A hierarchy of behavioral equivalences in the $pi$-calculus with noisy channels

A hierarc h y of b eha vioral equiv alences in the π -calculus with n o isy c hannels Y ongzhi Cao 1 , 2 , ∗ 1 Institute of Softwar e, Scho ol of Ele ctr onics Engine ering and Computer Scienc e Peking University, Beijing 100871, China 2 Key L ab or atory of High Conﬁdenc e Softwar e T e chnolo gies (Peking University) Ministry of Educ ation, China E-mail: caoyz@pku.edu.cn Abstract The π -ca lculu s is a pro cess algebra where agen ts interact by s e n ding comm unication links to eac h other via noiseless comm unication c h an n el s . T aking in to accoun t the realit y of n o isy c h a n nels, an extension of the π - calculus, called the π N -calculus, h as b een in tro duced recen tly . In this p a p er, we presen t an early trans itional semantic s of the π N - calculus, which is not a directly tr anslated v ersion of the late seman tics of π N , and then extend six kinds of b eha vioral equiv alences consisting of reduction bisimilarit y , barb ed bisimilarit y , barb ed equiv alence, barb ed co n gruence, bisimilarit y , and full bisimilarit y in to the π N -calculus. Suc h b eha vioral equ iv alences are cast in a hierarc h y , which is helpful to verify b eha vioral equiv alence of t wo agen ts. In p articular, w e sh ow that d ue to the noisy nature of c hann els, the coincidence of bisimilarit y and barb ed equiv alence, as well as the coincidence of full bisimilarit y and barb ed congruence, in the π -calculus do es not hold in π N . Keywor ds: π -calculus, π -calculus with noisy c hann els, b arb ed equ iv alence, barb ed con- gruence, bisimilarit y . 1 In tro d uction The n eed for formal metho ds in the sp eciﬁcation of concurrent systems has increas- ingly b ecome w ell accepted. P articular interest has b een dev oted to Petri n ets [36, 39], CSP [22, 23], A CP [10], CCS [28, 29], and the π -calculus [32]. The last o n e d u e to Milner et al. wa s develo p ed in the late 1980s with th e goal of analyzing the b eha vior of mobile systems, i.e., systems wh ose comm unication top ology can c h ange dynamically , and it turn s out to b e the un ique one among the aforemen tioned calculi that can ex- press mobilit y directly . T he π -calculus has its ro ots in CCS, namely CCS with mobilit y , in tro d uced by Engb erg and Nielsen [16], while the capacit y of dynamic r econﬁguration of logica l comm un ication structure giv es th e π -calculus a muc h greater expressiv en ess than CCS. In the π -calculus, all distinctions b et ween v ariables and constan ts are remo ve d , com- m un ication c h annels are id en tiﬁed by names, and compu tation is r ep resen ted purely as the comm unication of names across c hann els. The transfer o f a name betw een t w o agen ts (pro cesses) is ther efore the f undamental computational step in the π -calculus. ∗ Supp orted in part b y the National F ound ation of Natural Sciences of C h ina under Gran t s 6050 5011, 6049632 1, and 60736 011. 1 The basic (monadic) π -calculus allo ws only communicatio n o f c hann el names. There are t wo extensions of su c h a communication capabilit y: O ne is the p oly adic π -calculus [30] that supp orts comm un ication of tuples, needed to mo del passing of co mp lex m es- sages; the other is the higher-ord er π -calculus [40] that supp orts the communicat ion of pro cess abstractions, needed for mo deling soft ware comp osition within the calc u lus itself. Interesti n gly , b oth of th em can b e faithfully translated in to the basic π -calculus. As w e see, comm unication is a ke y in gredien t of the π -calculus. It is wo r th noting that all comm un ication c hann els in the π -calculus are implicitly pr esupp osed to b e noiseless. This means that in a comm unication along suc h a channel, the receiv er will alw a ys g et exactly wh at the sender deliv ers . Ho wev er, it is usually not the case in the real world, where comm un ication channels are often not completely reliable. Recen tly , Ying [54] to ok in to accoun t the noise of c hannels, an idea ad vocated in his earlier pap er [51], and prop osed a new v ersion of the π -calculus, called the π N -calculus. Su ch a calculus has the same s y ntax as that of the π -calculus. T he n ew feature of π N arises from a fundamental assumption: Comm u n ication c h annels in π N ma y b e noisy . This means that what is receiv ed at the receiving end of a c hannel ma y b e diﬀerent from what was emitted. According to a b asic idea of S hannon’s information theory [45] that noise can b e describ ed in a statistic wa y , the n oisy channels in π N w as formalized in [54] as follo ws : Firstly , lik e in π , all (noisy and noisel ess) comm un ication c hann els are iden tiﬁed b y names. Secondly , to describ e noise, ev ery pair of (c hannel) names x and y is asso ciated with a probability distribution p x ( ·| y ) o v er the output alphab et (here it is ju s t the set of names), where f or any name z , p x ( z | y ) ind icates th e p robabilit y that z is receiv ed from c hannel x when y is s ent along it. Finally , based on the probabilit y information arising from the noisy c h annels, a late probabilistic transitio n al seman tics of π N is p resen ted. The essen tial diﬀerence b et ween this semant ics and that of π is mainly caused by the actions p erformed by an outpu t agent xy .P . In π , this agen t has a single capabilit y of sending y via c h annel x , expressed as the transition xy .P xy − → P , whic h implies th at the same name y will b e receiv ed at the r eceiving end of the c hannel. How ev er, b ecause of n oise, in the π N -calculus the corresp onding tran s ition would b e xy .P xz − → p x ( z | y ) P . This pr obabilistic transition indicates that although the name in tentio n ally sent b y the agen t is y , the name at the receiving end of the c h annel x ma y be the name z , diﬀeren t from y , with the probability p x ( z | y ). W e refer the reader to Section 1.2 in [54] for a comparison b et wee n this mo del of noisy c hannels and the existing lite r ature [1, 2, 3, 4, 5, 6, 8, 9, 20, 25, 27, 38] includ ing some w orks ab out other formal mo dels with unr eliable comm un ication c hann els. It is well kno wn that b ehavio r al equiv alences play a v ery imp ortan t r ole in pr o cess algebras b ecause they p ro vide a form al d escription that one system imp lemen ts another. Tw o ag ents are deemed equiv alen t w hen they “hav e the same b ehavio r ” f or some suitable notion of b eha v ior. In terms of th e π -calculus, v arious b ehavi oral equiv alences ha ve b een studied extensiv ely; examples are [7, 11, 12, 18, 32, 34, 37, 40, 41, 42]. In [54], some concepts of appr o ximate b isimilarit y and equiv alence in CCS [50, 52, 53] w ere generalized in to the π N -calculus; suc h b eh avioral equiv alences inv olv e quant itativ e i n formation— probabilit y , since the agen t in π N is r epresen ted by a probabilistic transition system. T o our knowledge , except for th is work there are n o probab ilistic ve r sions of b eha v ioral equiv alences in the π -calculus a n d its v ariant s, although pr obabilistic extensions of the π -calculus [19, 20, 46] were in tro duced. The p urp ose of this pap er is to extend some classical b eha vioral equiv alences related 2 to (str on g) b arb ed equiv alence and (strong) b isim ilarity into π N and cast them in a hierarc hy . F ollo wing the model of noisy c hannels in [54], we d ev elop an early transitional seman tics of the π N -calculus whic h mak es the study of b eha vioral equ iv alences somewh at simpler. It is wo r thy of note, ho wev er, th at unlike in the π -calculus, this seman tics is not a directly tran s lated version of the late semant ics of π N in [54]. Surprisin gly , we ha ve found that not all b ound names in π N are compatible with alpha-con version when w e remov e th e strong assumption in [54] that free names and b ound names are distinct. As a resu lt, we ha ve to add a ru le for inputting b ound names to the corresp onding early seman tics of π . In addition, to hand le transitions of π N w ell, w e group the transitions according to their sources. ✻ ✻ ✻ Barb ed Equiv alence ( ˙ ≈ ) = Bisimilarity ( ∼ ) Barb ed Congruence ( ˙ ≃ ) = F ull Bisimilarity ( ≃ ) Barb ed Bisimilarity ( ˙ ∼ ) Reduction Bisimilarity ( ≏ ) (a) Hi erarc h y i n the π -calculus Barb ed Equiv alence ( ˙ ≈ ) Barb ed Congruence ( ˙ ≃ ) F ull Bisimilarity ( ≃ ) ✐ ✻ ✶ ✻ ✐ Bisimilarity ( ∼ ) ✶ Barb ed Bisimilarit y ( ˙ ∼ ) Reduction Bisimilarity ( ≏ ) (b) Hierarc h y in the π N -calculus Figure 1: Hierarc hy of b eha vioral equiv alences, where an arro w A → B expr esses that A is strictly included in B Based up on the early transitional semanti cs of π N , we then extend six kinds of b e- ha vioral equiv alences consisting of redu ction bisimilarit y , barb ed b isimilarit y , barb ed equiv alence, barb ed congruence [34], b isim ilarity , and fu ll b isimilarit y [32, 33] in to th e π N -calculus. All these equiv alences are deﬁn ed through certain bisim u lations inv olving in ternal action and discriminating p o we r . Because of noisy c h annels, a transition ma y o ccur with a certain p robabilit y , and th u s all the b isim u lations are deﬁ n ed quant ita- tiv ely . Some basic prop erties of these equiv alences hav e b een inv estigated. Finally , w e concen trate on a hierarch y of these b eha vioral equiv alences. In particular, du e to th e noisy nature of channels, th e co in cidence of bisimilarit y and barb ed equiv alence wh ic h holds in the π -calculus, as w ell as the coincidence of f ull bisimilarit y and barb ed con- gruence, fails in the π N -calculus. This hierarch y is sho wn in Fi gur e 1, toge th er with a corresp ondin g hierarch y in π (see, f or example, [43]). Clearly , the hierarc hy is helpfu l to v erify the b eha vioral equiv alence of t wo age nts: one can start a proof eﬀort at the middle tier; if this succeeds, one can switc h to a ﬁ ner equiv alence; otherwise, one can switc h to a coarser equiv alence. The remainder of this pap er is stru ctured as follo ws. W e brieﬂy review some basics of the π -calculus in Section 2. After recalling the formal framewo r k of π N [54] in Section 3.1, w e deve lop the early transitional se m an tics o f the π N -calculus and present it in t wo d iﬀeren t forms in the r emainder of this section. Section 4 is dev oted to reduction bisimilarit y and barb ed bisimilarit y , equiv alence, and congruence. In the subsequent 3 section, th e ot h er tw o equiv alences, bisimilarity and full b isimilarit y , are explored. W e complete the hierarc h y of these behavioral equiv alences in Secti on 6 and conclude t h e pap er in Section 7. 2 π -calculus F or the con venience of the r eader, this section collects some useful fact s on the π - calculus f rom [43]. The synta x, the early transitional seman tics, and some notations of the π -calculus are presented in Sectio n 2.1. Section 2.2 brieﬂy reviews sev eral b eha vioral equiv alences of the π -calculus. W e refer the r eader to [31, 32, 35, 43] and the references therein for an elab orated explanation and the d ev elopmen t of the theory of π . 2.1 Basic deﬁnitions W e p resupp ose in the π -calculus a countably in ﬁnite set N of names ranged o v er b y a, b, . . . , x, y , . . . with τ 6∈ N , and such names will act as comm unication c hann els, v ariables, and d ata v alues. W e employ P , Q, R, . . . to s erv e as meta -v ariables of agen ts or pr o cess expressions. Pr o cesses ev olve by p erform ing actions, and the capabilities for action are exp ressed via the f ollo wing four kind s of pr eﬁxes : π ::= xy | x ( z ) | τ | [ x = y ] π , where xy , an output pr eﬁx , is capable of sending the n ame y via the name x ; x ( z ), an input pr eﬁx , is capable of r eceiving an y name via x ; τ , the silent pr eﬁx , is an in ternal action; a n d [ x = y ] π , a match pr eﬁx , has the capabilit y π wh enev er x and y are the same name. W e no w r ecall the syn tax of the π -calculus. Deﬁnition 2.1. The pr o c e sses and the summations of the π -calculus are giv en r esp ec- tiv ely b y P ::= M | P | P ′ | ( ν z ) P | ! P M ::= 0 | π .P | M + M ′ . In the deﬁnition ab o ve, 0 is a designated p ro cess symb ol that can do n othing. A pr eﬁx π .P h as a single ca p abilit y expressed by π ; the agen t P cannot p ro ceed until that capabilit y has b een exercised. A sum P + Q represent s an agen t that can enact either P or Q . A p ar al lel c omp osition P | Q r epresen ts the com bined b eha vior of P and Q executing in parallel, that is, P and Q can act indep endently , and ma y also comm unicate if one p erf orms an outpu t and th e other p erf orm s an inpu t along the same channel. The r estriction op erator ( ν z ) in ( ν z ) P acts as a static b inder for the name z in P . In addition, the inpu t preﬁx x ( z ) also bin ds the name z . The agen t ! P , called r eplic ation , can b e though t of as an inﬁn ite co mp osition P | P | · · · or, equiv alen tly , as a pro cess satisfying the equation ! P = P | ! P . Iterativ e or arb itrarily long b eh avior can also b e d escrib ed b y an alternativ e mec hanism, the so-called r ecursion. It turns out that rep lication can enco de the r ecursiv e deﬁnition (see, for example, Section 9.5 in [31]). Let u s in tro duce s ome syn tactic n otations b efore going forw ard. As mentio n ed ab o v e, b oth input pr eﬁ x and restriction bin d names, and w e can deﬁn e the b ound names bn( P ) as those with a boun d occurr ence in P and the fr e e names fn ( P ) as those with a n ot 4 b ound o ccurr ence. W e write n( P ) for the names of P , namely , n( P ) = fn( P ) ∪ bn( P ), and sometimes use the abbreviation f n( P , Q ) for fn( P ) ∪ fn( Q ). A substitution is a f unction from names to names that is th e identi ty except on a ﬁn ite set. W e write { y /x } f or th e substitution that maps x to y and is iden tit y for all other names, and in general { y 1 , . . . , y n /x 1 , . . . , x n } , where the x i ’s are pairwise distinct, for a fun ction that maps eac h x i to y i . W e u se σ to range ov er substitutions, and wr ite xσ , or sometimes σ ( x ), f or σ applied to x . The pr o cess P σ is P where all free n ames x are replaced by σ ( x ), with changes of some b oun d names (i .e., alpha-con v ersion) wherev er needed to av oid name captures. F or later n eed, we ﬁ x some notati onal conv entions: A sequence of distinct restric- tions ( ν z 1 ) · · · ( ν z n ) P is often abbreviated to ( ν z 1 · · · z n ) P , or just ( ν e z ) P when n is not imp ortant. W e sometimes e lid e a trailing 0 , writing α f or th e pro cess α. 0 , w here this cannot cause confu sion. W e also follo w generally used op erator precedence on pro cesses. F or our purp ose of in v estigating barb ed equ iv alence, we only recall th e ea r ly tran- sition rules here; the reader ma y refer to [32, 43] for the late one. Th e transition rules are n othing other than infer en ce ru les of lab eled transition relations on pro cesses. The transition relations are lab eled by the actions , of which there are f ou r kinds: the silen t action τ , inpu t actions xy , free output actions xy , and b oun d output actions x ( y ). The ﬁrst action is int ern al action, the second is receiving the name y via the name x , the third is sendin g y via x , and the last is s en ding a fresh name via x . Let α, β , . . . range o v er actions. W e write Act for the set of actio n s. If α = xy , xy , or x ( y ), then x is called the subje ct and y is called th e obje ct of α . The fr e e names and b ound names of an actio n α are giv en by fn( α ) =    ∅ , if α = τ { x, y } , if α = xy or xy { x } , if α = x ( y ) and bn( α ) =  ∅ , if α = τ , xy , or xy { y } , if α = x ( y ) . The set of names, n( α ), of α is f n ( α ) ∪ b n( α ). The transition r elation lab eled by α will b e wr itten as α − → . Th us, P τ − → Q will express that P can evolv e invisibly to Q ; P xy − → Q w ill express that P can r eceiv e y via x an d b ecome Q ; P xy − → Q will express th at P can send y via x and evolv e to Q ; and P x ( y ) − → Q w ill express th at P ev olv es to Q after sending a fresh name via x . W e are no w in th e position to revie w the lab eled transition seman tics of t h e π - calculus. The (e arly) tr ansition r e lations , { α − → : α ∈ Act } , are deﬁned b y the ru les in T able 1. Note that fou r rules are elided fr om the table: the symmetric form s Sum-R, P ar-R, Comm-R, and Close-R of Sum-L, Par-L, Comm-L, and Close-L, r esp ectiv ely . W e d o not discuss and illustrate the rules, and only remark that the side conditions in P ar-L, Par-R, Close -L, Close-R, Rep-Close can alw a ys b e satisﬁed b y changing the ob ject of a b ound-output action. 2.2 Beha vioral equiv alences In this sub section, w e recall the notio n s of reduction bisimilarity , barb ed bisimi- larit y , barb ed e q u iv alence, b arb ed co n gruence, bisimilarit y , and full b isim ilarity o f the π -calculus studied in [32 , 34, 40, 33, 43]. A hierarc hy of them is also recorded. 5 Out xy .P xy − → P Inp x ( z ) .P xy − → P { y /z } T au τ .P τ − → P Mat π .P α − → P ′ [ x = x ] π .P α − → P ′ Sum-L P α − → P ′ P + Q α − → P ′ P ar-L P α − → P ′ P | Q α − → P ′ | Q bn( α ) ∩ fn ( Q ) = ∅ Comm-L P xy − → P ′ Q xy − → Q ′ P | Q τ − → P ′ | Q ′ Close-L P x ( z ) − − → P ′ Q xz − → Q ′ P | Q τ − → ( ν z )( P ′ | Q ′ ) z 6∈ fn( Q ) Res P α − → P ′ ( ν z ) P α − → ( ν z ) P ′ z 6∈ n ( α ) Op en P xz − → P ′ ( ν z ) P x ( z ) − − → P ′ z 6 = x Rep-Act P α − → P ′ ! P α − → P ′ | ! P Rep-Comm P xy − → P ′ P xy − → P ′′ ! P τ − → P ′ | P ′′ | ! P Rep-Close P x ( z ) − − → P ′ P xz − → P ′′ ! P τ − → (( ν z )( P ′ | P ′′ )) | ! P z 6∈ f n( P ) T able 1: E arly transition r ules of the π -calculus Let us b egin with r ed uction bisimilarit y . Deﬁnition 2.2. A relation R is a r e duction bisimulation if whenever ( P , Q ) ∈ R , (1) P τ − → P ′ implies Q τ − → Q ′ for some Q ′ with ( P ′ , Q ′ ) ∈ R ; (2) Q τ − → Q ′ implies P τ − → P ′ for some P ′ with ( P ′ , Q ′ ) ∈ R . R e duction b isimilarity , denoted ≏ , is the union of all r eduction bisimulations. A somewhat stronger notio n than reduction bisimilarit y but still v ery weak is barb ed bisimilarit y , wh ic h tak es observ abilit y into account . T o express o b serv abilit y formally , w e need the noti on of observability pr e dic ate P ↓ θ , where θ is an arbitrary n ame or co-name, n amely θ ∈ N ∪ { a : a ∈ N } . W e sa y th at P ↓ a if P can p erf orm an inp ut action with su b ject a ; and P ↓ a if P can p erf orm an output action w ith sub j ect a . Deﬁnition 2.3. A relation R is a b arb e d bisimulation if whenever ( P , Q ) ∈ R , (1) P ↓ θ implies Q ↓ θ , and v ice v ersa; (2) P τ − → P ′ implies Q τ − → Q ′ for some Q ′ with ( P ′ , Q ′ ) ∈ R ; (3) Q τ − → Q ′ implies P τ − → P ′ for some P ′ with ( P ′ , Q ′ ) ∈ R . Barb e d b i similarity , written ˙ ∼ , is the u nion of all b arb ed bisimulati ons . Based up on barb ed bisimulation, we ha ve the follo wing deﬁn ition. Deﬁnition 2.4. Tw o p ro cesses P and Q are called b arb e d e quivalent , d enoted P ˙ ≈ Q , if P | R ˙ ∼ Q | R for any R . W e are going to in tro duce b arb ed congruence, whic h is stronger than b arb ed equiv- alence. T o this end , w e n eed an auxiliary notion. 6 Deﬁnition 2.5. Pr o c ess c ontexts C are giv en by th e syntax C ::= [ ] | π . C + M | C | P | P |C | ( ν z ) C | ! C . W e denote b y C [ P ] the result of ﬁlling the hole [ ] in the conte xt C w ith the pro cess P . The elementary c ontexts are π . [ ] + M , [ ] | P , P | [ ] , ( ν z )[ ] , and ![ ]. No w, we can make th e follo wing deﬁnition. Deﬁnition 2.6. Two pro cesses P and Q are said to b e b arb e d c ongruent , written P ˙ ≃ Q , if C [ P ] ˙ ∼C [ Q ] f or ev ery pro cess cont ext C . Let us contin ue to deﬁ n e bisimilarity and an asso ciated congruence, f u ll bisimilarit y . Deﬁnition 2.7. (Str ong) bisimilarity is the largest symmetric relation, ∼ , su c h that whenev er P ∼ Q , P α − → P ′ implies Q α − → Q ′ for some Q ′ with P ′ ∼ Q ′ . Deﬁnition 2.8. T w o pro cesses P and Q are ful l bisimilar , written P ≃ Q , if P σ ∼ Qσ for ev ery su bstitution σ . Finally , w e su m marize a hierarc hy of the b ehavio r al equ iv alences ab ov e, whic h is depicted in Figur e 1(a). Theorem 2.9. (1) ˙ ≃ ⊆ ˙ ≈ ⊆ ˙ ∼ ⊆ ≏ ; eac h of the inclusions can b e strict. (2) ∼ = ˙ ≈ and ≃ = ˙ ≃ . F or a pro of of the ab o ve theorem, the reader is r eferred to Sections 2.1 and 2.2.1 (Lemma 2.2.7 and Theorem 2.2.9) of [43]. W e remark that barb ed b isimilarit y , congru- ence, and equ iv alence w ere in tro du ced in [34], the basic theo r y of bisimilarit y and full bisimilarit y was established in [32, 33], and the assertion (2) of T heorem 2.9 was ﬁ rst pro ved in [40]. 3 π -calculus with noisy c h annels In the last section, we made an implicit assump tion that all communicati on c hann els in the π -calculus are noiseless. In the presen t sectio n , such an assump tion is remo ved and the π -calculus with noisy c hannels, the π N -calculus, is explored. The ﬁ rst subs ection is dev oted to introd ucing the original formal framew ork of the π N -calculus due to Ying [54], which is based on the late transitional sema ntics o f π N . A n early t r ansitional seman tics of π N is prop osed in S ection 3.2. T o cop e with transitions of π N w ell, w e group the transitions according to their sources in Section 3.3. 3.1 Late t r ansitional seman tics of π N This su b section reviews brieﬂy some basic notions of the π N -calculus from [54], including noisy c hann els and the late trans itional semanti cs. A fu ndamenta l assumption in the π N -calculus [5 4 ] is that communicatio n channels ma y b e noisy; th at is, their inpu ts are sub ject to certain distu rbances in trans mission. In other w ords, the comm u nication situation is conceiv ed as that an input is transmitted through a c hannel and the output is pr o duced at the end of th e channel, but the ou tp ut 7 is often n ot completely determined by the input. In Shannon’s information theory [45], a mathematical mo d el of c h annels fr om statistic comm u nication theory , the n oisy nature is usually describ ed by a p r obabilit y distr ib ution o v er the outpu t alphab et. This distribution of cours e dep ends on the in put and in addition it ma y dep en d on the in ternal state of the c hannel. A simpler but s till very v aluable cla s s of noisy c h an n els is memoryless channels, on w hic h the π N -calculus is b ased. In su c h c hannels, it is assumed that any output do es not dep end on the in ternal state of the channel, and moreo v er, the outp uts of an y t wo diﬀerent inputs are indep end en t. It turns ou t that a memoryless channel can b e completely c h aracterized by its c h annel matrix [ p ( y | x )] x,y where p ( y | x ) is the conditional p r obabilit y of outpu tting y wh en the inp ut is x , and the subscripts x and y run o ver all inp u ts and outputs, resp ectiv ely . Clearly , p ( y | x ) ≥ 0, and by deﬁn ition, we hav e that P y p ( y | x ) = 1 for any inp ut x . The syn tax of the π N -calculus is completely the sa m e a s that of t h e π -calculus. The essential d iﬀeren ce b etw een π N and π is that π N tak es the n oise of comm u nication c hannels into consideration, whic h means th at the receiv er cannot alwa ys get exactly what the sender deliv ers. As ment ioned ab o ve, the noisy c hannels in π N are assu med to b e memoryless, and thus we m a y s upp ose that eac h name x ∈ N has a channel matrix M x = [ p x ( z | y )] y , z ∈ N where p x ( z | y ) is the p robabilit y that the receiv er will get the name z at the output when the send er emits the name y along the channel x . F or the π -calculus, ther e are tw o kinds of transitional seman tics. In Section 2.1, the input rule is a transition x ( z ) .P xy − → P { y /z } , whic h expresses th at x ( z ) .P can receiv e the n ame y via x and ev olv e to P { y /z } . An action of the form xy records b oth the name us ed for receiving and the name receiv ed. The placeholder z is instan tiated early , namely when th e input b y the r eceiv er is inferred. Hence the name “early” seman tics. In the literature on π , the ﬁr s t w ay to treat the semant ics for inp ut, the late transitional seman tics [3 2 ], adopts the inp ut rule x ( z ) .P x ( z ) 7− → P , wh ere th e la b el x ( z ) con tains a placeholder z for the name to b e r eceiv ed, rather than the name itself. In this cont ext, the inp u t action of the form x ( z ) whic h replaces the action xy in the early transitional seman tics can b e instan tiated late, that is, it can b e instant iated when a communicatio n is inferred. Therefore, the early and late terminology is based up on when a name (placeholder) is instant iated in inferr ing an int er action. I n fact, there is a v ery close relationship b et ween the tw o kin d s of semantic s (see, for example, Lemma 4.3.2 in [43]) whic h allo ws us to f r eely use the early or late semantic s as con venien t. Let u s write Act l for the set of actions in the late transitional s eman tics, namely , Act l = { τ , x ( y ) , x y , x ( y ) : x, y ∈ N } . If α = x ( y ), w e set fn( α ) = { x } and b n( α ) = { y } . The structural op erational seman tics of π N in [54] is based up on the late transitional seman tics and is giv en b y a family of probabilistic transition relations α 7− → p ( α ∈ Act l , p ∈ (0 , 1]) displa yed in T able 2. The table omits the symm etric forms of Sum-L, P ar -L, Comm-L, and Close-L ; note also that the rule IDE for agent iden tiﬁers in [54] is replaced by the ru les Rep-Act, Rep-Comm, and Re p-Close since we are us ing an equiv alent notion, replications, instead of agen t identiﬁers in th e syntax. The arr o w 7− → 8 Out xy .P xz 7− → p P p = p x ( z | y ) > 0 Inp x ( z ) .P x ( z ) 7− → 1 P T au τ .P τ 7− → 1 P Ma t π .P α 7− → p P ′ [ x = x ] π .P α 7− → p P ′ Sum-L P α 7− → p P ′ P + Q α 7− → p P ′ P ar-L P α 7− → p P ′ P | Q α 7− → p P ′ | Q bn( α ) ∩ fn( Q ) = ∅ Comm-L P xy 7− → p P ′ Q x ( z ) 7− → 1 Q ′ P | Q τ 7− → p P ′ | Q ′ { y /z } Close-L P x ( z ) 7− → p P ′ Q x ( z ) 7− → 1 Q ′ P | Q τ 7− → p ( ν z )( P ′ | Q ′ ) Open P xy 7− → p P ′ ( ν y ) P x ( y ) 7− → p P ′ y 6 = x Res P α 7− → p P ′ ( ν z ) P α 7− → p ( ν z ) P ′ z 6∈ n ( α ) Rep-A c t P α 7− → p P ′ ! P α 7− → p P ′ | ! P Rep-Comm P xy 7− → p P ′ P x ( z ) 7− → 1 P ′′ ! P τ 7− → p P ′ | P ′′ { y /z }| ! P Rep-Close P x ( z ) 7− → p P ′ P x ( z ) 7− → 1 P ′′ ! P τ 7− → p ( ν z )( P ′ | P ′′ ) | ! P T able 2: L ate transition rules of π N α kind barb( α ) sub j( α ) ob j( α ) n( α ) ασ τ Silen t τ − − ∅ τ xy Input x x y { x, y } xσ y σ xy Noisy free ou tp ut x x y { x, y } xσ y σ x ( y ) Noisy b ound outpu t x x y { x, y } xσ ( y σ ) T able 3: T erminology and notation for actions is used to distingu ish the late relations from the early , and the probabilit y v alues p arise en tirely from the noise of comm u nication c h an n els. Th e Out r u le, which repr esen ts the noisy natur e of c h annels, is th e uniqu e one th at all d iﬀerences b et wee n π N and π come from. It means that the p r o cess xy .P of output pr eﬁx form sends the name y via the c hannel x , b ut what the receiv er gets at the output of this c hann el ma y not b e y du e to noise residing in it, and a name z will b e receiv ed with the probability p x ( z | y ). 3.2 Early transitional seman tics of π N F or the π -calculus, it turns out that the early trans itional semantics is somewhat simpler than the late one for in v estigating b eha vioral equiv alences. Th e reason is that the inp ut action in th e early s eman tics is instantia ted early and thus w e need n ot c heck all p ossible instant iations of a placeholder. In ligh t of th is, w e pa y our atten tion to the early tran s itional semanti cs of π N in this subsection. This semantics is n ot, ho wev er, a direct translation of T able 2, as we will see shortly . 9 Out xy .P xz − → p P p = p x ( z | y ) > 0 Inp x ( z ) .P xy − → 1 P { y /z } T au τ .P τ − → 1 P Mat π .P α − → p P ′ [ x = x ] π .P α − → p P ′ Sum-L P α − → p P ′ P + Q α − → p P ′ P ar-L P α − → p P ′ P | Q α − → p P ′ | Q Comm-L P xy − → p P ′ Q xy − → 1 Q ′ P | Q τ − → p P ′ | Q ′ Close-L P x ( y ) − − → p P ′ Q xy − → 1 Q ′ P | Q τ − → p ( ν y )( P ′ | Q ′ ) Res P α − → p P ′ ( ν z ) P α − → p ( ν z ) P ′ z 6∈ n ( α ) Op en-Ou t P xy − → p P ′ ( ν y ) P x ( y ) − − → p P ′ y 6 = x Op en-In p P xy − → 1 P ′ ( ν y ) P xy − → 1 P ′ y 6 = x Rep-Act P α − → p P ′ ! P α − → p P ′ | ! P Rep-Comm P xy − → p P ′ P xy − → 1 P ′′ ! P τ − → p P ′ | P ′′ | ! P Rep-Close P x ( y ) − − → p P ′ P xy − → 1 P ′′ ! P τ − → p ( ν y )( P ′ | P ′′ ) | ! P T able 4: Early transition rules of π N Lik e the late transitional s eman tics of π N , the early seman tics of π N is also giv en in terms of p robabilistic tr an s ition relations. A pr obabilistic trans ition in the π N is of the form P α − → p Q where P and Q are t wo pro cesses, α ∈ Act = { τ , xy , xy , x ( y ) : x, y ∈ N } , and p ∈ (0 , 1]. The in tuitiv e meaning of this transition is t h at t h e ag ent P p erforms acti on α and b ecomes Q , with probabilit y p . It should b e p oin ted out that a lthough the actions here are the same as t h ose in the early seman tics of π , th e meanings of them are not completely identi cal. More concretely , τ and xy still represent an in ternal action and an input of a name y alone c h an n el x , resp ectiv ely , but xy rep r esen ts output of a name y via a noisy c hannel x th at c h anges the int en d ed output of some name in to y with a certain pr obabilit y , and x ( y ) r epresen ts outpu t of a b ound name y via a n oisy channel x that c h an ges the intended output of some name into y with a c ertain probabilit y . T able 3 displa ys terminology and notati on p ertaining to the actions. Its columns list, resp ectiv ely , th e kind of an action α , the b arb of α , the subje ct of α , the obje ct of α , the set of names of α , and the eﬀect of applying a substitution to α ; some issues d iﬀeren t from those of π su c h as x ( y ) σ will b e explained su bsequently . The early transition rules of π N is presen t in T able 4. As b efore, we omit the symmetric forms of Sum-L, Pa r -L, Comm-L, and Close-L. Let us make a brief discussion ab out the rationale b ehind the design: 1) Since we follo w the assump tion of th e n oisy c h annels in [54], the O u t ru le is th e same as Out in T ab le 2. It sho w s that the action p erf ormed by xy .P is not xy but xz , and the probabilit y p x ( z | y ) that y b ecomes z in c hannel x is ind icated. Th is is th ou ght of as that n oise happ ens at the end of sending, not at the end of r eceiving. 2) All ru les except f or O u t and Op en-Inp are ju st simple imitations of the corre- 10 sp ond in g rules in the π -calculus. Nev ertheless, there are tw o diﬀerences: one is that a pr obabilit y parameter p is tak en in to accoun t, whic h is n ecessary for enco d in g the noise of c hann els; the other is that the side conditions in Par-L, C lose-L, and Rep-Close are elided. The latter arises entirely from that the condition bn( α ) ∩ fn( Q ) = ∅ is not required wh en consid er in g the left p arallel comp osition P | Q . The reason for r emo ving the cond ition bn( α ) ∩ fn( Q ) = ∅ arises from the follo w ing consideration. Recall that in π N [54] it was supp osed that free names and b ound names are distinct. This assu mption incon veniences the use of some transition rules such as Op en-Ou t in our con text. R ecall also that in π the side condition bn ( α ) ∩ fn( Q ) = ∅ of inferring P | Q can b e easily satisﬁed b y utilizing a lp ha-con ve r sion on P . Ho w ever, this con ve r sion must in v olv e the congruent equatio n ( ν z ) P ≡ ( ν w ) P { w /z } , where w 6∈ n( P ). It is unfortunate that suc h a w ell kn o wn and widely us ed equation in th e concurrency comm unity seems to b e impr acticable for π N . T o see this, let us examine a sp eciﬁc example. Sup p ose that p x ( y | y ) = p 0 , p x ( a | y ) = p 1 , p x ( b | y ) = p 2 , where p 0 + p 1 + p 2 = 1.W e c ho ose P def = ( ν a ) xy | x ( w ) .w z and Q def = ( ν b ) xy | x ( w ) .w z . Assume that ( ν z ) P ≡ ( ν w ) P { w/z } w ith w 6∈ n( P ) holds in π N . Then it is clear that P ≡ Q , and thus we can iden tify P with Q . B y the early transition rules of π N , we get without b r eaking th e side condition bn( α ) ∩ fn ( Q ) = ∅ that P τ − → p 0 y z , P τ − → p 1 ( ν a ) az , P τ − → p 2 bz ; Q τ − → p 0 y z , Q τ − → p 1 az , Q τ − → p 2 ( ν b ) b z . Because ( ν a ) az and ( ν b ) bz are inactiv e, while xz and bz are capable of sendin g z , this forces that p 1 = p 2 = 0, and thus p 0 = 1. It means that the channel x is n oiseless when outputting y ; this is absurd because x and y can b e tak en arbitrarily , including noisy c hannels. In ligh t of the previous d iscussion, it seems b etter to do aw ay with the congru en t equation ( ν z ) P ≡ ( ν w ) P { w/z } , w 6∈ n ( P ). As a resu lt, we cannot keep the side condi- tion bn( α ) ∩ fn( Q ) = ∅ , b ecause otherwise the asso ciativ e la w among pro cess in teractions w ould b e violate d . T o see th is, one ma y consider the pro cesses x ( z ) | (( ν y ) x y | y w ) and ( x ( z ) | ( ν y ) x y ) | y w . Since the asso ciativit y is a v ery imp ortant pr op erty of mobile systems that the π -calculus has intended to understand , we would n ot lik e to d estro y it. As a consequence, the s ide conditions in Par-L, Close- L, and Rep-Close are remo ved, and this yields that the p rop osed early trans itional seman tics of π N is somewhat d iﬀeren t f r om that of π when considering only noiseless c h an n els. Recall also that in the π -calculus, a priv ate name in a pr o cess P is lo cal, meaning it can b e used only f or comm un ication b et ween comp onen ts within P . Such a pr iv ate name cannot immediately b e used as a p ort for comm u nication b et ween P and its en vironment ; in fact, b ecause P ma y r ename its p riv ate n ames, these names are not known b y the en vironment. Note, ho wev er , that not allo wing the congru ence ( ν z ) P ≡ ( ν w ) P { w/z } , together with the noise of c hann els, mak es the priv ate names in π N somewhat public. More concretely , sin ce a p ro cess P in π N cannot rename its p riv ate names, these names ma y app ear in the en vironment of P . In addition, every name ma y b e confused with an y other n ame b ecause of the n oise of c h annels. Nev ertheless, p riv ate names are needed in π N since a priv ate name, sa y z , can a t least b e used to preclude a pro cess from comm unicating with its en vironment via the p ort z . 11 3) An Op en-Inp ru le is added. This arises fr om t wo aspects of consideration: One is that if a c h annel is capable o f inputting, then it s hould ha ve the abilit y of arbitrary inputting. In other w ord s, if P xy − → 1 Q , then restricting y to P sh ould n ot p rev ent the c hannel x from inpu tting y . The other asp ect is that if an agen t can receiv e a name from outside, then the name ma y b e though t of as op en and the scop e of the r estriction ma y b e extended. T echnicall y , the Op en-In p ru le deriv es fr om the inv alidation of the congruen t equation ( ν z ) P ≡ ( ν w ) P { w/z } , w 6∈ n( P ), b ecause without the equation, b ound names cannot b e c h anged and the free name in an in put ma y clash with b ound names. W e remark that su ch a ru le is n ot necessary in the π -calculus, since all the b ound n ames in π can b e r enamed by alpha-con ve r sion. Let u s con tinue in tro ducing some notions. As in π , the input pr eﬁx x ( z ) and the restriction ( ν z ) bin d the name z . In view of 2) ab o v e, in the π N -calculus w e need to diﬀeren tiate b etw een the b oun d n ames in ( ν z ) .Q and x ( z ) .Q . A b oun d n ame z is calle d str ongly b ound in P if it lies within some sub-term x ( z ) .Q of P . As sh o wn in Deﬁn ition 3.3 (1), we p ermit of changing a strongly b ound n ame in to a fr esh name, which is called str ong alpha-c onversion . Any b ou n d name that is not strongly b ound is said to b e we akly b ound . An o ccur rence of a name in a pro cess of π N is fr e e if it is not b ou n d. F or instance, in P def = ( ν s ) x ( z ) . ( ν z ) y z .xs , the n ame s is w eakly b ound, z is str ongly b ound , and x and y are free. W e den ote the free names, b oun d n ames, strongly b ound names, and wea k ly b ound names in a pro cess P b y fn( P ), bn ( P ), sbn ( P ), and wbn( P ), resp ectiv ely . W e no w deﬁne the eﬀect of applying a su bstitution σ to a pro cess P in π N . Since w eakly b oun d names cannot b e conv erted, the pro cess P σ is P where all free n ames and weakly b ound names x are replaced b y xσ , with stron g alpha-con ve r sion wherev er needed to a v oid captures. Th is means that strongly b oun d names are c hanged suc h that whenev er x is replaced b y xσ then the so obtained o ccurrence of xσ is not strongly b ound . F or instance, ( a ( x ) . ( ν b ) xb.c y . 0 ) { x, c/y , b } = a ( z ) . ( ν c ) z c.cx. 0 . Clearly , according to the ab o ve deﬁn ition of substitution, w e hav e th e follo wing f act. Lemma 3.1. F or an y substitution σ , (1) 0 σ = 0 ; (2) ( π .P ) σ = π σ.P σ ; (3) ( P + Q ) σ = P σ + Qσ ; (4) ( P | Q ) σ = P σ | Qσ ; (5) (( ν z ) P ) σ = ( ν z σ ) P σ ; (6) (! P ) σ =! P σ . Since the n otion of free names is not suﬃcien t for deﬁning str uctural congruen ce, we in tro d uce an e x tend ed notio n of free n ames. Motiv ated b y a similar n otion in [54], w e deﬁne the set of noisy f ree names to b e the set of all free n ames in a pro cess and those names pr o duced by noise when sending free names. F ormally , w e ha ve the follo wing. 12 Deﬁnition 3.2. The set of noisy f r e e na mes , d en oted fn ∗ ( P ), is deﬁned in ductiv ely as follo ws: (1) fn ∗ ( 0 ) = ∅ ; (2) fn ∗ ( xy .P ) = { x } ∪ { z ∈ N : p x ( z | y ) > 0 } ∪ fn ∗ ( P ); (3) fn ∗ ( x ( z ) .P ) = { x } ∪ S y ∈ N (fn ∗ ( P { y /z } ) \{ y } ); (4) fn ∗ ( τ .P ) = fn ∗ ( P ); (5) fn ∗ ([ x = y ] π .P ) = { x, y } ∪ fn ∗ ( π .P ); (6) fn ∗ ( P + P ′ ) = fn ∗ ( P | P ′ ) = fn ∗ ( P ) ∪ fn ∗ ( P ′ ); (7) fn ∗ (( ν z ) P ) = fn ∗ ( P ) \{ z } ; (8) fn ∗ (! P ) = fn ∗ ( P ). Although fn ∗ ( P ) is an extension of fn( P ), it is not n ecessarily that fn ∗ ( P ) ⊇ fn( P ). F or e x amp le, assume that p x ( z | y ) = 1. Then we see that fn ∗ ( xy . 0 ) = { x, z } , w h ile fn( xy . 0 ) = { x, y } . If it is required that p x ( y | y ) > 0 for all x, y ∈ N , then we ind eed ha ve that fn ∗ ( P ) ⊇ fn( P ). W e can now deﬁn e structural congruence as follo ws. Deﬁnition 3.3. Tw o process expressions P and Q in the π N -calculus are structur al ly c ongruent , d enoted P ≡ Q , if we can transform one in to th e other b y us in g the follo win g equations (in either direction): (1) x ( z ) .P ≡ x ( w ) .P { w/z } if w is fresh in P . (2) Reorderin g of terms in a summation. (3) M + 0 ≡ M , P | 0 ≡ P , P | Q ≡ Q | P , and P | ( Q | R ) ≡ ( P | Q ) | R. (4) ( ν z )( P | Q ) ≡ P | ( ν z ) Q if z 6∈ fn ∗ ( P ), ( ν z ) 0 ≡ 0 , and ( ν z )( ν w ) P ≡ ( ν w )( ν z ) P . (5) [ x = x ] π .P ≡ π .P . (6) [ x = y ] π .P ≡ 0 if x and y are d istinct. (7) ! P ≡ P | ! P . The noisy free names of t wo structurally congruen t p ro cesses are clearly related by the follo wing fact. Lemma 3.4. If P ≡ Q can b e in ferred without u sing (5) and (6) in Deﬁnition 3.3, then fn ∗ ( P ) = fn ∗ ( Q ). W e also mak e an observ ation on the n oisy free names of pr o cesses app earing in the same transition. Lemma 3.5. Let P α − → p P ′ b e a transition in π N . (1) If α = xy , then x, y ∈ fn ∗ ( P ) and fn ∗ ( P ′ ) ⊆ fn ∗ ( P ). 13 (2) If α = xy , then x ∈ fn ∗ ( P ) and fn ∗ ( P ′ ) ⊆ fn ∗ ( P , y ). (3) If α = x ( y ), th en x ∈ fn ∗ ( P ) and fn ∗ ( P ′ ) ⊆ fn ∗ ( P , y ). (4) If α = τ , then fn ∗ ( P ′ ) ⊆ fn ∗ ( P ). Pr o of. The p ro of is carried out by in duction on the d ep th of inference P α − → p P ′ . W e need to consider all kinds of trans ition rules that are p ossible as the last rule in deriving P α − → p P ′ . The assertions (1) and (2) can b e v eriﬁed dir ectly , th e p ro of of (3 ) needs (1), and the pro of of (4) needs the ﬁr st thr ee. This is a lo n g b ut r outine case analysis, so the details are omitted. W e end this subsection b y providing s ome image-ﬁniteness prop erties of p robabilistic transitions. T o this end, w e sup p ose that outputting a name can only gives rise to ﬁ nite noisy names, that is, we adopt th e follo wing conv ention: Con ven t ion 3.6. F or an y x, y ∈ N , the set { y i : p x ( y i | y ) > 0 } is ﬁ n ite. The follo wing facts ab out inpu t and output actions are th e corresp ondin g results of Lemmas 1.4.4 and 1.4.5 in [43]. Lemma 3.7. (1) If P xy − → 1 P ′ and y 6∈ fn( P ) ∪ wbn( P ), then P xz − → 1 P ′ { z /y } for any z . (2) If P xy − → 1 P ′ and z 6∈ f n( P ) ∪ wb n( P ), then there is P ′′ suc h that P xz − → 1 P ′′ and P ′′ { y /z } = P ′ . (3) F or any P and x , there exist P 1 , . . . , P n and y 6∈ fn( P ) ∪ wbn( P ) suc h that if P xz − → 1 P ′ then P ′ = P i { z /y } for some P i . (4) F or an y P , there are only ﬁnitely man y x such th at P xy − → 1 P ′ for some y and P ′ . Pr o of. The ﬁrst t wo assertions can b e easily p ro ved by in duction on inference, and the last tw o by ind uction on P . Similar to the ab ov e, w e ha ve a resu lt ab out output actions. Lemma 3.8. (1) F or any P , there are only ﬁnitely many q u adruples x, y , p , P ′ suc h that P xy − → p P ′ . (2) F or any P , there are only ﬁnitely many q u adruples x, y , p , P ′ suc h that P x ( y ) − → p P ′ . Pr o of. All the t wo assertions are pr o ve d by ind uction on P . The pr o of of (1) needs Con ven tion 3.6, and the pro of of (2) uses the f act that we akly b ound names cannot b e con ve r ted. 3.3 T ransition groups of π N In the last subsection, the primitive transition r u les of π N ha ve b een established . A clo s er examination, ho wev er , shows that at least tw o confusions may arise from the present ation of these r ules. T o a void this, we in tro d uce another pr esentati on , transition groups, in this subs ection. 14 T o illustr ate our motiv ation, let u s consider a simple example: T ak e P def = xy + xz , and sup p ose that p x ( y | y ) = 0 . 7 , p x ( z | y ) = 0 . 1 , p x ( s | y ) = 0 . 1 , p x ( t | y ) = 0 . 1; p x ( y | z ) = 0 . 5 , p x ( z | z ) = 0 . 3 , p x ( s | z ) = 0 . 1 , p x ( w | z ) = 0 . 1 . By the transition rules of π N , we see that P xy − → 0 . 7 0 , P xy − → 0 . 5 0 , P xs − → 0 . 1 0 , and P xt − → 0 . 1 0 . A t this p oin t, tw o confusions arise: One is th at there are t wo pr obabilistic transitio n s P xy − → 0 . 7 0 and P xy − → 0 . 5 0 with the same s ou r ce and target pro cesses and the same action, but d iﬀeren t p robabilit y v alues. In [54], it is though t that the probability of transition P xy − → 0 is either 0 . 7 or 0 . 5, but w hic h of th em is n ot exactly kno wn and the c hoice b et wee n 0 . 7 and 0 . 5 is made by the environmen t. Th is un derstanding th at comes from the idea of imp recise pr obabilit y studied widely by the co mmunities of Statistics and Artiﬁcial Intellig ence (see, for example, [49]) is v ery natural. Th e limit is that more uncertain ties are inv olv ed in inference. Another wa y to deal with this problem in the literature on probabilistic p ro cesses is to mo d ify probabilistic transitio n systems. T his is done b y adding up al l p ossible v alues of transitio n probabilit y w ith the s ame source and target agen ts a n d the same action, and then normalizing them if necessary (for instance, see [47, 26, 48, 51]). Once again, this kind of mo diﬁcation highly complicates the theory of probabilistic pr o cesses. The other co n fusion is that t h e information o n the origins of some probabilistic transitions is lost. As for the ab o ve example, th e information that P xt − → 0 . 1 0 deriv es from the ﬁrst sub-term of P is lost. In addition, we cannot determine from whic h the transition P xs − → 0 . 1 0 arises. There is a con venien t presentat ion for alleviating the confusions. In fact, we ma y group all transitions having the s ame origin. Regarding the example ab ov e, we m ay s a y that P has t wo trans ition groups P { xy − → 0 . 7 0 , xz − → 0 . 1 0 , xs − → 0 . 1 0 , xt − → 0 . 1 0 } and P { xy − → 0 . 5 0 , xz − → 0 . 3 0 , xs − → 0 . 1 0 , xw − → 0 . 1 0 } . Notice that the ag ent P def = xy + xz h as a nondeterministic choice b etw een xy and xz , and it is usu ally though t that suc h a c hoice is made by the en vironment, so w e ma y think that th e choice b et we en the transition groups is also made b y the environmen t. In this w ay , the ﬁrst confusion is completely excluded, and if we hav e observ ed an output xt and the environmen t do es not change her c hoice, th en the pro cess has a smaller probabilit y of outputting xz and th e ou tp ut xs arises necessarily from th e ﬁrs t sub -term of P . In general, w e use P { α i − → p i Q i } i ∈ I to represent a group of probabilistic transitions P α i − → p i Q i , i ∈ I , satisfying P i ∈ I p i = 1; P { α i − → p i Q i } i ∈ I is called a tr ansition gr oup . W e omit the ind exing set I w henev er I is a singleton. In fact, some representa tions analogous to the tr an s ition group ha ve already b een us ed in the literature (see, for example, [44, 20, 46, 15 ]). 15 T o manipulate transition groups, we need th e op erator ⊎ deﬁned as follo w s : { α i − → p i Q i } i ∈ I ⊎{ β − → p Q } = ( { α i − → p i + p Q i } i ∈ I , if Q = Q i and β = α i for some i { α i − → p i Q i } i ∈ I ∪ { β − → p Q } , otherwise; { α i − → p i Q i } i ∈ I ⊎ { β j − → p j Q j } j ∈ J =  { α i − → p i Q i } i ∈ I ⊎ { β j − → p j Q j }  ⊎ { β k − → p k Q k } k ∈ J \{ j } . Mo v eo ver, if in the same transition group, there are P α i − → p i Q i and P α j − → p j Q j with α i = α j and Q i = Q j , then w e sometimes com bin e them in to a sin gle one P α i − → p i + p j Q i and delete j from the indexing set I . F or instance, the transition groups P { xy − → 0 . 4 Q } ⊎ { xy − → 0 . 6 Q } , P { xy − → 0 . 4 Q, xy − → 0 . 6 Q } , and P { xy − → 1 Q } mean the same thing. The follo wing result is helpful to group the transitions of π N . Lemma 3.9. Let P { α i − → p i Q i } i ∈ I b e a transition group derived from the early transition rules of π N in T ab le 4. Then all α i , i ∈ I , ha ve the same b arb. Pr o of. It is obvious by the in f erence rules in the transitional semant ics of π N . In li ght o f Lemma 3 .9 , w e can sa y that a transition group has a barb, whic h i s deﬁned as the common barb arising fr om the actions of pr obabilistic transitions in the transition group. A tran s ition group is called an output tr ansition gr oup (resp ectiv ely , input tr ansition gr oup ) if its b arb is of the form x (resp ectiv ely , x ). Using Lemma 3.9, the transition rules in T able 4 are group ed in T able 5. F or later need, we in tro duce one more notation. A function µ from Ω to the closed unit in terv al [0 , 1] is called a pr ob ability distribution on Ω if P x ∈ Ω µ ( x ) = 1. By D (Ω) w e den ote the set of all p robabilit y distrib utions on the set Ω. If µ ∈ D (Ω × Γ) , ω ∈ Ω, and S ⊆ Γ, we deﬁn e µ ( ω , S ) = P s ∈ S µ ( ω , s ). Observe that eac h transition group, say P { α i − → p i Q i } i ∈ I , giv es rise to a probabilistic distribution µ on Act × P r oc , wher e µ is deﬁ n ed by µ ( α, Q ) =  p, if P α − → p Q b elongs to P { α i − → p i Q i } i ∈ I 0 , otherwise . Therefore, w e sometimes write P − → µ or P barb ( α i ) − − − − − − → µ for P { α i − → p i Q i } i ∈ I , where barb( α i ) is the barb of the trans ition group. W e end this subsection with tw o prop erties of tr ansition groups. The ﬁrs t one corresp onds to the image-ﬁniteness of transition r elations in π . Here, b y image-ﬁniteness w e mean that for an y pro cess P and action α in π , there are only ﬁnitely many pro cesses Q su c h that P α − → Q . Lemma 3.10. Keep Conv en tion 3.6. Then for ev ery α ∈ Act and P ∈ P r oc , the set { µ | { α }× P roc : P − → µ } is ﬁn ite, wh ere µ | { α }× P roc is the restriction of µ to { α } × P r oc . Pr o of. It follo w s immediately from Lemmas 3.7 and 3.8. The other prop erty of transition groups is concerned with applying su bstitution to transitions. T o state it, we need the next deﬁnition. 16 Out xy .P { xy i − − → p i P } i ∈ I I = { i : p i = p x ( y i | y ) > 0 } T au τ .P { τ − → 1 P } Inp x ( z ) .P { xy − → 1 P { y /z }} Mat π .P { α i − → p i P i } i ∈ I [ x = x ] π .P { α i − → p i P i } i ∈ I Sum-L P { α i − → p i P i } i ∈ I P + Q { α i − → p i P i } i ∈ I P ar-L P { α i − → p i P i } i ∈ I P | Q { α i − → p i P i | Q } i ∈ I Comm-L P { xy i − − → p i P i } i ∈ I ′ ⊎ { x ( y i ) − − − → p i P i } i ∈ I ′′ Q { xy i − → 1 Q i } P | Q { τ − → p i P i | Q i } i ∈ I ′ ⊎ { τ − → p i ( ν y i )( P i | Q i ) } i ∈ I ′′ Res-Out P { xy i − − → p i P i } i ∈ I ′ ⊎ { x ( y i ) − − − → p i P i } i ∈ I ′′ ( ν y j ) P { xy i − − → p i ( ν y j ) P i } i ∈ I ′ \{ j } ⊎ { x ( y i ) − − − → p i ( ν y j ) P i } i ∈ I ′′ \{ j } ⊎ { x ( y j ) − − − → p j P j } y j 6 = x Res-Inp P { xy − → 1 P ′ } ( ν z ) P { xy − → 1 ( ν z ) P ′ } z 6 = x, y Res-T au P { τ − → p i P i } i ∈ I ( ν z ) P { τ − → p i ( ν z ) P i } i ∈ I Op en-In p P { xy − → 1 P ′ } ( ν y ) P { xy − → 1 P ′ } y 6 = x Rep-Act P { α i − → p i P i } i ∈ I ! P { α i − → p i P i | ! P } i ∈ I Rep-Comm P { xy i − − → p i P i } i ∈ I ′ ⊎ { x ( y i ) − − − → p i P i } i ∈ I ′′ P { xy i − → 1 P ′ i } ! P { τ − → p i P i | P ′ i | ! P } i ∈ I ′ ⊎ { τ − → p i ( ν y i )( P i | P ′ i ) | ! P } i ∈ I ′′ T able 5: T ransition group ru les of π N 17 Deﬁnition 3.11. (1) A s ubstitution σ is said to b e c onsistent with w eakly b ound n ames of a pro cess P if z σ ∈ wb n( P σ ) implies z 6∈ fn ∗ ( P ). (2) A su bstitution σ is said to b e c omp atible with a c hannel x ∈ N if for an y y , z ∈ N it holds that p xσ ( u | y σ ) = X z σ = u p x ( z | y ) . W e remark that the d eﬁnition of compatibilit y h ere is a sp ecial case of Deﬁn ition 1 in [54]. A transition group u nder a consisten t and compatible su b stitution has the follo wing prop ert y . Prop osition 3.12. Let P { α i − → p i P i } i ∈ I b e a trans ition group. S upp ose that σ is a substitution satisfying the follo wing conditions: 1) σ is consisten t with w eakly b ound n ames of all pro cesses app earing in the inference for deriving the transition group; 2) σ is compatible with the sub jects of output actions app earing in the inference for deriving the transition group. Then, there is a transition group P σ { α i σ − → p i P i σ } i ∈ I . Pr o of. It follo ws from a case b y case c hec k of all p ossible transition groups P { α i − → p i P i } i ∈ I . T he condition 1) is us ed to compute pr obabilit y of P σ α i σ − → P i σ , and the cond ition 2) is required when restricting P . W e only co n sider the case o f O u t and omit the remainder. In the case of Out, we ma y assume that P = xy .R , p x ( y i | y ) = p i , α i = xy i , and P i = R . Then th e trans ition group is xy .R { xy i − → p i R } i ∈ I . Th is giv es a transition group xσ y σ.Rσ { xσ u j − → q j Rσ } j ∈ J b y the deﬁ nition of su bstitution, wh er e J = { j : q j = p xσ ( u j | y σ ) > 0 } . It follo ws from th e condition 2) that q j = p xσ ( u j | y σ ) = X y i σ = u j p x ( y i | y ) = X i ∈ I j p x ( y i | y ) = X i ∈ I j p i , where I j = { i ∈ I : y i σ = u j } . Therefore, P σ { α i σ − → p i P i σ } i ∈ I = xσ y σ.Rσ { xσ y i σ − → p i Rσ } i ∈ I = xσ y σ.Rσ { xσ u j − → P i ∈ I j p i Rσ } j ∈ J = xσ y σ.Rσ { xσ u j − → q j Rσ } j ∈ J , as desired. 4 Barb ed equiv alence Ha ving built the transition group rules, we can no w turn to sev eral b eha vioral equiv- alences in π N . As mentioned in the last section, collecting th e p robabilistic trans itions arising f r om a noisy c hann el in to a group can allevia te some confusions, so the transi- tion group pr o vides a us efu l wa y of deﬁn ing b eha vioral equiv alences. Of course, other 18 w ays whic h do not use the transition group are p ossible (for example, Deﬁn ition 4 in [54]). Recall that in π N t wo kin ds of actions, n oisy fr ee output xy and n oisy b oun d output x ( y ), ha ve the same barb x . Clearly , not all obser vers can detect whether an emitted name is b ound. It is therefore natur al to ask wh at is the eﬀect of redu cing this discriminatory p o wer. In this section, the concepts ab out b eha vioral equiv alences are based up on the assumption that the observe r is limited to seeing wh ether an action is enabled on a given c hannel; b ehavio r al equiv alences with a m ore p o werful observ er will b e studied in the next section. Because the b eha vior of a pro cess in π N is eviden tly dep end en t on the noise pr ob ab ility distribution, it is b etter to parameterize a b eha vioral equiv alence with the n oise. Ho wev er , for simp licit y w e assume that all pro cesses in a deﬁnition of b eha vioral equiv alence are consid ered under the same noisy en v ir onmen t, that is, th e c h annel matrix of ev ery name is ﬁ xed. Let us start w ith s ome b asic notions. T he tr ansitive closur e of a binary relation R on P r oc is the minimal transitive relation R ∗ on P r oc that contai n s R , that is, if ( P , Q ) ∈ R ∗ , then there exist P 0 , . . . , P n ∈ P r oc satisfying that P = P 0 , Q = P n , and ( P i − 1 , P i ) ∈ R for i = 1 , . . . , n . Thus, if R and S are tw o equiv alence relations on P r oc , then so is ( R ∪ S ) ∗ . It turn s out that ( R ∪ S ) ∗ is the smallest equ iv alence relation con taining both R and S . F or any equiv alence relation R on P r oc , w e denote b y P r oc/ R th e set of equ iv alence classes in d uced b y R . The deﬁn ition b elo w only tak es the in ternal action into accoun t. Deﬁnition 4.1. An equiv alence relation R on P r oc is a r e duction bisimulation if when - ev er ( P , Q ) ∈ R , P τ − → µ implies Q τ − → η f or s ome η satisfying µ ( τ , C ) = η ( τ , C ) for an y C ∈ P r oc/ R . Note that the ab o ve r eduction b isim u lation and also other su bsequent notions on b ehavio r al equiv alences are deﬁn ed in the same st yle as Larsen-Sko u ’s p robabilistic bisim u lation [26], that is, th e probabilities of reac hin g an equiv alence class ha ve to b e computed. Recall that in a non-probab ilistic setting w e simply require that P τ − → P ′ implies Q τ − → Q ′ and ( P ′ , Q ′ ) ∈ R , that is, it is enough that the agen t Q has a p ossi- bilit y to imitate the step of the agen t P . Ho we ver, if we w ork in a p r obabilistic setting, in addition to n eed that the second agent is able to imitate the ﬁr st one, it is also rea- sonable to require that he do es it with the s ame probabilit y . In other words, w e hav e to consider all the p ossible wa ys to imitate the execution of the action, and thus w e ha v e to add the probabilities asso ciated with these possib ilities. Corresp ondingly , summing up the pr obabilities of reac h ing an equiv alence class replaces the condition ( P ′ , Q ′ ) ∈ R in a non-probabilistic setting. It should b e noted that the treatmen t of pr obabilities here is sligh tly diﬀeren t fr om that of λ -bisim u lation in [54], where a higher probabilit y of an action is allo we d to sim u late the same action. In the literature, lu mping equiv alence, a notion on Mark ov c hains to aggrega te state sp aces [14, 21], was also deﬁned b y su mming up the p r obabilities of r eac hing an equiv alence class; it needs not to consider the lab els of transitions and is diﬀerent from our deﬁnitions on b eha vioral equiv alences in π N . Because the union of equiv alence relations ma y not b e an equiv alence relation, u nlik e in π , the union of reduction b isim ulations is not a reduction b isim ulation in general. Nev ertheless, we ha ve the follo wing resu lt. Prop osition 4.2. Let ≏ = ( S i R i ) ∗ , where R i is a r ed uction bisimulat ion o n P r oc . Then ≏ is the largest redu ction bisimulatio n on P r oc . 19 Pr o of. It suﬃ ces to show th at ≏ is a reduction bisimulatio n o n P r oc . Su pp ose that ( P , Q ) ∈ ≏ and P τ − → η 0 . Then th ere are P 0 , . . . , P n ∈ P r oc and reduction bisim ulations R 1 ′ , . . . , R n ′ suc h that P = P 0 , Q = P n , a n d ( P i − 1 , P i ) ∈ R i ′ for i = 1 , . . . , n . By deﬁnition, there exist η 1 , . . . , η n suc h that P i τ − → η i and η i − 1 ( τ , C i ′ ) = η i ( τ , C i ′ ) for i = 1 , . . . , n and any C i ′ ∈ P r oc/ R i ′ . Note that for an y C ∈ P r oc/ ≏ and R i ′ , it follo ws from R i ′ ⊆ ≏ that C = S j C i ′ j for some C i ′ j ∈ P r oc/ R i ′ , and moreo v er, S j C i ′ j is a disjoin t union sin ce every C i ′ j is an equiv alence class. W e thus hav e that η i − 1 ( τ , C ) = η i − 1 ( τ , [ j C i ′ j ) = X j η i − 1 ( τ , C i ′ j ) = X j η i ( τ , C i ′ j ) = η i ( τ , C ) for eac h i = 1 , . . . , n . This means that η 0 ( τ , C ) = η n ( τ , C ). He n ce, ≏ is a reduction bisim u lation, as desired. The reduction bisimulatio n ≏ is called r e duction bisi milarity . In other wo r ds, P and Q are r e duction bisimilar if ( P , Q ) ∈ R for some redu ction bisim u lation R . As a p ro cess equ iv alence, reduction bisimilarit y is ser ious ly defectiv e. F or example, it relates an y t wo pr o cesses that hav e no in tern al actions, suc h as xa and y a . T o obtain a satisfactory pro cess equiv alence, it is therefore n ecessary to allo w more to b e observ ed of pro cesses. In the sequel, we use θ to rang ov er barb s but τ . Based up on Lemma 3.9, w e ha ve the follo wing deﬁn ition. Deﬁnition 4.3. The observability p r e dic ate ↓ θ in the π N -calculus is d eﬁned as follo ws : (1) P ↓ a if P has an input transition group with sub ject a . (2) P ↓ a if P has an output transition group with sub ject a . T aking observ ab ility in to consid eration, w e mo d ify the notion of reduction bisimu- lation as follo ws. Deﬁnition 4.4. An equiv alence relation R on P r oc is a (str ong) b arb e d bisimulation if whenev er ( P , Q ) ∈ R , (1) P ↓ θ implies Q ↓ θ ; (2) P τ − → µ implies Q τ − → η for some η satisfying µ ( τ , C ) = η ( τ , C ) for any C ∈ P r oc/ R . Analogous to P r op osition 4.2, we ha ve the follo wing fact. Prop osition 4.5. Let ˙ ∼ = ( S i R i ) ∗ , where R i is a barb ed b isim ulation on P r oc . T hen ˙ ∼ is the largest barb ed bisimulat ion on P r oc . Pr o of. Since b arb ed bisim ulations are redu ction bisimulatio n s, w e see that ˙ ∼ is a redu c- tion b isimulation by Pr op osition 4.2. F or an y ( P, Q ) ∈ ˙ ∼ , it is clear that P ↓ θ implies Q ↓ θ . Thereby , ˙ ∼ is a barb ed bisim u lation, an d moreo ver, it is the largest o n e since it includes all barb ed bisimulat ions on P r oc . 20 The barb ed bisimulatio n ˙ ∼ is called (str ong) b arb e d b isimilarity ; w e sa y that P and Q are (str ong) b arb e d bisimilar , d enoted P ˙ ∼ Q , if ( P, Q ) ∈ R f or some b arb ed bisim u lation R . It follo ws readily from deﬁnition that barb ed bisimilarity is prop erly included in reduction b isimilarit y . In addition, the fact b elo w is also ob vious. Lemma 4.6. If P ≡ Q , then P ˙ ∼ Q . Lik e reduction bisimilarity , barb ed bisimilarit y is not satisfactory as a pro cess equiv- alence as well . Neve r th eless, it will underp in t wo goo d relations, barb ed equiv alence and barb ed congruence. Let us b egin with barb ed equiv alence. Deﬁnition 4.7. Two pro cesses P an d Q in π N are called (str ong) b arb e d e quivalent , denoted P ˙ ≈ Q , if P | R ˙ ∼ Q | R for any R . It follo ws directly from the ab ov e deﬁnition that ˙ ≈ ⊆ ˙ ∼ . On the other hand, there are a large num b er of counter-examples to sho w that ˙ ∼ * ˙ ≈ . F or example, if P def = xa.y b and Q def = xa , then for an y channel matrix of x , w e ha ve that P ˙ ∼ Q b y deﬁnition. Ho w ever, if one tak es R def = x ( w ), then there exists P | R { τ − → 1 y b } , and moreo ver, it cannot b e matc hed by Q | R b ecause the uniqu e transition group Q | R { τ − → 1 0 } ha ving τ as barb yields y b ˙ ≁ 0 . Hence, P | R ˙ ≈ Q | R d o es not hold. F urther , we hav e the follo w ing fact that will b e of use later. Lemma 4.8. (1) If P ˙ ≈ Q , then ( ν z )( P | R ) ˙ ≈ ( ν z )( Q | R ) for an y R and z . (2) Let S b e an equiv alence relation included in ˙ ∼ . If for any R and z , ( P , Q ) ∈ S implies (( ν z )( P | R ) , ( ν z )( Q | R )) ∈ S , then S ⊆ ˙ ≈ . Pr o of. F or (1) , assume th at P ˙ ≈ Q . By deﬁn ition, w e see that P | R ˙ ≈ Q | R for an y R . Moreo v er, it is s traigh tforwa r d to sh o w th at ( ν z ) P ˙ ≈ ( ν z ) Q for an y z . Therefore, th e assertion (1) holds. F or (2), supp ose that ( P, Q ) ∈ S . Then we see by the h yp othesis of S that for an y R , ( P | R , Q | R ) ∈ S ⊆ ˙ ∼ . Hence, P | R ˙ ∼ Q | R for any R . So P ˙ ≈ Q , and th u s S ⊆ ˙ ≈ , ﬁnish in g the pr o of. Finally , we int r o duce th e concept of b arb ed congruence. Deﬁnition 4.9. Two p ro cesses P and Q in π N are (str ong) b arb e d c ongruent , denoted P ˙ ≃ Q , if C [ P ] ˙ ∼C [ Q ] for ev ery pro cess context C . In other wo r d s, tw o terms are barb ed congruen t if the agen ts obtained by placing them into an arbitrary conte xt are b arb ed bisimilar. Th e follo win g remark clariﬁes the relationship b et wee n barb ed equiv alence and barb ed congruence Remark 4.10. By deﬁnition, w e see that barb ed congru en t pro cesses in the π N -calculus are barb ed equiv alent , namely , ˙ ≃ ⊆ ˙ ≈ . In fact, this inclusion is strict. Th e follo win g example serve s : Let us tak e P def = x ( w ) . [ w = y ] τ and Q def = x ( w ) . [ w = z ] τ , and supp ose that all comm unication c hannels, except for x , are noiseless. W e al s o assume that the c hann el matrix of x is giv en by p x ( y | y ) = 0 . 5 , p x ( z | y ) = 0 . 5; p x ( y | z ) = 0 . 5 , p x ( z | z ) = 0 . 5; p x ( s | s ) = 1 for an y s 6 = y , z . 21 It follo w s readily that for an y R ∈ P r oc with R { α i − → p i R ′ i } i ∈ I , if α i 6∈ { xy , xz , x ( y ) , x ( z ) } , then P | R ˙ ∼ Q | R . Moreo v er, if there exists α i ∈ { xy , xz , x ( y ) , x ( z ) } , then an y tr an s ition group of R having α i as an action must b e one of the follo wing forms: (1) R { xy − → 0 . 5 R ′ 1 } ⊎ { xz − → 0 . 5 R ′ 1 } ; (2) R { x ( y ) − → 0 . 5 R ′ 2 } ⊎ { xz − → 0 . 5 ( ν y ) R ′ 2 } ; (3) R { xy − → 0 . 5 ( ν z ) R ′ 3 } ⊎ { x ( z ) − − → 0 . 5 R ′ 3 } ; (4) R { x ( y ) − → 0 . 5 ( ν z ) R ′ 4 } ⊎ { x ( z ) − − → 0 . 5 ( ν y ) R ′ 4 } . F or th e form (1), we hav e that P | R { τ − → 0 . 5 τ | R ′ 1 } ⊎ { τ − → 0 . 5 R ′ 1 } an d Q | R { τ − → 0 . 5 τ | R ′ 1 } ⊎ { τ − → 0 . 5 R ′ 1 } , whic h means that P | R ˙ ∼ Q | R . F or the form (2), we h a ve that P | R { τ − → 0 . 5 ( ν y )( τ | R ′ 2 ) } ⊎ { τ − → 0 . 5 ( ν y ) R ′ 2 } and Q | R { τ − → 0 . 5 τ | ( ν y ) R ′ 2 } ⊎ { τ − → 0 . 5 ( ν y ) R ′ 2 } . This yields that P | R ˙ ∼ Q | R because of ( ν y )( τ | R ′ 2 ) ≡ τ | ( ν y ) R ′ 2 . Th e form (3 ) is similar to that of the form (2), and w e ca n also ge t that P | R ˙ ∼ Q | R . F or the form (4), w e see that P | R { τ − → 0 . 5 ( ν y )( τ | ( ν z ) R ′ 4 ) } ⊎ { τ − → 0 . 5 ( ν y , z ) R ′ 4 } and Q | R { τ − → 0 . 5 ( ν z )( τ | ( ν y ) R ′ 4 ) } ⊎ { τ − → 0 . 5 ( ν y , z ) R ′ 4 } . Because ( ν y )( τ | ( ν z ) R ′ 4 ) ≡ τ | ( ν y , z ) R ′ 4 ≡ ( ν z )( τ | ( ν y ) R ′ 4 ), we get that P | R ˙ ∼ Q | R . S um- marily , we ha ve that P | R ˙ ∼ Q | R for any R , and th us P ˙ ≈ Q . On the other hand, let C = x ( y ) . [ ] | xs.xs . Then C [ P ] = x ( y ) .x ( w ) . [ w = y ] τ | xs.xs and C [ Q ] = x ( y ) .x ( w ) . [ w = z ] τ | xs.xs . It is easy to c h ec k that C [ P ] ˙ ≁ C [ Q ], and thus P ˙ ≃ Q do es n ot hold, as desired.  5 Bisimilarit y As men tioned in the last section, b eha vioral equiv alences und er a p o we r ful observ er that can d iﬀerentiate b et we en noisy free output and n oisy b ound output are inv estigate d in this section. W e b egin with a classical notion, b isim ulation. Deﬁnition 5.1. An equiv alence relation R on P r oc is a (str ong) bi si mulation if when- ev er ( P , Q ) ∈ R , P − → µ implies Q − → η f or s ome η satisfying µ ( α, C ) = η ( α, C ) for an y α ∈ Act and C ∈ P r oc/ R . The next result giv es the largest bisim ulation. Prop osition 5.2 . Let ∼ = ( S i R i ) ∗ , w here R i is a bisimulat ion on P r oc . Then ∼ is the largest bisim u lation on P r oc . 22 Pr o of. It is s im ilar to that of P rop osition 4.2. The largest bisimula tion ∼ is called (str ong) b isimilarity . In other words, P and Q are (str ong) b i similar , w ritten P ∼ Q , if ( P , Q ) ∈ R for some bisim ulation R . As an immediate consequence of Deﬁnitions 4.4 and 5.1, we h a v e the follo w ing. Lemma 5.3. An y t wo b isimilar pr o cesses are barb ed bisimilar, i.e., ∼ ⊆ ˙ ∼ . The remark b elo w tells us that bisimilarit y is not included in barb ed congruence. Remark 5.4. Ju st lik e in the π -calculus, b isimilar pro cesses in the π N -calculus ma y not b e b arb ed congruen t, that is, ∼ * ˙ ≃ . T he follo win g coun ter-example serv es: Let P def = au | b ( v ) and Q def = au.b ( v ) + b ( v ) .a u , and supp ose, for simp licit y , that there is a channel x w ith p x ( b | b ) = 1. Then we see that P ∼ Q . How ev er, the con text C def = ( x ( a ) . [ ]) | xb yields that C [ P ] ˙ ≁ C [ Q ]. Therefore, P ˙ ≃ Q do es n ot h old.  The follo wing notion will provide an equiv alen t c h aracterizat ion of bisimilarit y . Deﬁnition 5.5. A family of binary relations ∼ n on P r oc stratifying the bisimilarit y is deﬁned indu ctiv ely as f ollo ws: (1) ∼ 0 is the un iv ersal relation on pro cesses. (2) F or any 0 < n < ∞ , ( P , Q ) ∈ ∼ n if (2.1) P − → µ implies Q − → η for some η satisfying µ ( α, C n − 1 ) = η ( α, C n − 1 ) for an y α ∈ Act and C n − 1 ∈ P r oc/ ∼ n − 1 , and (2.2) Q − → η implies P − → µ for some µ s atisfying µ ( α, C n − 1 ) = η ( α, C n − 1 ) for an y α ∈ Act and C n − 1 ∈ P r oc/ ∼ n − 1 . (3) ( P , Q ) ∈∼ ∞ if ( P , Q ) ∈∼ n for all n < ∞ . Observe that ev ery ∼ n is an equ iv alence relation, so the notation P r oc/ ∼ n − 1 in the ab o ve deﬁnition m akes sen s e. Note also that ∼ 0 , ∼ 1 , . . . , ∼ ∞ is a decreasing sequence of relations. The result b elo w giv es another wa y to chec k b isimilarit y . Prop osition 5.6. P ∼ Q if and only if P ∼ ∞ Q . Pr o of. W e ﬁ r st pr o v e the n ecessit y . By deﬁnition, we only need to sh o w that ∼⊆∼ n for all n < ∞ . Proceed by indu ction on n . The case n = 0 is trivial. Assume that ∼⊆∼ n − 1 . F or any ( P , Q ) ∈∼ and P − → µ , it follo ws f rom the d eﬁnition of ∼ th at there exists η suc h that Q − → η and µ ( α, C ) = η ( α, C ) for an y α ∈ Act and C ∈ P r oc/ ∼ . By in duction h yp othesis ∼ ⊆∼ n − 1 , w e see that µ ( α, C n − 1 ) = η ( α, C n − 1 ) f or any α ∈ Act and C n − 1 ∈ P r oc/ ∼ n − 1 . Cons equen tly , ∼ ⊆∼ n , as desired. No w, let us sho w the s uﬃciency . It is e n ough to pro v e that ∼ ∞ is a bisimulation. Supp ose that P ∼ ∞ Q and P − → µ . T h en for eac h n < ∞ , there is η n suc h that Q − → η n and µ ( α, C n ) = η n ( α, C n ) for an y α ∈ Act and C n ∈ P r oc/ ∼ n . By Lemma 3.10, the num b er of distin ct η n ’s is ﬁnite, so th ere is η ′ suc h th at η ′ = η n for inﬁ n itely man y n . Since ∼ 0 , ∼ 1 , . . . , ∼ ∞ is a decreasing sequence of equ iv alence relations, w e get that µ ( α, C n ) = η ′ ( α, C n ) for an y C n ∈ P r oc/ ∼ n . This give s r ise to ( P , Q ) ∈ ∼ n for all n < ∞ , whic h means that ( P, Q ) ∈ ∼ ∞ . T herefore, ∼ ∞ is a bisimulation, ﬁn ishing th e pro of. 23 F or later need, let us pause to dev elop a “bisim ulation up to” tec h nique. Firstly , w e mak e the follo wing deﬁn ition. Deﬁnition 5.7. A binary symmetric relation R on P r oc is a bisimulation up to ∼ if whenev er ( P , Q ) ∈ R , P − → µ implies Q − → η for some η satisfying µ ( α, C ) = η ( α, C ) for an y α ∈ Act and C ∈ P r oc/ ( R∪ ∼ ) ∗ . The follo wing fact sho ws that an y bisim u lation up to ∼ relation is a bisimulation, as exp ected. Prop osition 5.8. If R is a bisimulation up to ∼ , then R ⊆∼ . Pr o of. Let S = ( R∪ ∼ ) ∗ . F or an y ( P , Q ) ∈ S , there exist P 0 , . . . , P n suc h that P 0 = P , P n = Q , and ( P i − 1 , P i ) ∈ R∪ ∼ for i = 1 , . . . , n . If ( P i − 1 , P i ) ∈ R , then P i − 1 − → µ i − 1 implies P i − → µ i for some µ i satisfying µ i − 1 ( α, C ) = µ i ( α, C ) for any α ∈ Act and C ∈ P r oc/ S . If ( P i − 1 , P i ) ∈ ∼ , then P i − 1 − → µ i − 1 implies P i − → µ i for some µ i satisfying µ i − 1 ( α, C ′ ) = µ i ( α, C ′ ) for an y α ∈ Act and C ′ ∈ P r oc/ ∼ . T his, together w ith the fact ∼⊆ S , yields that µ i − 1 ( α, C ) = µ i ( α, C ) for any α ∈ Act and C ∈ P roc/ S . As a result, for any P 0 − → µ 0 , there are µ 1 , . . . , µ n suc h that for i = 1 , . . . , n , P i − → µ i and µ i − 1 ( α, C ) = µ i ( α, C ) for an y α ∈ Act and C ∈ P r oc/ S . Th er efore, S is a bisim u lation, and thus R ⊆∼ . T his completes the pro of. W e now establish a bisim u lation up to ∼ relation w hic h will b e u sed in the next section. Lemma 5.9. Let R = { (( ν e z )( P | R ) , ( ν e z )( Q | R )) : e z are arbitrary names , and P , Q, R ∈ P r oc with P ∼ Q } . Then R is a bisimulat ion up to ∼ . Pr o of. Clearly , R is an equiv alence r elation. T o c heck that it is a bisimulat ion up to ∼ , w e app eal to a case analysis on th e last rules applied in the inference of rela ted tran- sitions. It needs to examine all inductiv e steps through combinatio n s o f the transition group rules o f Par, Comm, Res-Out, Res-Inp, Res-T au, and Op en-Inp. Th is is a long and routine argumen t, so w e omit the details here. T o obtain a congruen ce based on actions, w e mak e th e follo wing deﬁ n ition. Deﬁnition 5.10. Tw o pro cesses P and Q in π N are called (str ong) ful l bisimilar , d e- noted P ≃ Q , if P σ ∼ Qσ for any su b stitution σ . Lik e in π , bisimilarit y in π N is not pr eserv ed by subs titution, as illustrated b elo w. Remark 5.11. It follo ws immediatel y from deﬁ nition that fu ll bisimilar p ro cesses are necessarily bisimilar, i.e., ≃⊆∼ . Nev ertheless, ∼ * ≃ . In other w ord s, there are some bisimilar pr o cesses th at are not full bisimilar. F or example, let P def = au | b ( v ) and Q def = au.b ( v ) + b ( v ) .a u . Then f or any noisy c h annels, w e alw ays ha ve that P ∼ Q . Ho we ver, the sub stitution σ deﬁn ed by σ ( x ) =  a, if x = b x, otherwise giv es r ise to that P σ = au | a ( v ) and Qσ = au.a ( v ) + a ( v ) .au . Obviously , P σ ≁ Qσ , an d th us P and Q are not fu ll bisimilar.  24 6 A hierarc h y of b eha vioral equiv alences In the previous t wo sections, w e ha ve introdu ced several b ehavio ral equ iv alences. Some simple inclusion rela tionsh ips among them h a ve b een established. In this sec - tion, we consu mmate the relationships and then giv e a hierarc hy of these b eha vioral equiv alences. The follo wing theorem sho ws that bisimilar pro cesses are b arb ed equiv alent. Theorem 6.1. F or an y t wo pro cesses P and Q in π N , if P ∼ Q , th en P ˙ ≈ Q . Pr o of. By Lemmas 5.8 and 5.9, w e see that P ∼ Q implies ( ν z )( P | R ) ∼ ( ν z )( Q | R ) for an y R and z . Usin g th e fact ∼⊆ ˙ ∼ obtained in Lemma 5.3, we get from Lemma 4.8 (2) that ∼⊆ ˙ ≈ , th u s ﬁnish ing the pro of. Remark 6.2. In the π -calculus, it is w ell kno w n that strong barb ed equiv alence coin- cides with strong bisimilarity; see, for example, Theorem 2.2.9 in [43]. Ho w eve r , in the π N -calculus the co nv erse of the ab o ve theo r em is n ot true in general. In o th er w ords, t wo barb ed equiv alen t pr o cesses ma y n ot b e bisimilar. F or example, let us consider the barb ed equ iv alent pro cesses P def = x ( w ) . [ w = y ] τ and Q def = x ( w ) . [ w = z ] τ in Remark 4.10, and kee p the h yp othesis of related c hann el matrices. Then w e ﬁn d that P { xy − → 1 τ } and Q { xy − → 1 0 } . This shows us that P ≁ Q . Obvio u sly , this non -coincidence of barb ed equiv alence and bisimilarit y arises f rom the noise of channel x .  W e con tinue to discuss the relationship b et ween barb ed congruence and full b isimi- larit y . T o this end, we n eed one more concept. Deﬁnition 6.3. An equiv alence relation R on pr o cesses is said to b e a pr o c ess c ongru- enc e if ( P , Q ) ∈ R implies ( C [ P ] , C [ Q ]) ∈ R for eve r y pr o cess con text C . The follo wing is an easy consequence, whic h is useful for c hec king pro cess congruence. Prop osition 6.4. An equiv alence relation R is a pr o cess congruence if and only if it is preserve d b y all element ary context s. The n ext ob s erv ation giv es a basic pr o cess congruence; its pro of follo ws immediately from the d eﬁnition of structur al congruence. Prop osition 6.5. T h e structural congruence ≡ is a p r o cess congruence. As an immediate consequence of Deﬁnition 4.9, Pr op osition 6.5, and Lemma 4.6, w e ha ve the follo win g. Corollary 6.6. If P ≡ Q , then P ˙ ≃ Q . F or subs equ en t need, we show that ≃ is also a pro cess congruence. Lemma 6.7. ≃ is a p ro cess congruence, and m oreo v er, it is the largest pro cess congru- ence included in ∼ . 25 Pr o of. W e ﬁrst show that ≃ is a p ro cess congruen ce. Sup p ose that P ≃ Q , i.e., P σ ∼ Qσ for ev ery substitution σ . By Prop osition 6 .4 , we only need to prov e that for every elemen tary con text C and an y sub stitution σ , C [ P σ ] ∼ C [ Qσ ]. F or C = π . [ ] + M , if the preﬁx π is of form xy , τ , or [ x = y ] π , then it follo w s directly f r om P σ ∼ Qσ th at C [ P σ ] ∼ C [ Qσ ]. In the ca se C = x ( z ) . [ ] + M , we see that C [ P σ ] = x ( z ) .P σ + M and C [ Qσ ] = x ( z ) .Qσ + M . Because z is not str ongly b ound in P σ or Q σ , w e get by P ≃ Q that for any z ′ , P σ { z ′ /z } ∼ Qσ { z ′ /z } . This give s rise to x ( z ) .P σ ∼ x ( z ) .Qσ , and th us C [ P σ ] ∼ C [ Qσ ]. By carryin g out an analysis of the transition group rules r elated to comp osition, restriction, and replicati on , it is routine to c hec k that C [ P σ ] ∼ C [ Qσ ] holds for the other four element ary context s , and we do not go int o the details. By d eﬁnition, w e see that ≃⊆∼ . W e no w v erif y that ≃ is the largest pro cess congru- ence included in ∼ . Let ∼ ∗ b e an arbitrary p r o cess congruen ce includ ed in ∼ . Supp ose that P ∼ ∗ Q and σ = { y 1 , . . . , y n /x 1 , . . . , x n } . Without loss of generalit y , we assume that there is a noiseless channel s with s 6∈ fn ∗ ( P σ, Qσ ), and set C def = ( ν s )( sy 1 . . . . .sy n | s ( x 1 ) . . . . .s ( x n ) . [ ]) . Then C [ P ] ∼ ∗ C [ Q ], hence C [ P ] ∼ C [ Q ]. Notice that C [ P ] { ( τ − → 1 ) n ( ν s ) P σ ≡ P σ } , where ( τ − → 1 ) n is the n -fold comp osition of τ − → 1 , therefore C [ Q ] { ( τ − → 1 ) n ∼ P σ } . But C [ Q ] { ( τ − → 1 ) n ( ν s ) Qσ ≡ Qσ } only , so P σ ∼ Qσ . W e thus get that P ≃ Q , and hence ∼ ∗ ⊆≃ , as d esired. Based on the previous lemmas, we can pr o v e the next r esult. Theorem 6.8. F or an y t wo pro cesses P and Q in π N , if P ≃ Q , th en P ˙ ≃ Q . Pr o of. W e see f r om Lemma 6.7 that ≃ is a pro cess congruence included in ∼ . Since ∼⊆ ˙ ≈ b y T heorem 6.1, ≃ is a pr o cess congruence included in ˙ ≈ . By deﬁn ition, ˙ ≃ is the largest pro cess congruence included in ˙ ≈ , therefore we ha v e that ≃⊆ ˙ ≃ , ﬁnishing the pro of of the theorem. It would b e exp ected that ˙ ≃ ⊆≃ . Ho wev er , this is not true, as we s h all s ee. Remark 6.9. Lik e Theorem 6.1, th e con ve r se of the ab o ve theorem is not true in general, that is, t wo barb ed congruen t p ro cesses ma y not b e full bisimilar. Ev en t wo barb ed congruent pr o cesses ma y not b e bisimilar in the π N -calculus. F or example, tak e P def = ( ν y ) xy .xy and Q def = ( ν y ) xy . ( ν y ) xy , and sup p ose, for simplicit y , that the channel x is n oiseless. It is easy to c hec k by ind uction on context C that C [ P ] ˙ ∼ C [ Q ] holds for an y context . Consequently , P ˙ ≃ Q b y deﬁ nition. Nev ertheless, notice that there is a transition group P { x ( y ) − → 1 xy } and the only transition group of Q making P ∼ Q p ossible is Q { x ( y ) − → 1 ( ν y ) xy } . But it is obvious that xy ≁ ( ν y ) xy . Hence, P ≁ Q .  Finally , based on our resu lts in Sections 4–6, w e sum marize the h ierarc hy of the b ehavio r al equiv alences in the π N -calculus, whic h is d epicted in Figure 1 (b ). Theorem 6.10. In the π N -calculus, (1) ≃⊆ ˙ ≃ ⊆ ˙ ≈ ⊆ ˙ ∼ ⊆ ≏ and ≃⊆∼ ⊆ ˙ ≈ ; eac h of the inclusions can b e s trict. (2) Neither ∼⊆ ˙ ≃ nor ˙ ≃ ⊆∼ holds. 26 7 Conclusion This pap er is d ev oted to a hierarc hy of b eha vioral equ iv alences in the π N -calculus, the π -calculus with n oisy c h annels. Firs t, we hav e devel op ed an early transitional semantic s of the π N -calculus and pro vid ed t w o present ations of the transition rules. It is w orth noting that this seman tics is not a directly translat ed v ersion of the late seman tics of π N in [54 ], and we hav e found that not all b ound n ames are compatible w ith alpha- con ve r sion in the noisy en vironment, whic h is a striking dissimilarit y b etw een π N and π . A s a r esu lt, an O p en-Inp rule for inputting b ound names is required. Then we ha ve in tro du ced some notions o f b eha vioral equiv alences in π N , includin g reduction bisimilarit y , barb ed bisimilarit y , barb ed equiv alence, b arb ed congruence, bisimilarit y , and full b isimilarit y . Some basic p rop erties of them h a ve also b een stated. Finally , w e ha ve established an in tegrated hierarch y of these b eha vioral equiv alences. I n p articular, b ecause of the noisy natur e of c hannels, the coincidence of bisimilarit y and barb ed equiv alence, as w ell as the coincidence of full bisimilarit y and barb ed congruence, in th e π -calculus do es not hold in π N . There are some limits and pr oblems arising fr om the p resen t w ork w hic h are w orth further studying. W e only present a hierarc hy of strong b eha vioral equiv alences; the cor- resp ond ing wea k v ersion that ignores in visible inte r nal actions is a researc h topic. Some algebraic la ws and axiomatizatio n s of these b ehavio r al equiv alences are interesting prob- lems for futur e researc h. A hierarc hy of b eha vioral equiv alences in some su b calculus (for example, asyn c hronous π -calculus where async hr onous observers are less discrimin ating than syn c hronous observers [7, 13, 17, 18, 24]) is yet to b e addr essed. Note that the con ve r se statemen ts of Theorems 6.1 and 6.8 cann ot h old. Hence, a necessary and su ﬃ- cien t cond ition for the con v ers e statemen ts to b e true is desirable. 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