [23], extended bundle [22] or dul event structure (Definition 4.1) is lso described by n precursor event structures, but not vice vers. From [21] we know tht every set of event trces described by n event structure of cited clss is lso described by dul event structure. Tht set of event trces described by dul event structure cn lso be described by n precursor event structure is shown by mpping the dul event structure E d = (E d, d, d,l d ) to Ω(E d ) = (E d,,,,l d ) where = {({e },e) e d e e = e } nd

18 18 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption = {(X,e) P f (E d ) E d X d P(E d ) : X d d e X X d }. It is esily checked tht Ω(E d ) PREES. In the following, we show tht E d nd Ω(E d ) hve the sme set of event trces. Suppose (e 1,...,e n ) is n event trce of E d. Then i < n : {e 1,...,e i } e i+1, since i < n : X i : X i d e i+1 X i {e 1,...,e i } hve to hold. Furthermore, i < n : X i {e 1,...,e i } : (X i e i+1 ), since i < n : j < i : (e i+1 d e j+1 ) hve to hold. Hence, (e 1,...,e n ) Tr e (Ω(E d )). Suppose (e 1,...,e n ) Tr e (Ω(E d )) then for ll i we hve the existence of X i {e 1,...,e i } such tht X i e i+1. Thus, X d P(E d ) : X d d e i+1 X i X d. Hence, X d : X d d e i+1 X d {e 1,...,e i }, which shows the cuslity constrint. Furthermore, X i {e 1,...,e i } : (X i e i+1) holds. Hence j < i : (e i+1 d e j+1 ). Thus, (e 1,...,e n ) is n event trce of E d. We showed tht PREES cn describe ll sets of event trces tht re describble by dul event structures. On the other hnd, the set of event trces obtined from E 4 of Figure 2 cnnot be described by dul event structure. Theorem 4.2. Let (V, V, l) be termintion sensitive, lbelled set of event trces. Then (V, V, l) is non-empty, prefix closed, history-order independent, termintion-order independent nd interrupt free if nd only if E PREES : V = Tr e (E) V = Tr e (E). We verify every direction seprtely: : Let E PREES. The non-emptiness, prefix closedness history-order independence nd the termintion order independency is esily seen. Now ssume (e 1,...,e n+1 ) Tr e (E) (1) (e 1,...,e m,e) Tre (E) (2) (e 1,...,e n+1,e) / Tr e (E) (3) {e 1,...,e m} {e 1,...,e n } {e 1,...,e n } {e 1,...,e q } From the definition of event trces of E nd (3) we obtin ( i < n + 1 : X i {e 1,...,e i } : (X i e i+1 )) (4) ( X {e 1,...,e n+1 } : (X e)) (5) ( i < n + 1 : X i {e 1,...,e i } : X i e i+1 ) (6) ( X {e 1,...,e n+1 } : X e) (7) From (1) we obtin tht (4) nd (6) is not possible. Furthermore, (5) cnnot be vlid, since (2) holds. Thus (7) hs to hold, i.e. there is X {e 1,...,e n+1 } such tht X e. But this X shows tht (e 1,...,e q,e) / Tre (E), hence the interrupt freeness is proved. The mximlity of the terminted event trces is n immedite consequence of the definition of Tr e (E). : Let (V,V,l) hve the stted properties. Define E = (U,,,T,l) by

19 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption 19 = {({e 1,...,e n },e) e 1,...,e i : {e 1,...,e i } {e 1,...,e n } (e 1,...,e i,e) V (e 1,...,e n ) V (e 1,...,e n,e) / V} {({e},e) e U} = {({e 1,...,e n },e) (e 1,...,e n,e) V} T = {{e 1,...,e n } (e 1,...,e n ) V}. It is esily seen tht E PREES. In the following, we show tht for ll e 0,...,e n we hve ( (e 0,...,e n ) V (e 0,...,e n ) Tr e (E) ) ( (e 0,...,e n ) V (e 0,...,e n ) Tr e (E) ). This is done by induction on n. (e 0,...,e n ) V (e 0,...,e n ) Tr e (E): The cse if n {0,1} is esily seen. Now suppose (e 0,...,e n+2 ) V. Then (e 0,...,e n+1 ) V, since V is prefixed closed. And so by induction i < n + 1 : X i {e 0,...,e i } : (X i e i+1 ). Now let X {e 0,...,e n+1 }. Assume tht X e n+2. Then there exists e 0,...,e j nd e 0,...,e k such tht X = {e 0,...,e k } {e 0,...,e j } {e 0,...,e k } (e 0,...,e j,e n+2) V (e 0,...,e k ) V (e 0,...,e k,e n+2) / V. From the interrupt freeness of V we obtin (e 0,...,e n+2 ) / V, which is contrdiction. Hence, X {e 0,...,e n+1 } : (X e n+1 ). The cuslity constrints re n immedite consequence of the prefix closedness of V. By the mximlity constrint on V we get (e 0,...,e n+1 ) / V. Hence, (e 0,...,e n+1 ) / Tr e (E) by induction. Therefore, for ll X {e 0,...,e n+1 } : X / T, which estblish the termintion constrint. Hence (e 0,...,e n+2 ) V. (e 0,...,e n ) V (e 0,...,e n ) Tr e (E): The cse n = 0 is esily seen. Suppose (e 0,...,e n+1 ) Tr e (E). Then we hve (e 0,...,e n ) V by induction, since (e 0,...,e n ) Tr e (E). From the cuslity constrint of n event trce, we obtin tht e 0,...,e i : {e 0,...,e i } {e 0,...,e n } (e 0,...,e i,e n+1) V. Assume tht (e 0,...,e n+1 ) / V. Then by the definition of E we hve {e 0,...,e n } e n+1. Hence (e 0,...,e n+1 ) / Tr e (E), which is contrdiction. Thus, (e 0,...,e n+1 ) V. (e 0,...,e n ) V (e 0,...,e n ) Tr e (E): Let (e 0,...,e n ) V then we hve lredy shown tht (e 0,...,e n ) Tr e (E). Thus, (e 0,...,e n ) Tr e (E) by the definition of T. (e 0,...,e n ) V (e 0,...,e n ) Tr e (E): Let (e 0,...,e n ) Tr e (E). Then by definition there is X {e 0,...,e n 1 } : (X {e n }) T. Hence, there exists e 0,...,e q such tht (e 0,...,e q,e n) V {e 0,...,e q } = X. From the termintion order independency we obtin (e 0,...,e n ) V. Theorem 4.3. Suppose E PREES. Then for ll e 1,...,e n U we hve (e 1,..,e n ) Tr e (E) E [e1 ]...[e n] is defined. First we stte tht E [e1 ]...[e n] = (E,,,T,l ) with E = {e E Z {e 1,...,e n } : (X e )} = {(Z,e ) e E Z E Z : Z e Z = Z \ {e 1,...,e n }} = {(X,e ) e E X E X : X e X = X \ {e 1,...,e n }} T = {X X E X T : X = X \ {e 1,...,e n }} l = l E

20 20 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption if it is defined. This cn be shown by induction on n, which is omitted here. The min sttement is shown by induction on n, where the bse cse n = 0 is esily seen. : If (e 1,..,e n+1 ) Tr e (E) then (e 1,..,e n ) Tr e (E). Hence, by induction E [e1 ]...[e n] is defined. From the non disbling condition of event trces of E we obtin e n+1 E, from the well cuslity constrint we get e n+1 nd from the termintion constrint we obtin tht E [e1 ]...[e n] PREES. Thus E [e1 ]...[e n+1 ] is defined. : If E [e1 ]...[e n+1 ] is defined then E [e1 ]...[e n] is defined nd so by induction (e 1,..,e n ) Tr e (E). Therefore, it is only left to show tht X {e 1,...,e n } : (X e n+1 ), which hs to be the cse, since otherwise e n+1 / E, X {e 1,...,e n } : X e n+1, which hs to be the cse, since otherwise e n+1 is not enbled in E [e1 ]...[e n] nd X {e 1,...,e n } : X / T, which hs to be the cse, since otherwise E [e1 ]...[e n] / PREES. Theorem 4.4. The set of ll prees ordered by is n ω-complete prtil order, where the lest upper bound of n ω-chin (E i ) i IN is given by i E i = ( i E i, i i, i i, i T i, i l i). It is esily seen tht is prtil order with (,,,, ) s its lest element. Furthermore, E = i E i is prees. In the following we only consider T. The cses nd follow nlogously. upper bound: Obviously, T j i T i. Let X i E i such tht X E j nd i : X T i. Thus X E i nd from E j E i or E i E j we get X T j, s required. lest upper bound: Let E be prees such tht E i E for ll i IN. Then i E i E. Let X T. Then j : X T j. Hence, X T, since E j E. Let X T such tht X i E i. Since X is finite, there exists j IN such tht X E j. Hence, X T j. Then by definition X T. Proposition 4.2. Suppose E 1, E 2, E PREES such tht E 1 E 2 = then Tr(E 1 +E 2 ) = Tr(E 1 )+Tr(E 2 ) Tr(E 1 ; E 2 ) = Tr(E 1 ) ; Tr(E 2 ) Tr(E 1 [>E 2 ) = Tr(E 1 )[>Tr(E 2 ) Tr(E 1 A E 2 ) = Tr(E 1 ) A Tr(E 2 ) Tr(Lb(E,f) = Lb(Tr(E),f) Tr(E \\A) = Tr(E 1 ) \\A We present the proof of Tr(E 1 A E 2 ) = Tr(E 1 ) A Tr(E 2 ). The other cses re simpler nd hence re omitted. Suppose (ẽ 0,...,ẽ n ) Tr e (E 1 A E 2 ). Define for i n: { { w (i) 1 = e ǫ if ẽ i = (e, ) otherwise w (i) 3 = e if ẽ i = (e,e ) ǫ otherwise

21 Fecher, Mjster-Cederbum / Event Structures for Arbitrry Disruption 21 Now, we will rgue tht w (0) 1 w(0) 3 w (n) 1 w(n) 3 Tr e (E 1 ): the cuslity nd the conflict constrint is esily seen. Suppose the termintion constrint is violted, then by the definition of A ll events corresponds to E 1 re disbled, hence (ẽ 0,...,ẽ n ) / Tr e (E 1 A E 2 ). Contrdiction. w (i) 2 nd w (i) 4 cn be nlogously defined. Hence, (ẽ 0,...,ẽ n ) Tr(E 1 ) A Tr(E 2 ) is n immeditely consequence of the definitions of w (i) j. It is lso strightforwrdly checked tht every termintion trce of Tr e (E 1 A E 2 ) is lso termintion trce of Tr(E 1 ) A Tr(E 2 ). Suppose ŵ (0) 1 ŵ(0) 2 ŵ(0) ŵ (n) 1 ŵ(n) 2 ŵ(n) Tr(E 1 ) A Tr(E 2 ). We use induction on n. If ŵ (n) 1 = ǫ then ŵ (0) 1 ŵ(0) 2 ŵ(0) ŵ (n) 1 Tr e (E 1 A E 2 ) by induction. If ŵ (n) 1 = (e, ) then the index of the cuslity set obtined in E 1 cn be used to obtin cuslity set in Tr e (E 1 A E 2 ). Suppose ŵ (0) 1 ŵ(0) 2 ŵ(0) ŵ (n) 1 / Tr e (E 1 A E 2 ) becuse of conflicts. But this cnnot be the cse, becuse otherwise w (0) 1 w(0) 3 w (n) 1 Tr e (E 1 ) would be contrdicted by the disbling or by the termintion constrint. The termintion constrint of ŵ (0) 1 ŵ(0) 2 ŵ(0) ŵ (n) 1 cn be esily seen. Hence, ŵ (0) 1 ŵ(0) 2 ŵ(0) ŵ (n) 1 Tr e (E 1 A E 2 ). The other cses, ŵ (n) 2 nd ŵ (n), re shown nlogously. B. Proof of Theorem The min problem in verifying Theorem 4.4 is to hndle recursion, since it will be hrd to ppropritely relte n ction execution in the SOS-trnsition system to n ction execution in the corresponding PREES-denottion. The technique used in [4] cnnot be pplied, since there recursion is hndled vi metric spces, nd not vi complete prtil orders. This hs the consequence tht it is not even possible to give denottion to ungurded processes, since the distnce function will not be contrctive in these cses. The technique used in [18] hs the disdvntge tht event identifiers re introduced t the lnguge level nd therefore finite systems cn never be obtined if recursion is present. In order to solve the problem of relting ction execution to event execution, we first introduce n event-bsed SOS-trnsition system, where every ction execution is nnotted by the event-nme of its PREES-denottion. Then we show tht the event-bsed trnsition system of process decl, B is bisimilr to (EXP, Act, decl,b) nd lso to (PREES, Act,, [ decl,b ]). Hence, Theorem 4.4 follows by the trnsitivity of the bisimilrity [25]. The dvntge of our pproch is tht we cn hndle ungurded processes. B.1. Event Bsed Trnsition System. Let EXP e be the set tht contins exctly the elements generted by C ::= B C;B C [>B C A C C[f] C\\A C i where B EXP, f F L, i {1,2} nd A Obs. Let PA e = (Vr EXP) EXP e. Plese note tht vribles re still ssigned to the stndrd expressions. In Figure 4 the event-bsed trnsition rules decl EXPe (Act U) (EXP e { }) re presented. In the following C denotes n element of EXP e { } nd B denotes n element of EXP { }.