Strong Bisimulation. Overview. References. Actions Labeled transition system Transition semantics Simulation Bisimulation

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1 Strong Bisimultion Overview Actions Lbeled trnsition system Trnsition semntics Simultion Bisimultion References Robin Milner, Communiction nd Concurrency Robin Milner, Communicting nd Mobil Systems 32

2 Actions nd Sttes Actions: We presuppose n infinite set N of nmes; we use, b, to rnge over N. Then we introduce the set N = { N }, which we cll co-nmes. We ssume tht N nd Nre disjoint, nd we denote their union N N, by L, the set of lbels (the kind of lbels, which identifies the buttons on our blck boxes). For the moment, the set L nd Σ coincide. Conceptul chnges: Wht mtters bout string s - sequence of ctions - is not whether it drives the utomton into n ccepting stte (since we cnnot detect this by interction) but whether the utomton is ble to perform the sequence of s interctively. A lbeled trnsition system cn be thought of s n utomton without strt or ccepting sttes. Any stte cn be considered s the strt. 33

3 Generl Automton,b,c b,c q3,b q0 q2 c b q1 c 34

4 Lbeled Trnsition System A lbeled trnsition system over ctions Σ is pir (Q, T ) consisting of: set Q = {q 0, q 1, } of sttes, ternry reltion T (Q Σ Q), known s trnsition reltion. If (q,, q ) T we write q q', nd we cll q the source nd q the trget of the trnsition. Alterntive definition: (S, T, { t : t T } ) S is set of sttes T is set of trnsition lbels t S S is trnsition reltion for ech t T. 35

5 LTS nd Automton An LTS cn be though of s n utomton without strt or ccepting sttes. By omitting the strt stte, we gin the freedom to consider ny stte s the strt. Ech selection of strt defines different utomton, but is bsed upon the sme LTS. 36

6 Strong Simultion - Ide In 1981 D. Prk proposed new pproch to define the equivlence of utomtons - bisimultion. Given lbeled trnsition system there exists stndrd definition of bisimultion equivlence tht cn be pplied to this lbeled trnsition system. The definition of bisimultion is given in coinductive style tht is, two systems re bisimulr if we cnnot show tht they re not. Informlly, to sy system S1 simultes system S2 mens tht S1 s observble behvior is t lest s rich s tht of S2. 37

7 Strong Simultion - Definition Let (Q, T ) be n lbeled trnsition system, nd let S be binry reltion over Q. Then S is clled strong simultion over (Q, T ) if, whenever p S q, If p α p' then there exists q Q such tht q α q' nd p S q. We sy tht q strongly simultes p if there exists strong simultion S such tht p S q. 38

8 Strong Simultion - Exmple S1: q0 S2: 25ct 25ct q1 p1 25ct te te q2 q4 p2 coffee q3 Clim: The sttes q0 nd p0 re different. Therefore, the systems S1 nd S2 should not be considered equivlent. p0 25ct p3 25ct p4 coffee p5 39

9 Defining S If we define S = {(p0, q0), (p1, q1), (p3, q1), (p2, q4), (p4, q2), (p5, q3)} then S is strong simultion; hence S1 strongly simultes S2. To verify this, for every pir (p, q) S we hve to consider ech trnsition of p, nd show tht it is properly mtched by some trnsition of q. However, there exists no strong simultion R tht contins the pir (q1, p1), becuse one of q1 s trnsition could never be mtched by p1. Therefore, the sttes q0 nd p0 re different, nd the systems S1 nd S2 re not considered to be equivlent. 40

10 Strong Bisimultion The converse R -1 of ny binry reltion R is the set of pirs (y, x) such tht (x, y) R. Let (Q, T ) be n lbeled trnsition system, nd let S be binry reltion over Q. Then S is clled strong bisimultion over (Q, T ) if both S nd its converse S -1 re strong simultions. We sy tht tht the sttes p nd q re strongly bisimulr or strongly equivlent, written p ~ q, if there exists strong bisimultion S such tht p S q. 41

11 Digrms The condition for S to be strong bisimultion cn be expressed in digrms: if p S q then for some q, q p p S q Thus q strongly simultes p, or p is strongly simulted by q, mens tht whtever trnsition pth p tkes, q cn mtch it by pth, which retins ll of p s options. 42

12 Bisimultion - Bord Gme Checking the equivlence of interctive systems cn be considered bord gme between two persons, the unbeliever, who thinks tht S1 nd S2 re not equivlent, nd the believer, who thinks tht S1 nd S2 re equivlent. The underlying strtegy of this gme is tht the (demonic) unbeliever is trying to perform trnsitions, which the cnnot be mtched by the (ngelic) believer. The unbeliever loses if there re no trnsitions left for either systems, wheres the believer loses, if he cnnot mtch move mde by the unbeliever. 43

13 Working With Simultions Wht do we do with (bi)simultions? Exhibiting (bi)simultion: guessing reltion S tht contins (p,q) Checking (bi)simultion: checking tht given reltion S is in fct (bi)simultion There exist lgorithms nd tools (e.g. CWB) tht cn generte reltions tht by construction stisfy the property of being (bi)simultion. Results on (semi-)decidbility re very importnt for such tools. 44

14 Checking Bisimultion S1: p1 b p2 S1 ~ S2? To construct S strt with (p0, q0) nd check whether S2 cn mtch ll trnsitions of S1: S = { (p0, q0), (p1, q1), (p3, q1), (p2, q2), (p4, q3) } p0 p3 c p4 System S2 cn simulte system S1. Now check, whether S -1 is simultion or not: S -1 = { (q0, p0), (q1, p1), (q1, p3), (q2, p2), (q3, p4) } q0 S2: q1 b c q2 q3 Strt with (q0, p0) S -1. 1: q0 hs one trnsition tht cn be mtched by two trnsitions of S1 (trget p1 nd p3, respectively) nd we hve (q1, p1) S -1 nd (q1, p3) S -1. 2: q1 hs two trnsitions b nd c, which, however, cnnot be ppropritely mtched by the relted sttes p1 nd p3 of system S1 (p1 hs only b trnsition whilst p3 hs only c trnsition). We hve, therefore, S1 ~ / S2. 45

15 Linking Sttes b p0 b p1 p2 q0 q2 b q1 S = {(p0,q0), (p0,q2), (p1,q1), (p2,q1)} 46

16 ~ is n Equivlence Reltion p ~ p p ~ q implies q ~ p p ~ q nd q ~ r imply p ~ r 47

17 Reflexivity Let Q be process nd Id Q = {(p,p) p Q}. For reflexivity, it is enough to show tht Id Q is bisimultion. Proof: Suppose Id Q = {(p,p) p Q}. We hve to show tht for ll (p,p) Id Q, if p p' α, then there exists q such tht p α q' nd (p,q ) Id Q. Now, let p Id Q p, if p p' α, then we hve to find stte q Q such tht p q' α nd p Id Q q. By ssumption, p Q, we tke q = p, hence p α p', nd by definition of Id Q, we hve p Id Q p, s required. Finlly, since Id Q = Id Q -1, Id Q is bisimultion. q.e.d. 48

18 Symmetry For symmetry, we hve to show tht if S is bisimultion then so is its converse S -1. However, this is obvious from the definition of bisimultion. 49

19 Trnsitivity S 1 S 2 = {(p, r) q exists with (p, q) S 1 nd (q, r) S 2 } Proof: Let (p, r) S 1 S 2. Then there exists q with (p, q) S 1 nd (q, r) S 2. ( ) If p α p', then since (p, q) S 1 there exists q nd q α q' nd (p, q ) S 1. Furthermore, since (q, r) S 2 there exists r with r α r' nd (q, r ) S 2. Due to the definition of S 1 S 2 it holds tht (p, r ) S 1 S 2 s required. ( ) similr to ( ). 50

20 Fct ~ is the lrgest strong bisimultion, tht is, ~ is strong bisimultion nd includes ny other such. Assume tht ech S i (i=1,2, ) is strong bisimultion. Then U S is strong bisimultion. i I i Let ech S i (i=1,2, ) be strong bisimultion. We hve to show tht U S is strong bisimultion. i I i Let (p,q) U S i I i. If p α p', then since (p,q) S i, 1 i n, there exists q S i with q α q' nd (p,q ) S i nd (p,q ) U S. By symmetry, the converse holds s well. i I i 51

21 Bisimultion - Summry Bisimultion is n equivlence reltion defined over lbeled trnsition system, which respects non-determinism. The bisimultion technique cn be used to compre the observble behvior of intercting systems. Note: Strong bisimultion does not cover unobservble behvior, which is present in systems tht hve opertors to define rection (tht is, internl ctions). 52