Game Characterizations of Process Equivalences

Transcription

1 Gme Chrcteriztions of Process Equivences Xin Chen nd Yuxin Deng Deprtment of Computer Science nd Engineering, Shnghi Jio Tong University, Chin Abstrct In this pper we propose hierrchy of gmes tht ows us to me systemtic comprison of process equivences by chrcterizing process equivences s gmes The we-nown iner/brnching time hierrchy of process equivences cn be embedded into the gme hierrchy, which not ony provides us with refined nysis of process equivences, but so offers guidnce to ining interesting new process equivences Introduction A gret mount of wor in process gebr hs centered round process equivences s bsis for estbishing system correctness Usuy both specifictions nd impementtions re written s process terms in the sme gebr, where specifiction describes the expected high-eve behviour of the system under considertion nd n impementtion gives the detied procedure of chieving the behviour An pproprite equivence is then chosen to verify tht the impementtion conforms to the specifiction In the st three decdes, ot of process equivences hve been deveoped to cpture vrious spects of system behviour They usuy fit in the iner/brnching time hierrchy [0]; see Figure for some typic process equivences Process equivences cn often be understood from different perspectives such s ogics nd gmes For exmpe, bisimution equivence cn be chrcterized by Hennessy-Miner ogic [] nd the mod mu-ccuus [] Equivences which re weer thn bisimution equivence in the iner/brnching time hierrchy cn be chrcterized by some sub-ogics of Hennessy-Miner ogic [3] It is so we-nown tht bisimution equivence cn be chrcterized by bisimution gmes [6] between n ttcer nd ender in n eegnt wy; two processes re bisimir if nd ony if the ender of bisimution gme pyed on the processes hs history free winning strtegy Bisimution gmes cme from Ehrenfeucht-Frïssé gmes tht were originy introduced to determine expressive power of ogics [9] To some extent gmes cn be considered s descriptive nguges ie ogics In mny cses we cn design gme directy from the semntics of prticur ogic such tht the gme cptures the ogic For exmpe, the bisimution gme with infinite durtion is n Ehrenfeucht-Frïssé gme Supported by the Ntion 973 Project (003CB37005) nd the Ntion Ntur Science Foundtion of Chin ( ) Supported by the Ntion Ntur Science Foundtion of Chin ( )

2 bisimution equivence -nested simution equivence redy simution equivence possibe-futures equivence redy trce equivence fiure trce equivence rediness equivence simution equivence fiure equivence competed trce equivence trce equivence Fig The iner/brnching time hierrchy [0]

3 tht cptures Hennessy-Miner ogic [6], nd the fixed point gme tht ows infinite fixed point nd mod moves cptures the mod mu-ccuus [8] Gmes indeed offer new sights into od probems, nd sometimes et us understnd these probems esier thn before In this pper we provide systemtic comprison of different process equivences from gme-theoretic point of view More precisey, we present gme hierrchy (cf Figure 4) which hs more refined structure thn the process equivence hierrchy in Figure Viewing the hierrchies s prti orders, we cn embed the process equivence hierrchy into the gme hierrchy becuse ech process equivence cn be chrcterized by corresponding css of gmes Moreover, there re gmes tht do not correspond to ny existing process equivences This ind of gmes woud be usefu for guiding us to ine interesting new process equivences To ine gmes, we me use of gme tempte tht is bsicy n bstrct two-pyer gme eving concrete moves unspecified Then we ine few types of moves Instntiting the gme tempte by different combintions of moves genertes different gmes We compre the gmes using preorder which sys tht G G if pyer II hs winning strtegy in G impies she hs winning strtegy in G The preorder provides us with net mens to compre process equivences Suppose G nd G chrcterize process equivences nd, respectivey Then we hve tht G G if nd ony if, ie is corser retion thn The rest of the pper is orgnized s foows Section briefy recs the initions of beed trnsition systems nd bisimutions In Section 3, we design sever inds of moves nd gme tempte in order to ine gmes In Section 4, we present two gme hierrchies, with or without considering terntions of moves, nd we combine them into fin hierrchy In Section 5, we show tht the iner/brnching time hierrchy cn be embedded into our gme hierrchy Section 6 concudes nd discusses some future wor Preiminries We presuppose countbe set of ctions Act = {, b, } Definition A beed trnsition systems (LTS) is tripe (P, A, ), where P is set of sttes, A Act is set of ctions, P A P is trnsition retion As usu, we write P Q for (P,, Q) nd we extend the trnsition retion to trces in the stndrd wy, eg P 0 Pn if P t 0 P P P n n P n, where t = n An LTS (P, A, ) is imge-finite if for P P the set {P P P, for some A} is finite In this pper we ony consider imge-finite LTSs Insted of drwing LTSs s grphs, we use CCS processes to represent the LTSs generted by their opertion semntics [4] We sy two processes re isomorphic if their LTSs re isomorphic

4 Definition A binry retion R is bisimution if for (P, Q) R nd Act, () whenever P P, there exists Q Q such tht (P, Q ) R, nd () whenever Q Q, there exists P P such tht (P, Q ) R We ine the union of bisimutions s bisimirity, written Bisimirity cn be pproximted by sequence of inductivey ined retions The foowing inition is ten from [4], except tht is repced by r The mening of the superscript r wi be cer in Section 5 Definition 3 Let P be the set of processes, we ine r 0 = P P, P r n+ Q, for n 0, if for t Act, () whenever P t P, there exists Q t Q such tht P r n Q, () whenever Q t Q, there exists P t P such tht P r n Q The inition of for 0 is simir to the previous one, except tht we repce t with where Act For imge-finite LTSs, it hods tht = n 0 r n = n 0 n 3 Gme Tempte We briefy review the bisimution gmes [8] A bisimution gme G (P, Q) strting from the pir of processes (P, Q) is round-bsed gme with two pyers Pyer I, viewed s n ttcer, ttempts to show tht the initi sttes re different wheres pyer II, viewed s ender, wishes to estbish tht they re equivent A configurtion is pir of processes of the form (P i, Q i ) exmined in the i-th round, nd (P, Q) is the configurtion for the first round Suppose we re in the i-th round The next configurtion (P i+, Q i+ ) is determined by one of the foowing two moves: : Pyer I chooses trnsition P i P i+ nd then pyer II chooses trnsition with the sme be Q i Q i+ []: Pyer I chooses trnsition Q i Q i+ nd then pyer II chooses trnsition with the sme be P i P i+ Pyer I wins if she cn choose trnsition nd pyer II is unbe to mtch it within rounds Otherwise, Pyer II wins If = then there is no imittion on the number of rounds Beow we ine four other moves tht wi give rise to vrious gmes ter on Definition 4 (Moves) Suppose the current configurtion is (P, Q), we ine the foowing inds (or sets, more precisey) of moves

5 t : Pyer I performs nonempty ction sequence t = Act from P, P P P nd then pyer II performs the sme ction sequence from Q, Q Q Q Pyer I seects some j nd sets the configurtion for the next round to be (P j, Q j ) [t]: Pyer I performs nonempty ction sequence t = Act from Q, Q Q Q nd then pyer II performs the sme ction sequence from P, P P P Pyer I seects some j nd sets the configurtion for the next round to be (P j, Q j ) r- t : Pyer I performs nonempty ction sequence t = Act from P, P P P nd then pyer II performs the sme ction sequence from Q, Q Q Q The configurtion for the next round is (P, Q ) r-[t]: Pyer I performs nonempty ction sequence t = Act from Q, Q Q Q nd then pyer II performs the sme ction sequence from P, P P P The configurtion for the next round is (P, Q ) For the se of convenience, we ine some unions of the moves bove: t := t [t] r := r- t r-[t] := [] M is the set of moves Cery, moves re speci r- t moves nd r- t moves re speci t moves We hve simir observtion for box modities r- t t [] r-[t] [t] We now introduce the concept of terntion for gmes; it hs n intimte retion with quntifier terntion in ogics Definition 5 (Aterntion) An terntion consists of two successive moves such tht one of them is in t nd the other is in [t] The number of terntions in gme is the number of occurrences of such successive moves in the gme Note tht bisimution gmes hve no restriction on their terntion numbers Definition 6 (Extr conditions) Given round-bsed gme nd set α which is the set of moves pyer I cn me in the gme, n extr condition cn be one of the foowing, for some m M, m: The gme is extended with one more round, where pyer I cn ony me move in m Moreover, pyer I cn me move in m α in ech round, but the gme hs to be finished regrdess of the remining rounds, which impies tht if pyer I fis to me pyer II stuc by this move, she oses

6 m: Simir to the cse for m, except tht if pyer I mes move in m α to end the gme, the st two moves must be n terntion Therefore, this condition coud not be ppied to 0-round gme c 0 : In the beginning of the gme, dedoc processes rechbe from P 0 nd Q 0 re coored C 0 In ech round, the two processes in the reted configurtion shoud be in the sme coor (or neither of them is coored), otherwise pyer II oses We now ine gme tempte which is intuitivey n bstrct gme in the sense tht concrete gmes cn be obtined from it by instntiting its moves Definition 7 (Gme tempte) The gme tempte n-γ α,β (P, Q) with n 0 denotes -round gme between pyer I nd pyer II with the strting configurtion (P, Q) such tht the foowing conditions re stisfied The number of terntions in the gme is t most n; it is omitted when there is no restriction on the number of terntions β is n extr condition; it is omitted when there is no extr condition 3 Pyer I cn ony me move in α M in ech round if β is neither m nor m Otherwise, pyer I cn so me move in m α in ech round, but if she cnnot me pyer II stuc by this move, she oses 4 The pyers winning conditions re simir to those in bisimution gmes Notice tht -round bisimution gmes cn be ined by Γ Athough ot of gmes cn be ined by vrious combintions of n, α nd β; this pper miny focuses on some typic ones Given gme Γ α,β (P, Q), we sy pyer I (resp pyer II) wins Γ α,β (P, Q) if pyer I (resp pyer II) hs winning strtegy in it, nd we bbrevite the gme to Γ α,β if the strting configurtion is insignificnt 4 Gme Hierrchy To fciitte the presenttion, we cssify our gmes into two hierrchies with respect to preorder retion between gmes; one hierrchy counts terntions of moves nd the other does not count We show tht the retions in the hierrchies re correct Then we combine the two hierrchies into one, by introducing some new retions At st, we prove tht no more non-trivi retions cn be dded into the fin hierrchy We sh see in Section 5 tht the hierrchy of process equivences in Figure cn be embedded into this hierrchy of gmes The preorder retion between gmes is ined s foows Definition 8 Given two gmes G nd G, we write G G, if for ny processes P nd Q, pyer II wins G (P, Q) = pyer II wins G (P, Q) Here is indeed preorder s this is inherited from ogic impiction We write G G if G G nd G G

7 4 Gme Hierrchy I We propose the gme hierrchy I in Figure Its correctness is stted by the next theorem Theorem In Figure, if G G then G G The rest of this subsection is devoted to proving Theorem Let α, α M nd β be n extr condition The foowing sttements cn be derived from Definition 7 immeditey: () Γ0 α = Γ0 α () Γ α,β 0 = Γ α,β 0 (3) For 0, Γ α,α = Γ+ α (4) For 0, Γ α,t = Γ α,r Since t contins r, r contins, nd if two processes P, Q do not hve the sme coor, they cn be distinguished in round by move in, we get the foowing sttement: Γ α = Γ α,c0 = Γ α, = Γ α,, for α {, r, t} Lemm For ny processes P nd Q, the foowing sttements re equivent: () P Q () pyer II wins Γ (P, Q) (3) pyer II wins r (P, Q) (4) pyer II wins Γ (P, t Q) Proof It is trivi tht Γ Γ r Γ, t so we hve (4) (3) () Observe tht Γ is excty the -round bisimution gme, which mens () () (cf [6]) We now show () (4) Assume P Q, we construct winning strtegy for pyer II for the gme Γ t (P, Q): in ny round, suppose the configurtion is (P i, Q i ) If pyer I performs P i Pi Pi, then pyer II cn respond with Q i Qi Qi, such tht P ij Q ij for j Cery, whtever configurtion for the next round pyer I seects, she cnnot win the gme Lemm yieds the immedite corory tht Γ = Γ r = Γ t Lemm () Γ+ Γ r, () Γ r Γ t for Γ t, for Proof Since r t, both () nd () cn be esiy derived Lemm 3 Γ r Γ r,c0 Γ r, Γ r, Γ+ r for Proof It is esy to see tht Γ r Γ r,c0 Γ r, Γ+ r nd Γ r Γ r, Γ r, Γ+ r r,c0 We now prove Γ Γ r, by induction on Given two processes P, Q, suppose pyer II wins Γ r, (P, Q) We show tht pyer II wins Γ r,c0 (P, Q) s we

8 Γ Γ r Γ t Γ Γ r, Γ t, Γ, Γ r, Γ t, Γ,c 0 Γ r,c 0 Γ t,c 0 Γ r Γ t Γ Γ r, Γ t, Γ r 3 Γ t 3 Γ 3 Γ r, Γ t, Γ, Γ r, Γ t, Γ,c 0 Γ r,c 0 Γ t,c 0 Γ r Γ t Γ Γ r, Γ t, Γ, Γ r, Γ t, Γ,c 0 Γ r,c 0 Γ Γ r Γ,c 0 0 Γ 0 Fig Gme hierrchy I

9 = From the ssumption, pyer II wins Γ r, (P, Q) The gme Γ r,c0 (P, Q) hs just one round nd dedoc processes rechbe from P nd Q re coored C 0, nd the other processes re uncoored (Cery both P nd Q re coored C 0 or neither of them is coored) We distinguish four cses Cse : Pyer I performs P t P, where t Act is nonempty ction sequence nd P is coored Pyer II cn perform Q t Q such tht Q is coored Otherwise there is some Act nd pyer I cn me pyer II stuc by performing P t P for some P in the first round of Γ r, (P, Q), which contrdicts the ssumption Cse : Pyer I performs P t P, where t Act is nonempty ction sequence nd P is uncoored C 0 Pyer II cn perform Q t Q such tht Q is uncoored C 0 Otherwise, in Γ r, (P, Q), pyer I cn me pyer II stuc by ming move in [] in the second round, contrdicting the ssumption Cse 3: Pyer I performs Q t Q, where t Act is nonempty ction sequence nd Q is coored This cse is simir to Cse Cse 4: pyer I performs Q t Q, where t Act is nonempty ction sequence nd Q is uncoored C 0 This cse is simir to Cse > We now pyer II wins Γ r, (P, Q) In the first round of Γ r,c0 (P, Q), whenever pyer I performs some ction sequence from P (resp Q) to P (resp Q ), pyer II cn wys perform the sme ction sequence from Q (resp P) to Q (resp P ) such tht both P nd Q re coored C 0, or neither of them is coored Otherwise, in Γ r, (P, Q), pyer I cn me pyer II stuc in the second round In the second round of Γ r,c0 (P, Q), the gme becomes Γ r,c0 (P, Q ) nd by induction pyer II wins the Γ r,c0 (P, Q ) Simir to Lemms nd 3, the other retions iustrted in Figure cn be proven, thus Theorem is estbished 4 Gme Hierrchy II The gmes in Section 4 do not count terntions of moves, which re ten into ccount in this section For simpicity, we re not going to discuss the gmes ined from those in Figure by restricting the number of terntions Insted, we focus on the gmes in which the pyers cn ony me moves in To further simpify the exposition, Figure 3 ony iustrtes gme hierrchy where the number of terntions n is restricted to 0 nd However, in the rest of the pper the emms cover n 0 From Definitions 6 nd 7, the retions iustrted in Figure 3 re pprent, so we omit the proof of the theorem beow Theorem In Figure 3, if G G then G G

10 0-Γ 0-Γ,c 0 0-Γ, -Γ Γ 0-Γ 0-Γ, -Γ Γ 0-Γ, Γ, 0-Γ,c 0 Γ,c 0 0-Γ 0-Γ, -Γ Γ 0-Γ 3 0-Γ, -Γ 3 Γ 3 0-Γ, Γ, 0-Γ,c 0 Γ,c 0 0-Γ Γ Γ, Γ,c 0 Γ Γ,c 0 0 Γ 0 Fig 3 Gme hierrchy II

11 0-Γ 0-Γ,c 0 0-Γ, -Γ Γ Γ r Γ t 0-Γ 0-Γ, -Γ Γ Γ r, Γ t, 0-Γ, Γ, Γ r, Γ t, 0-Γ,c 0 Γ,c 0 Γ r,c 0 Γ t,c 0 Γ r Γ t 0-Γ 0-Γ, -Γ Γ Γ r, Γ t, Γ r 3 Γ t 3 0-Γ 3 0-Γ, -Γ 3 Γ 3 Γ r, Γ t, 0-Γ, Γ, Γ r, Γ t, 0-Γ,c 0 Γ,c 0 Γ r,c 0 Γ t,c 0 Γ r Γ t 0-Γ Γ Γ r, Γ t, Γ, Γ r, Γ t, Γ,c 0 Γ r,c 0 Γ Γ r Γ,c 0 0 Γ 0 Fig 4 The whoe gme hierrchy

12 43 The Whoe Gme Hierrchy We now combine gme hierrchies I nd II into singe hierrchy, s described in Figure 4 Simir to Figure 3, we hve not drwn the gmes with terntions exceeding, but our emms beow cover them In the combined gme hierrchy, we hve the new retions, Γn+ t n-γ, Γ t,c0 n+,c0, n-γ n-γ the emm beow; the others cn be proven nogousy, Γ t, n+ for n 0 We give proof of Γ t n+ n-γ in Lemm 4 Γ t n+ n-γ for n 0 Proof We prove the sttement by induction on n Assume pyer II wins n- Γ (P 0, Q 0 ) for some processes P 0 nd Q 0 n = 0 Suppose pyer I performs P 0 P P (resp Q 0 Q Q ) Since pyer II wins 0-Γ (P 0, Q 0 ), she cn respond with Q 0 Q Q (resp P 0 P P ) Hence, pyer II wins Γ t(p 0, Q 0 ) n > 0 From the ssumption, pyer II wins n-γ (P 0, Q 0 ) In the first round of Γn+(P t 0, Q 0 ), if pyer I performs P 0 P P (resp Q 0 Q Q ), pyer II cn respond with Q 0 Q Q (resp P 0 P P ), such tht for ech P i nd Q i, where i, pyer II wins (n )-Γ (P i, Q i ) By induction, Γn t (n )-Γ, pyer II wins Γn t(p i, Q i ) for ny i Hence, pyer II wins Γn+ t (P 0, Q 0 ) We re in position to stte the min resut of the pper Theorem 3 () In Figure 4, if G G then G G () No more retions cn be dded to the gme hierrchy in Figure 4, except for those derived from the trnsitivity of The first sttement foows from Theorems, nd Lemm 4 provided we coud show tht ( ) In Figure 4, if G G then G G The rest of this section is devoted to proving (*) nd the second sttement of Theorem 3 by providing counterexmpes to prove the invidities of some retions For tht purpose, it suffices to estbish Lemms 5 to 7 beow Lemm 5 For, () Γ r Γ () Γ t, Γ r Proof () We ine the processes beow: Exmpe A = A nd A i = { 0 if i = 0 A i if i > 0

13 Consider Γ (A, A ), in ech round pyer I cn ony perform ction from one process, nd pyer II cn wys respond propery, since both A nd A cn perform ction for times Then pyer II wins Γ (A, A ) But pyer I wins Γ r (A, A ), she performs n ction sequence t = + from A in the first round, pyer II fis to respond to such sequence from A, since the process cn ony perform ction for times () Consider the foowing processes Exmpe P 0 = b0, Q 0 = c0, P i+ = (P i + d0) + (Q i + e0), Q i+ = (P i + e0) + (Q i + d0) It is not difficut to prove tht pyer II wins Γ r(p +, Q + ) by induction on = This cse is esy > We distinguish five sub-cses Cse : Pyer I performs P + (P +d0) Then pyer II cn perform Q + (Q + d0) By induction pyer II wins Γ r (P, Q ) nd thus she so wins Γ r (P + d0, Q + d0) Cse : Pyer I performs P + (Q +e0) Then pyer II cn perform Q + (P + e0) Simir to the previous cse, pyer II wins Γ r (Q + e0, P + e0) Cse 3: Pyer I performs Q + (Q + d0) Then pyer II cn perform P + (P + d0) The rest is simir to Cse Cse 4: Pyer I performs Q + (P +e0) Then pyer II cn perform P + (Q + e0) The rest is simir to Cse t Cse 5: If pyer I performs P + P t (resp Q + Q ) for some t Act t nd t >, pyer II cn wys respond with Q + Q t (resp P + P ) such tht P nd Q re isomorphic On the other hnd, pyer I wins Γ t, (P +, Q + ) A winning strtegy is to perform P + (P + d0) (P + d0) (b0 + d0), where ech process pssed in the sequence cn perform ction d nd the st process cn perform ction b But pyer II fis to perform such n ction sequence from Q + nd wi become stuc in the second round Simir to Lemm 5, the next two emms cn be proven by providing pproprite counterexmpes See Appendix A for their detied proofs Lemm 6 For, () 0-Γ,c0 Γ t () 0-Γ, Γ t,c0 (3) 0-Γ+ t, Γ

14 (4) 0-Γ,c0 Γ, Lemm 7 For n 0, () Γ,c0 n+ n-γ () Γ,,c0 n+ n-γ (3) (n + )-Γn+3, n-γ 5 Chrcterizing Process Equivences In this section we revisit some importnt process equivences in the iner/brnching time hierrchy showed in Figure Definition 9 Given gme G nd process equivence, we sy is chrcterized by G if for ny processes P, Q, it hods tht P Q iff pyer II wins G(P, Q) Theorem 4 () Trce equivence is chrcterized by Γ r () Competed trce equivence is chrcterized by Γ r,c0 (3) Fiures equivence is chrcterized by Γ r, (4) Fiure trce equivence is chrcterized by Γ t, (5) Redy trce equivence is chrcterized by Γ t, (6) Rediness equivence is chrcterized by Γ r, (7) Possibe-futures equivence is chrcterized by Γ r Proof We ony prove (5) nd the others cn be proven nogousy Suppose P nd Q re redy trce equivent, written P RT Q, we prove tht pyer II wins Γ t, (P, Q) In the first round, if pyer I performs some trce t from P or Q, then pyer II considers t s redy trce, since she hs fu nowedge of pyer I s move Cery, in the second round pyer I cnnot me pyer II stuc Conversey, suppose pyer II wins Γ t, (P, Q) It is pprent tht P, Q hve the sme redy trces, nd then P RT Q Simir to the pproximtion of bisimirity (cf Definition 3), we cn ine simirity S, competed simirity CS, redy simirity RS, -nested simirity S, nd their pproximnts We write, where 0, for the pproximnts of Lemm 8 For 0, () Γ chrcterizes () Γ r chrcterizes r (3) 0-Γ chrcterizes S (4) 0-Γ,c0 chrcterizes CS (5) 0-Γ, chrcterizes RS Due to c of spce we do not ist the initions of those process equivences; they cn be found in [0]

15 (6) -Γ chrcterizes S Proof A the sttements cn be esiy proven by induction on, so we omit them Since we re deing with imge-finite LTSs, the next theorem foows from Lemm 8 Theorem 5 () Simution equivence is chrcterized by 0-Γ () Competed simution equivence is chrcterized by 0-Γ,c0 (3) Redy simution equivence is chrcterized by 0-Γ, (4) -nested simution equivence is chrcterized by -Γ Furthermore, new equivences cn be ined using the gmes in Figure 4 For exmpe, we cn ine new equivence using gme Γ t which is stronger thn possibe-futures equivence nd redy trce equivence, but weer thn -nested simution equivence In ddition, from the gme hierrchy, we ern the retionship between the pproximnts of bisimirity, simirity, competed simirity etc For exmpe, possibe-futures equivence is stronger thn S, but is incomprbe with S 3 Hence, the gme hierrchy is interesting in tht it offers n intuitive wy of compring vrious process equivences 6 Concuding Remrs We hve presented hierrchy of gmes tht ows us to compre process equivences systemticy in gme-theoretic wy by chrcterizing process equivences s gmes The hierrchy not ony provides us with refined nysis of process equivences, but so offers guidnce to ining interesting new process equivences The wor cosey reted to ours is [5] which provides Stiring css of gmes to chrcterize vrious process equivences The methodoogy dopted in the current wor is different becuse we exmine in systemtic wy the theory of gmes tht coud chrcterize typic equivences in the process equivence hierrchy Pying our ttention to the nysis of process equivences is for the purpose of studying the compexity of equivence checing We now tht mode checing cn be considered in gme-theoretic wy [7], but the compexity depends on prticur modes Simir phenomen exist for equivence checing However, equivence checing is much hrder thn mode checing, nd sometimes it cnnot be done in simir wys Further investigtion in this respect woud be interesting Acnowedgement We thn Ensho Shen, Yunfeng To nd Chodong He for interesting discussions We so thn the nonymous referees for their constructive comments

16 References [] M Hennessy nd R Miner Agebric ws for nondeterminism nd concurrency Journ of the ACM, 3():37 6, 985 [] D Kozen Resuts on the proposition mu-ccuus Theoretic Computer Science, 7: , 983 [3] A Kucer nd J Esprz A ogic viewpoint on process-gebric quotients In Proceedings of the 8th Annu Conference of the Europen Assocition for Computer Science Logic, voume 683 of Lecture Notes in Computer Science, pges Springer, 999 [4] R Miner Communiction nd concurrency Prentice-H, Inc, 989 [5] S K Shu, H B H III, nd D J Rosenrntz HORNSAT, mode checing, verifiction nd gmes (extended bstrct) In Proceedings of the 8th Interntion Conference on Computer Aided Verifiction, voume 0 of Lecture Notes in Computer Science, pges 99 0 Springer, 996 [6] C Stiring Mod nd tempor ogics for processes Notes for Summer Schoo in Logic Methods in Concurrency, 993 [7] C Stiring Loc mode checing gmes In Proceedings of the 6th Interntion Conference on Concurrency Theory, voume 96 of Lecture Notes in Computer Science, pges Springer, 995 [8] C Stiring Gmes nd mod mu-ccuus In Proceedings of the nd Interntion Worshop on Toos nd Agorithms for Construction nd Anysis of Systems, voume 055 of Lecture Notes in Computer Science, pges 98 3 Springer, 996 [9] W Thoms On the Ehrenfeucht-Frïssé gme in theoretic computer science In Proceedings of the 3rd Interntion Joint Conference CAAP/FASE on Theory nd Prctice of Softwre Deveopment, voume 668 of Lecture Notes in Computer Science, pges Springer, 993 [0] R J vn Gbbee The iner time-brnching time spectrum (extended bstrct) In Proceedings of the st Interntion Conference on Concurrency Theory, voume 458 of Lecture Notes in Computer Science, pges Springer, 990 A Proofs from Section 43 Proof of Lemm 6 Proof () We ine the foowing processes: Exmpe 3 P = b0 + 0, Q = b0, P i+ = (P i + r(q i ) + r(p i ) + Q i ), Q i+ = (P i + r(q i )) + (r(p i ) + Q i ) where r is n injective renming function tht mps ctions in P, Q to some fresh ctions We prove tht pyer II wins Γ t (P, Q ) = This cse is trivi

17 > In the first round of Γ t(p, Q ), we hve the foowing sub-cses: Cse : Pyer I performs P (P +r(q )+r(p )+Q ) By induction, whenever pyer II responds with Q (P +r(q )) or Q (r(p )+Q ), she cn wys continue the gme for t est ( ) rounds Cse : Pyer I performs Q (P +r(q )) nd pyer II performs P (P +r(q )+r(p )+Q ) The rest is simir to Cse Cse 3: Pyer I performs Q (r(p )+Q ) nd pyer II performs P (P +r(q )+r(p )+Q ) The rest is simir to Cse t Cse 4: Pyer I performs P P t (resp Q Q ), where t Act, t pyer II cn respond with Q Q t (resp P P ) such tht P nd Q re isomorphic By induction hypothesis, whenever pyer I chooses some configurtion for the next round, pyer II cn wys continue the gme for t est ( ) rounds The fct tht pyer I wins 0-Γ,c0 (P, Q ) cn be proven simiry = Pyer I performs P 0 nd 0 is coored C0, but pyer II fis to me proper response > Pyer I performs P (P + r(q ) + r(p ) + Q ) Then pyer II hs two wys to respond Cse : Pyer II responds with Q (P + r(q )) Then in the second round the configurtion is (P + r(q ) + r(p ) + Q, P +r(q )) If pyer I performs some ction from r(p ), pyer II cn respond with the sme ction from r(q ) By induction, pyer I wins 0-Γ,c0 (P, Q ) by performing some ction from P in the first round, nd obviousy it is so the cse for 0-Γ,c0 (r(p ), r(q )) Hence, pyer I cn me pyer II stuc in the -th round nd does not need ny terntion Cse : Pyer II responds with Q (r(p ) + Q ) The rest is simir to Cse () Define the foowing processes: Exmpe 4 P = b0 + c0, Q = (b0 + c0), P i+ = (P i + r(q i ) + r(p i ) + Q i ), Q i+ = (P i + r(q i )) + (r(p i ) + Q i ) Pyer II wins Γ t,c0 (P, Q ), but pyer I wins 0-Γ, (P, Q ) The proofs re simir to prt () (3) Define the foowing processes: Exmpe 5

18 P = (b0 + c0) + b0 + c0, Q = b0 + c0, P i+ = (P i + r(q i ) + r(p i ) + Q i ), Q i+ = (P i + r(q i )) + (r(p i ) + Q i ) Pyer II wins Γ t, (P, Q ), but pyer I wins 0-Γ+ (P, Q ) The proofs re simir to prt () (4) We give the foowing exmpe Exmpe 6 P 0 = b0, where A i+ is ined in Exmpe P i+ = P i + A i+, The sttement tht pyer II wins Γ, (P, A ) cn be proven by induction on, s we did in the proofs of previous emms We observe tht pyer I wins 0-Γ,c0 (P, A ), since she cn perform P P P P P0 in the first rounds, nd pyer II hs to respond with A A A A A0 Since P 0 is not dedoc process, it is not coored C 0, but A 0 is coored C 0 Proof of Lemm 7 Proof () Consider the foowing processes: Exmpe 7 P 0 = b0, Q 0 = 0, P i+ = (P i + Q i ), Q i+ = P i + Q i Pyer II wins n-γ (P n+, Q n+ ) for 0, since pyer I needs t est (n+) terntions to distinguish P n+ nd Q n+ without cooring It foows tht pyer I wins n-γ (P n+, Q n+ ) But pyer I wins Γ,c0 n+ (P n+, Q n+ ) becuse she cn me pyer II stuc in the st round () We ine the foowing processes: Exmpe 8 P = (b0 + c0) + c0, Q = (b0 + c0), P i+ = (P i + Q i ), P i+ = P i + Q i

19 Pyer II wins n-γ,c0(p n+, Q n+ ), since (n+) terntions re needed for pyer I in the first (n + ) rounds to distinguish the two processes Pyer I wins Γ, n+ (P n+, Q n+ ) becuse she cn me (n + ) terntions in the (n + ) rounds of the gme (3) Consider foowing processes: Exmpe 9 P 0 = (b0 + 0), Q 0 = (b0 + 0) + 0, P i+ = Q i, Q i+ = P i + Q i Pyer II wins n-γ, (P n, Q n ), since in the first (n+) rounds, pyer I need (n+) terntions in order to prevent pyer II from ming two processes in the configurtion for the next round isomorphic, nd she so needs one more round, but no more terntion, to me pyer II stuc We so showed winning strtegy for pyer I in (n + )-Γn+3 (P n, Q n )