Trnsition systems (motivtion) Course Modelling of Conurrent Systems ( Modellierung neenläufiger Systeme ) Winter Semester 2009/0 University of Duisurg-Essen Brr König Tehing ssistnt: Christoph Blume In the following we will look t trnsition systems, whih n e used to represent the ehviour of system in very diret nd expliit wy. 2 3 Brr König Course Modelling of Conurrent Systems Ations nd sequenes of tions We use the following nottion: At: set of tomi tions, often denoted y,,,.... We will sometimes use the internl tion τ, whih should e invisile from outside. At : the set of ll finite words over the lphet At. At ω : the set of ll infinite words over the lphet At. An infinite word w At ω n e represented y mpping w : N0\{0} At. At = At At ω : the set of ll finite nd infinite words over At. Pref(L): given lnguge L At we define the set of ll prefixes of L s follows: Brr König Course Modelling of Conurrent Systems 24 Reltions We use the following nottion for reltions: A reltion R etween the sets A, B is suset of A B, i.e., R A B. Let (, ) A B e pir with (, ) R. Then we lso write R (in words: is relted to ). Pref(L) = {u At w L, v At : w = uv}. Brr König Course Modelling of Conurrent Systems 25 Brr König Course Modelling of Conurrent Systems 26
Trnsition systems Trnsition system (definition) Let At e fixed set of tions. A trnsition system T = (Z, ) over At onsists of A set Z of sttes nd set Z At Z of trnsitions etween sttes. A trnsition system is lled finite if the stte set s well s the set of trnsitions is finite. Insted of (, l, ) we will in the following write l. Furthermore we will revite... n y...n. In ddition ε holds for every stte. Given stte we write if there exists stte with nd if there is no suh stte. Trnsition systems (exmples) A lssil exmple: the te/offee-mhine We wnt to model very simple mhine tht outputs te or offee fter hs een inserted nd utton hs een pressed, n show fulty ehviour nd my potentilly ehve non-deterministilly. Brr König Course Modelling of Conurrent Systems 27 Trnsition systems (exmples) Brr König Course Modelling of Conurrent Systems 28 Trnsition systems (exmples) offee te hnge utton(offee) utton(te) utton(offee) utton(te) offee te A te/offee-mhine. A mhine tht gives k hnge. We distinguish etween input tions of the form nd output tions (lso lled otions) of the form. Brr König Course Modelling of Conurrent Systems 29 Brr König Course Modelling of Conurrent Systems 29
Trnsition systems (exmples) Trnsition systems (exmples) error error offee te utton(offee) utton(te) offee utton(offee) error utton(te) te A mhine with n error. The ourrene of n error is tully rther n internl tion nd ould lterntively e modelled with τ. An (unfir) mhine with fulty ehviour whih my enter the error stte fter hs een inserted. Brr König Course Modelling of Conurrent Systems 29 Trnsition systems (exmples) Brr König Course Modelling of Conurrent Systems 29 Trnsition systems (exmples) repir error offee te utton(offee) utton(te) offee utton(offee) te utton(te) A mhine with n error stte tht n e repired. A mhine with non-deterministi ehviour tht mkes hoie of everges for the user. Brr König Course Modelling of Conurrent Systems 29 Brr König Course Modelling of Conurrent Systems 29
Deterministi trnsition systems Deterministi trnsition system (definition) A trnsition system T = (Z, ) is lled deterministi, if for every stte Z: Remrks: Whenever nd 2, then = 2. All te/offee-mhines, prt from the lst, re deterministi. Opposed to deterministi finite utomt we do not require for deterministi trnsition systems tht every tion is fesile in every stte. Trnsition systems (exmples) The Dining Philosophers prolem onsiders proesses (= philosophers) nd resoures (= forks): Three philosophers re seted t round tle nd there is fork etween eh pir of philosophers sitting side y side. Philosophers eventully get hungry nd need oth djent forks in order to et. Eh philosopher piks up oth forks (in ny order) t n ritrry moment in time, ets nd, fter eting, puts k oth forks. P F3 F F2 P2 P3 Brr König Course Modelling of Conurrent Systems 30 Trnsition systems (exmples) Brr König Course Modelling of Conurrent Systems 3 Trnsition systems (exmples) Question The intention is tht the system runs forever nd never termintes. Cn the system reh dedlok stte in whih no tions re possile? We solve this question y drwing the orresponding trnsition system (t lest prtilly). Sttes: 3-tuples of the form (, 2, 3), where i symolies the stte of fork Fi. It holds tht: {, P, P2} (F is not ssigned, ssigned to P or to P2) 2 {, P2, P3} (nlogous mening) 3 {, P3, P} (nlogous mening) Hene the trnsition system hs 3 3 = 27 sttes. Ations: t(pi, Fj): philosopher Pi tkes fork Fj. ei: philosopher Pi ets. ri: philosopher Pi returns oth forks. Brr König Course Modelling of Conurrent Systems 32 Brr König Course Modelling of Conurrent Systems 33
Trnsition systems (exmples) Exerpt from the trnsition system: r (,, ) t(p, F) (P,, ) t(p, F3) t(p2, F2) Trnsition systems (exmples) Exerpt from the trnsition system: r (,, ) t(p, F) (P,, ) t(p, F3) t(p2, F2) e (P,, P) (P, P2, ) e (P,, P) (P, P2, ) t(p2, F2) t(p3, F3) t(p2, F2) t(p3, F3) e2 e (P, P2, P) t(p2, F) (P2, P2, ) (, P2, ) r (P, P2, P3) e2 e (P, P2, P) t(p2, F) (P2, P2, ) (, P2, ) r (P, P2, P3) Dedlok! No more trnsitions re possile. Brr König Course Modelling of Conurrent Systems 34 Trnsition systems (exmples) Possile solutions for the dedlok prolem: Avoid dedloks: A philosopher hs to tke oth forks simultneously. Introdution of left-hnded nd right-hnded philosophers. A left-hnded philosopher tkes the left fork first, right-hnded philosopher tkes the right fork first. If we hve left-hnded s well s right-hnded philosophers, no dedloks n our. Order the resoures: F < F2 < F3. The smller resoure with respet to the ordering, the erlier it hs to e tken. Reognie nd resolve dedloks: dedloks hve to e found (y monitoring proess, vi timeouts,... ) nd to e resolved ordingly. Brr König Course Modelling of Conurrent Systems 34 Behviourl equivlenes (tre equivlene) Lnguge of stte (definition) The lnguge of stte is the set of ll words over At, whih orrespond to pths originting from. There re three distint possiilities to define lnguges: S() = {w At Z : w } S ω () = {23 At ω S () = S() S ω () 2 3 2... } The lnguge of stte is lso lled its set of tres. Tre equivlene (definition) Two sttes, Z re lled S-tre equivlent whenever S() = S( ). Anlogously we define S ω - nd S -tre equivlene. Brr König Course Modelling of Conurrent Systems 35 Brr König Course Modelling of Conurrent Systems 36
Behviourl equivlenes (tre equivlene) Tre equivlene hs the following property: Whenever, re S -tre equivlent, then they re lso S- nd S ω -tre equivlent. Reson: S () = S ( ) implies S() = S () At = S ( ) At = S( ). Anlogously: S ω () = S () At ω = S ( ) At ω = S ω ( ). Behviourl equivlenes (tre equivlene) However: Whenever, re S-tre equivlent sind, then they re not neessrily S ω -tre equivlent (nd vie vers). Counterexmples: Sttes, re S-equivlent, ut not S ω -equivlent. (From only finite pths originte.)... Sttes, re S ω -equivlent, ut not S-equivlent.... Brr König Course Modelling of Conurrent Systems 37 Behviourl equivlenes (filures equivlene) Brr König Course Modelling of Conurrent Systems 38 Behviourl equivlenes (filures equivlene) Motivtion: the sttes, of the following two te/offee-mhines re tre equivlent. offee utton(offee) te utton(te) offee utton(offee) te utton(te) However, one does not relly wnt to onsider these two sttes s equl. In one se the mhine works orretly, in the other the mhine keeps the without giving k everge. Therefore: tre equivlene is not suffiient. We lso need wy of expressing tht the rehle sttes llow respetively disllow the sme tions. Brr König Course Modelling of Conurrent Systems 39 Filures equivlene (Definition) Let T = (Z, ) e trnsition system. The set of filure pirs of stte Z is defined s: F() = {(w, A) At P(At) Z : w nd for ll A} Two sttes, Z re lled filure equivlent whenever F() = F( ). Brr König Course Modelling of Conurrent Systems 40
Behviourl equivlenes (filures equivlene) Behviourl equivlenes (filures equivlene) Motivtion: the sttes, of the two following te/offee-mhines re tre equivlent. offee utton(offee) te utton(te) offee utton(offee) te utton(te) In the exmple F( ) ontins the pir (, {utton(offee), utton(te)}) whih is not ontined in F(). Remrks: If pir (w, A) is ontined in the set F(), then we know for every A A tht lso (w, A ) F(). (We lso sy: F() is losed under set inlusion.) Hene it is suffiient to onsider pirs (w, A) where A is mximl. The definition of tivtion equivlene with the following sets A() is not suffiient to distinguish the orret mhine from the fulty one. A() = {(w, A) At P(At) Z : w nd for ll A} Brr König Course Modelling of Conurrent Systems 4 utomt Proess luli Petri nets Grph trnsformtion Introdution Trnsition systems Bühi Behviourl equivlenes (filures equivlene) Another exmple for filures equivlene: It holds tht: F() = {(ε, A) A {, }} {(, A) A {,, }} {(, A) A {,, }} {(, A) A {,, }} = F( ) nd, re hene filure equivlent. Brr König Course Modelling of Conurrent Systems 43 Brr König Course Modelling of Conurrent Systems 42 Behviourl equivlenes (isimilrity) There is nother importnt equivlene whih is slightly finer thn filures equivlene nd usully esier to mehnie: isimilrity or isimultion equivlene. Intuitively we require tht the two sttes, re le to mutully simulte eh other. Bisimultion (definition) Let T = (Z, ) e trnsition system. A reltion R Z Z on sttes is lled isimultion if for every pir (, 2) R nd for every tion At: for every with there exists 2 with 2 2 nd (, 2 ) R. for every 2 with 2 2 there exists with nd (, 2 ) R. Brr König Course Modelling of Conurrent Systems 44
Behviourl equivlenes (isimilrity) Behviourl equivlenes (isimilrity) Bisimilrity (Definition) Two sttes, 2 re lled isimilr if there exists isimultion R with (, 2) R. In this se we write 2. The reltion is lled isimilrity or isimultion equivlene. Exmples: offee te utton(offee) utton(te) offee te utton(offee) utton(te) The two sttes, re not isimilr. Brr König Course Modelling of Conurrent Systems 45 Behviourl equivlenes (isimilrity) Brr König Course Modelling of Conurrent Systems 46 Behviourl equivlenes (isimilrity) Exmples: offee te offee te utton(offee) utton(te) utton(offee) utton(te) The two sttes, re not isimilr. Exmples: d 5 d 2 6 7,, 3 4 8 The two sttes =, = 5 re isimilr. Bisimultion: R = {(, 5), (2, 6), (2, 7), (3, 8), (4, 8)} d Brr König Course Modelling of Conurrent Systems 47 Brr König Course Modelling of Conurrent Systems 48
Behviourl equivlenes (isimilrity) Behviourl equivlenes (isimilrity) Bisimilrity n lso e hrteried s gme: Bisimultion gme Prtiipnts: Plyer I, Plyer II Equipment: 2 tokens, trnsition system Initil sitution: The two tokens re pled on the sttes,. Rules of one round: Plyer I hooses one of the two tokens nd mkes n ritrry (-)trnsition. Plyer II hs to tke the other token nd mke n -trnsition s n nswer. Importnt: In eh round Plyer I n hnge nd hoose the other token! Bisimultion gme (ontinution) Gme plying: Strting with the initil sitution one round fter the other is plyed, until one of the two plyers n not mke nother move. Winning ondition: Plyer I wins if he n mke move tht n not e simulted y Plyer II. Plyer II wins if he n simulte every move of Plyer I. (This n lso men tht the gme ontinues forever.) Brr König Course Modelling of Conurrent Systems 49 Behviourl equivlenes (isimilrity) Brr König Course Modelling of Conurrent Systems 50 Behviourl equivlenes (isimilrity) Corretness of the isimultion gme (proposition) Two sttes, re isimilr if nd only if Plyer II hs winning strtegy in the orresponding isimultion gme, where tokens re initilly pled on,. Remrks: This mens tht Plyer II hs to e le to win if he does not mke mistke. It does not men tht Plyer II will lwys win regrdless of the moves he mkes. The strtegy mentioned ove is isimultion reltion R whih n e used to show tht nd re isimilr. How to desrie winning strtegy for Plyer I? Plyer I mkes the first move. It is possile to set up deision tree tht reords whih moves hve to e mde y Plyer I depending on the nswering moves of Plyer II. The leves of the deision tree re the moves of Plyer I whih n not e nswered y Plyer II. Brr König Course Modelling of Conurrent Systems 5 Brr König Course Modelling of Conurrent Systems 52
Behviourl equivlenes (isimilrity) Behviourl equivlenes (isimilrity) Exmple (winning strtegy for Plyer I): 2 4 Properties of isimilrity: Bisimilrity is isimultion (proposition) 2 3 5 6 Let T = (Z, ) e trnsition system. The isimilrity on Z stisfies: = {R R Z Z, R is isimultion} The isimilrity is isimultion itself. Plyer I: 3 Plyer II: 4 5 4 6 Plyer I: 5 5 3 3 Plyer II: Depending on whether Plyer II nswers with 4 5 or with 4 6, it is neessry for Plyer I to hoose different move. Brr König Course Modelling of Conurrent Systems 53 Behviourl equivlenes (isimilrity) Brr König Course Modelling of Conurrent Systems 54 Behviourl equivlenes (isimilrity) Closure properties (proposition) Let T = (Z, ) e trnsition system nd let R, R2 e isimultions. Then the following reltions re isimultions s well: Id Z = {(, ) Z}. 2 R = {(2, ) (, 2) R} 3 RR2 = {(, 3) 2 : (, 2) R, (2, 3) R2} 4 R R2 The first three sttements of the proposition imply tht isimilrity is n equivlene reltion, i.e., it is reflexive, symmetri nd trnsitive. Similr to the minimition proedure for (deterministi) finite utomt, there exists method for determining isimilr pirs of sttes in trnsition system. Ide: Strt with very orse reltion 0 tht reltes ll possile sttes. Refine this reltion step y step nd onstrut reltions, 2,.... As soon s two susequent reltions ide ( n = n+) we hve found the isimilrity (t lest for finite trnsition systems). Tht is, we hve = n. Brr König Course Modelling of Conurrent Systems 55 Brr König Course Modelling of Conurrent Systems 56
Behviourl equivlenes (isimilrity) Method for determining isimilr pirs of sttes Input: A trnsition system T = (Z, ) Define 0 = Z Z. n+ Z Z, where n+ if nd only if for ll At: For every with there exists 2 suh tht 2 nd n 2. 2 For every 2 with 2 there exists suh tht und n 2. The method termintes s soon s n= n+. Output: n Behviourl equivlenes (isimilrity) Exmple: determine the isimilr pirs of sttes of the following trnsition system 2 3 4 6 7 5 Brr König Course Modelling of Conurrent Systems 57 Behviourl equivlenes (isimilrity) Brr König Course Modelling of Conurrent Systems 58 Behviourl equivlenes (isimilrity) If we represent the equivlene reltions i vi equivlene lsses, then we otin the following sequene 0,, 2= 3. 5 7 5 0 3 2 6 3 2 6 7 4 2 = 3 3 2 5 6 3 2 6 4 5 7 7 4 4 Lemm It holds tht: n is n equivlene reltion for ll n N0. 2 n implies m for ll m n. 3 implies n for ll n N0. 4 n= n+ implies n= m for ll m n. Brr König Course Modelling of Conurrent Systems 59 Brr König Course Modelling of Conurrent Systems 60
Behviourl equivlenes (isimilrity) Behviourl equivlenes (isimilrity) Proposition Let T = (Z, ) e trnsition system whih is finitely rnhing, i.e., for every stte the set is finite. { At: } Then we hve if nd only if n for ll n N0. In other words: = n N0 n. Corollry For every finite trnsition system we hve: The method for determining isimilr pirs of sttes lwys termintes nd 2 returns the orret isimilrity. This proposition does not hold for trnsition systems whih re not finitely rnhing. Brr König Course Modelling of Conurrent Systems 6 Behviourl equivlenes (isimilrity) Brr König Course Modelling of Conurrent Systems 62 Internl tions Remrks onerning the method for determining isimilr pirs of sttes: For n effiient implementtion the reltions 0,, 2,..., should not e stored expliitly. Espeilly this holds for the reltion 0 whih ontins ll pirs of sttes nd hs hene sie Z 2. An effiient implementtion represents the equivlene reltion i vi its equivlene lsses. At the eginning there is only one equivlene lss whih is then refined ordingly (see lso the previous exmple). In the following we regrd τ-tions s speil internl tions whih re invisile from the outside. This mens speifilly: An tion sequene τ τ... 2 looks from outside s if no tion hs een performed. In n tion sequene τ τ...... 2 n externl oserver only sees tion. Brr König Course Modelling of Conurrent Systems 63 Brr König Course Modelling of Conurrent Systems 64
Internl tions Internl tions Hene we define new wek trnsition reltion: Definition For set At of tions we define the following reltions: ε 2 if nd only if ( ) τ 2, i.e., τ τ... 2. For n At we hve 2 if nd only if ε ε 2. Stndrd trnsitions will in the following lso e lled strong trnsitions in order to distinguish them from wek trnsitions. Divergene A trnsition system T = (Z, ) with τ-trnsitions is lled divergent if there exists n infinite run τ τ 2 τ.... It is lled onvergent if there is no suh run. Brr König Course Modelling of Conurrent Systems 65 Internl tions Brr König Course Modelling of Conurrent Systems 66 Internl tions Behviourl equivlenes nd τ-trnsitions Wek tre equivlene, wek filures equivlene nd wek isimilrity re defined on wek trnsitions of the form. In this wy we otin orser ehviourl equivlenes, i.e., more sttes re relted to eh other. Exmple : 3 τ τ τ τ 2 4 5 7 6 8 The two sttes, re wekly tre equivlent. Brr König Course Modelling of Conurrent Systems 67 Brr König Course Modelling of Conurrent Systems 68
Internl tions Internl tions Exmple 2: 3 τ τ 2 4 5 7 τ 6 τ 8 The two sttes, re wekly tre equivlent s well. We will now onsider wek isimilrity in more detil: Wek isimultion (definition) Let T = (Z, ) e trnsition system. A reltion R Z Z on sttes is lled wek isimultion if for every pir (, 2) R nd for every At\{τ} {ε} we hve: For every with there exists 2 with 2 2 nd (, 2 ) R. For every 2 with 2 2 there exists with nd (, 2 ) R. Brr König Course Modelling of Conurrent Systems 69 Internl tions Brr König Course Modelling of Conurrent Systems 70 Internl tions Wek isimilrity (definition) Two sttes, 2 re lled wekly isimilr if there exists wek isimultion R with (, 2) R. In this se we write 2. The reltion is known s wek isimilrity or oservtionl equivlene. Remrk: the definition of isimilrity given ove is not very stisftory for prtil purposes, sine there re usully mny more strong thn wek trnsitions. Hene the following lterntive hrterition is preferle: Alterntive hrkterition of wek isimultion (proposition) A reltion R Z Z is wek isimultion if nd only if for every pir (, 2) R nd for every At: For eh with there exists 2 with 2 â 2 nd (, 2 ) R. For eh 2 with 2 2 there exists with â nd (, 2 ) R. We define â =, whenever At\{τ}, nd ˆτ = ε. Brr König Course Modelling of Conurrent Systems 7 Brr König Course Modelling of Conurrent Systems 72
Internl tions Exmple : τ τ τ 2 3 4 5 7 6 8 The two sttes, re wekly isimilr with isimultion R = {(, 3), (, 4), (, 5), (2, 6), (, 7), (, 8)}. Brr König Course Modelling of Conurrent Systems 73 Internl tions τ Internl tions Exmple 2: τ τ 2 3 4 5 7 6 8 τ τ The two sttes, re not wekly isimilr. Plyer I mkes move = 3 7 tht n e nswered y Plyer II only with = 2. Then Plyer I mkes move 2 nd stte 7 does not llow ny further (wek) -move. Brr König Course Modelling of Conurrent Systems 74 Behviourl equivlenes (omprison) Remrks: The lterntive hrterition of wek isimilrity is more onvenient for showing tht two sttes re wekly isimilr. (Sine Plyer I hs fewer possiilities.) Insted the usul hrterition is more onvenient for showing tht two sttes re not wekly isimilr. (Sine Plyer I hs more possiilities.) The other, less onvenient, hrterition works s well. However it might e neessry to invest more effort. We now show how the vrious ehviourl equivlenes re relted. We first restrit ourselves to strong trnsitions. However, the results hold nlogously lso for the wek equivlenes. Filures equivlene implies S-tre equivlene (proposition) Let, e two filure equivlent sttes, i.e., F() = F( ). Then, re lso S-tre equivlent, i.e., S() = S( ). Remrk: two filure equivlent sttes re not neessrily S ω -tre equivlent. Brr König Course Modelling of Conurrent Systems 75 Brr König Course Modelling of Conurrent Systems 76
Behviourl equivlenes (omprison) Behviourl equivlenes (omprison) Bisimilrity implies filures equivlene nd S ω -tre equivlene (proposition) Let, e two isimilr sttes, i.e., we hve. Then nd re lso filure equivlent nd S ω -tre equivlent. Finlly we ompre strong nd wek isimilrity: Strong isimilrity implies wek isimilrity Let, two (strongly) isimilr sttes, i.e., we hve. Then, re lso wekly isimilr, i.e.,. Reson: Every strong isimultion ist lso wek isimultion (where eh strong trnsition is simulted gin y strong trnsition). Brr König Course Modelling of Conurrent Systems 77 Behviourl equivlenes (omprison) Brr König Course Modelling of Conurrent Systems 78 Preorders Hene we hve the following sitution (eh rrow stnds for n implition): S-equ. S ω -equ. S -equ. fil. equ. isimilr wekly S-equ. wekly S ω -equ. wekly S -equ. wekly fil. equ. wekly isimilr Aprt from the ehviourl equivlenes onsidered so fr there re lso preorders, whih order sttes, depending on whether they exhiit more or less ehviour. The reltions tht we onsider here re not rel orders, ut only preorders (lso lled qusi-orders) sine they re reflexive nd trnsitive, ut usully not nti-symmetri. Brr König Course Modelling of Conurrent Systems 79 Brr König Course Modelling of Conurrent Systems 80
Preorders Preorders Preorders (definition) Let T = (Z, ) e trnsition system nd let, Z e two sttes. The sttes, re ordered with respet to (S-)lnguge inlusion, whenever S() S( ). (Anlogously: S ω -/S -lnguge inlusion) The orresponding preorder is lled tre preorder. The sttes, re ordered with respet to inlusion of the sets of filure pirs, whenever F() F( ). The orresponding preorder is lled filures preorder. The notion of simultion is otined from the definition of isimultion y using only the first of the two onditions. Simultion (definition) Let T = (Z, ) e trnsition system. A reltion R Z Z on sttes is lled simultion whenever for every pir (, 2) R nd for every tion At: For every with there exists 2 with 2 2 nd (, 2 ) R. We sy tht stte 2 simultes stte (in symols: 2) whenever there exists simultion R with (, 2) R. Remrk: 2 implies S() S(2) nd S ω () S ω (2) Brr König Course Modelling of Conurrent Systems 8 Preorders Remrks onerning simultion: Let 2, i.e., 2 simultes. Then it does not neessrily hold tht either F() F(2) or F() F(2). 2 Sine R = {(, 3), (2, 4)} is simultion, we hve 3. However F() F(3) does not hold (sine (, {}) F(3)), nd neither does F(3) F() (sine (, ) F()). 3 4 5 Brr König Course Modelling of Conurrent Systems 83 Brr König Course Modelling of Conurrent Systems 82 Preorders 2 nd 2 do not neessrily imply 2. Tht is, mutul similrity does not imply isimilrity. 2 We hve 3 sine {(, 3), (2, 3)} is simultion, nd 3 sine {(3, )} is simultion ist. However: 3. Intuitive explntion: the two sttes n only e reognied s isimilr if Plyer I swithes to the other token. However, this is not possile for mutul simultion. 3 Brr König Course Modelling of Conurrent Systems 84
Preorders 2 lwys implies 2 nd 2. This is used y the ft tht every isimultion is lso simultion. Brr König Course Modelling of Conurrent Systems 85