Strong bisimilarity can be opened

Strong bisimilarity can be opened Henning E. Andersen Hans Hüttel Karina N. Jensen June 7, 2002 Abstract We present an extension of the semantics of the π-calculus without match where strong bisimilarity in the extended semantics is a process congruence and coincides with open bisimilarity in the usual semantics. In fact, in the extended semantics, strong, late, early and open bisimilarity coincide and coincide with barbed congruence. This result leads to a simple axiom system for the finite fragment of the π-calculus without match. 1 Introduction One of the goals of a process calculus is to allow equational reasoning about processes, reducing questions of behavioural equivalence to equality in a suitable equational theory. An important requirement of a notion of behavioural equivalence is that it is a congruence, i.e. preserved under all syntactic operators of the calculus. Since the appearance of the π-calculus, various notions of bisimulation equivalence have been defined for the π-calculus. The obvious notion of bisimilarity is strong bisimilarity. Processes P and Q are strong bisimilar if any transition P α P can be matched by a transition Q α Q such that P and Q are strong bisimilar. While intuitively pleasing, strong bisimilarity is not a congruence, nor does it take the value-passing nature of the π-calculus into account. Late and early bisimilarity were therefore proposed by Milner et al. in [5]. Unfortunately, these notions of equivalence also fail to be congruences; in particular, they are not preserved by input prefix. The remedy for this is to identify and characterize the maximal process congruences contained in late and early bisimilarity. A third possibility is barbed bisimilarity [6] by Milner and Sangiorgi. Barbed bisimilarity also fails to be a process congruence; again, the remedy is to define the maximal process congruence contained in barbed bisimilarity. Department of Computer Science, Aalborg University, 9220 Aalborg Ø, Denmark. E-mail hans@cs.auc.dk 1

Subsequently, in [9], Sangiorgi proposed open bisimilarity which has the pleasant property of being preserved by all process constructs. The remedy is here to require that matching transitions must match under all name substitutions. Since then, a number of subcalculi have been identified for which the four notions of bisimilarity coincide. Most importantly, this is case for the asynchronous π-calculus [2, 4, 3]. Here, a process does not continue after an output action. Moreover, Sangiorgi has identified a private π-calculus where the equivalences coincide. In the private π-calculus, any output action must output a fresh name. In [1] Boreale and Sangiorgi consider the problem of giving a labelled characterization of barbed congruence for the π-calculus without the match construct. This calculus forms a natural subcalculus. In particular, programming languages based on the π-calculus and related calculi such as Pict [7] do not have a match construct. In this paper we also consider the π-calculus without match or mismatch. We present a new semantics of this calculus which has the property that strong bisimilarity in the new semantics is a process congruence and coincides with open bisimilarity in the usual semantics of the π-calculus for a calculus without the match and mismatch constructs. This result allows us to define a simple axiom system for open bisimilarity. Moreover, early congruence and barbed congruence still coincide, giving us a simple labelled characterization of barbed congruence in our extended semantics. 2 The π-calculus without match 2.1 Syntax We shall consider a version of the monadic π-calculus where the syntactic categories are names, N, ranged over by a, b, c,..., x, y, z..., actions, Act and processes, P π. Definition 2.1 The syntax of processes and actions is defined as follows: Actions: α ::= x(b) x a τ Processes: P ::= 0 α.p P 1 + P 2 P 1 P 2 (νx) P!P In an action x(b) or x a we call x the subject and a the object. Restriction and input prefix both act as name binders. The set of bound names in P is denoted by bn(p ). If processes P and Q are α-convertible, we write P α Q. A name occurrence which is not bound is free. We let n(p ) = fn(p ) bd(p ). A substitution is a function σ : N N which, simultaneously renames the free names in a process. We write σ = x = y if σ(x) = σ(y). 2

A process context is a process expression with a single hole: Contexts: C ::= [] α.c P + C C + P P C C P (νx) C!C If C is a process context, C[P ] denotes the process where P replaces the hole. A relation is a process congruence if it is preserved in all process contexts. Definition 2.2 Let = be an equivalence relation over P π. = is a process congruence if whenever P = Q we have that C[P ] = C[Q] for any process context C. 2.2 Structural congruence Structural congruence describes the identifications that must always be made, such as the order of parallel components being unimportant. Definition 2.3 Structural congruence over P π, denoted, is the least process congruence over P π, which satisfies the following clauses: P Q if P α Q!P P!P P + 0 P P + Q Q + P P + (Q + R) (P + R) + Q P 0 P P Q Q P P (Q R) (P Q) R (νx) 0 0 (νx)y P (νy)x P 2.3 Semantics (νx) (P Q) P (νx) Q, if x / fn(p ) The standard labelled transition semantics for the π-calculus is given by (P π, Act, ) where transitions are of the form P α Q.The transition relation is defined by the rules in table 1. 3 Bisimulation equivalences For any labelled transition semantics of the π-calculus we can define a number of bisimulation equivalences. 3

(Pre) α.p α P (Par) P α P P Q α P Q bn(α) fn(q) = (Sum) P α P P + Q α P (Res) P α P (νx) P α (νx) P x / n(α) (Com) P x a P Q x(b) Q P Q τ P {a/b}q (Str) P α P Q α Q P Q and P Q 3.1 Strong bisimulation Table 1: Standard labelled transition rules Definition 3.1 Let (P π, Act, ) be a labelled transition system and let R be a symmetric binary relation over P π. R is a strong bisimulation over (P π, Act, ) if P RQ implies that if P α P then there exists a Q, with the property that Q α Q and P RQ. Definition 3.2 P, Q P π are strong bisimilar, denoted P Q, if there exists a strong bisimulation R such that P RQ. 3.2 Late and early bisimilarity Late and early bisimilarity originated in [5] Definition 3.3 A symmetric binary relation R over P π is a late bisimulation, if P RQ implies that 1. if P x(b) P, then for some Q, Q x(b) Q and for all names a we have {a/b}p R{a/b}Q. 2. if P α P and α is any other action, for some Q, we have Q α Q and P RQ. Definition 3.4 Processes P and Q are late bisimilar, denoted P L Q, if there exists a late bisimulation R, such that P RQ. If one reverses the existential quantifiers in the input clause one obtains early bisimulation. Definition 3.5 A symmetric binary relation R over P π is an early bisimulation, if P RQ implies, that 4

1. if P x(b) P, for all names a, there exists a process Q, such that Q x(b) Q and {a/b}p R{a/b}Q. 2. if P α P for α is any other action, then there exists a process Q, such that Q α Q and P RQ. Definition 3.6 Processes P and Q are early bisimilar, denoted P E Q, if there exists an early bisimulation R, such that P RQ. Late bisimilarity implies early bisimilarity, which in turn implies strong bisimilarity. The inclusions are strict. Proposition 3.7 E L An essential property which a behavioural relation should satisfy is that it is preserved by all syntactic constructs. Neither strong, late nor early bisimilarity has this property as the following standard example shows. Example 3.8 Let x, y N be distinct names and x and y stand for x a.0 and y(b).0, respectively. We then have that x y L x.y + y. x as R = {( x y, x.y + y. x), (y, y), ( x, x), (0, 0)} is a strong bisimulation. However, a(x).( x y) E a(x).( x.y + y. x). The counterexample exploits that early bisimilarity is not preserved under name substitution. In particular, if a substitution σ has σx = σy, ȳ y ȳ.y + y.ȳ as ȳ y τ 0. Definition 3.9 Processes P and Q are late congruent, denoted P L Q, if for all substitutions σ we have that σp L σq. Definition 3.10 Processes P and Q are early congruent, denoted P E Q, if for all substitutions σ we have that σp E σq. 3.3 Barbed congruence Barbed bisimilarity originated in [6]. Definition 3.11 The barbs of a process P are defined by P m if P my P for some y P m if P m y P for some y Definition 3.12 A symmetric binary relation R på P π is a barbed bisimulation, if P RQ implies that 1. For all µ P µ iff Q µ 2. if P τ P, for some Q, we have Q τ Q and P RQ. 5

Definition 3.13 Processes P and Q are barbed bisimilar, denoted P Q, if there exists a barbed bisimulation R, such that P RQ. Barbed bisimilarity fails to be a congruence for the same reasons as late and early bisimilarity. Again, the solution is to define the congruence explicitly. Definition 3.14 Processes P and Q are barbed congruent, denoted P Q, if for all contexts C we have that C[P ] C[Q]. Barbed congruence coincides with early congruence [8]. Theorem 3.15 = E 3.4 Open equivalence Open bisimilarity, proposed by Sangiorgi [9], is born a congruence since the universal quantification over substitutions occurs within the clauses for matching transitions. Definition 3.16 A symmetric binary relation R på P π is a open bisimulation, hvis P RQ for alle substitutions σ, implies, at 1. if σp α P, then for some Q, σq α Q and P RQ. Definition 3.17 P, Q P π are open bisimilar, denoted P O Q, if there exists an open bisimulation R, such that P RQ. The congruence problem of Example 3.8 now disappears. Example 3.18 We noticed before that {x/y} x y allows a τ-action. At the same time, notice that {a/y} x.y + y. x cannot match this τ-action, no matter whether a = x or a x. Thus x y O x.y + y. x. A strong bisimulation which is closed under substitutions is in fact an open bisimulation. Definition 3.19 A binary relation R over P π is closed under a substitution σ, if P RQ implies, that σp RσQ. Theorem 3.20 Assume that a binary relation R over P π satisfies the following conditions: 1. R is a strong bisimulation. 2. R is closed under all substitutions. Then R is a open bisimulation. 6

(Com2) P x a e P Q y(b) e Q P Q (x=y)τ e P {a/b}q Table 2: The (Com2) rule The reverse implication is false; a counterexample can be found in [9, Section 3.3]. A main result of [9] is Theorem 3.21 Open equivalence is the maximal strong bisimulation, which is closed under all substitutions. Open bisimilarity respects all other process constructs [9], so we have Theorem 3.22 Open equivalence is a process congruence. However, even though we now no longer need to pass via a notion of congruence closure, we still have to live with a universal quantification over substitutions. The rest of our paper is devoted to overcoming this problem. 4 An extended semantics for the π-calculus The problem in example 3.8 boils down to the fact that the process x y is capable of a τ-action under some substitutions whilst x.y + y. x will never be able to perform a τ-action. We deal with this problem by extending the labelled transition semantics with the action(x = y)τ, henceforth referred to as a conditional τ - action, and a corresponding transition rule. Intuitively, P (x=y)τ P means that P τ P if x is instantiated to the same name as y. Definition 4.1 The extended semantics of the π-calculus is given by the labelled transition system (P π, Act {(x = y)τ}, e ), where e is the least relation closed under the rules in Tables 1 and 2. Strong, early, late and open bisimilarity are defined as before, the only difference being that we now consider the transition system (P π, Act {(x = ) y)τ}, e. The corresponding equivalences are denoted with the superscript e, so e.g. e is strong bisimilarity in the extended semantics. In the extended semantics, the congruence problem now goes away at the level of strong bisimilarity. 7

Example 4.2 x y og x.y + y. x are no longer strong bisimilar in the extended semantics, i.e. x y e x.y + y. x. For x y (x=y)τ e whereas x.y + y. x (x=y)τ e. 4.1 Operational correspondence To show our result we need to characterize the exact correspondence between the original and the extended semantics. Lemma 4.3 Let P P π. 1. If σp β P, either i P α e P, where β = σα and σp = P, or ii P (x=y)τ e P, where σ = x = y and σp = P. 2. If P (x=y)τ e P, for any substitution σ = x = y, we have that σp τ σp. 3. P α e P iff P α P, whenever α (x = y)τ. Lemma 4.4 tells us that transitions are preserved under substitution. Lemma 4.4 If P α e P,, then σp β e σp, where β = σα. 4.2 Strong bisimilarity is a process congruence We now show that in the extended semantics we have that strong bisimilarity is a process congruence. Our proof proceeds by establishing that strong bisimilarity in the extended semantics coincides with open bisimilarity according to the standard semantics. Since open bisimilarity is a congruence for the π-calculus without match, the result then follows. Theorem 4.5 In the π-calculus without match, e Q = O. Theorem 4.5 is shown via lemmas 4.6 and 4.7. Lemma 4.6 Let P, Q P π. If P O Q, then P e Q. Proof:We show that the relation R o = {(P, Q) : P O Q} is a strong bisimulation in the extended semantics. So let (P, Q) R o. Since (P, Q), for any substitution σ if σp β P, there exists aq, such that σq β Q andp O Q (1) 8

i.e. (P, Q ) R o. Since P og Q are open bisimilar, for any substitution σ transitions of P and σq can be matched. When σp β P in (1), by Lemma 4.3, this transition corresponds to one of the following two extended transitions: (a) P α e P, where σα = β and σp = P (b) P (x=y)τ e P, where σ = x = y and σp = P The actual case depends on σ. If the β-transition in (1) is not caused by σ, P α where σα = β,, the transition of case (a) is enabled. Conversely, if the β-transition is caused by σ, then β = τ and must correspond to the transition case (b). Also by Lemma 4.3 the transition σq β Q of (1) corresponds to one of the following two extended transitions: (a) Q α e Q, where σα = β and σq = Q (b) Q (x=y)τ e Q, where σ = x = y and σq = Q If the β-transition of (1) is not caused by σ, both P and Q are capable of an α-transition, where σα = β,. We thus get if P α e P there exists a Q, such that Q α e Q and P O Q where σα = β, σp = P and σq = Q. The existence of Q is guaranteed by (1). If the β-transition of (1) is caused by σ, then β = τ and as before if P (x=y)τ e P there exists a Q, such that Q (x=y)τ e Q og P O Q where σ = x = y, σp = P og σq = Q. It now only remains to be shown that P e Q, which corresponds to showing that (P, Q ) R o. Since (P, Q ) R o, P = σp og Q = σq, we get (σp, σq ) R o. I.e. σp O σq. Since open bisimilarity is a process congruence, we get that P O Q and consequently (P, Q ) R o. Lemma 4.7 Let P, Q P π. If P e Q, then P O Q. Proof:Consider the relation R u = {(P, Q) : P = σp 1 and Q = σq 1, where P 1 e Q 1 } 9

We show that R u is an open bisimulation. So assume (P, Q) R u. Since open bisimilarity in the extended semantics implies open bisimilarity in the standard semantics, we show that P and Q are open bisimilar in the extended semantics. By definition this amounts to showing that for any σ 1 if σ 1 P β e P, there exists a Q, such that σ 1 Q β e Q and P O Q (2) Let σ 1 be a substitution. We now consider the transitions of the processes σ 1 P og σ 1 Q of (2). As (P, Q) R u, we have that σ 1 P = σ 1 σp 1 = σ 2 P 1 and σ 1 Q = σ 1 σq 1 = σ 2 Q 1 for any σ, and that σ 2 = σ 1 σ og P 1 e Q 1. From the definition of strong bisimilarity we then get α If P 1 P 1, then there exists a Q α 1, such that Q 1 Q 1 and P 1 e Q 1 (3) Since lemma 4.4 is valid for any substitution we have that the α-transition α P 1 e P 1 in (3) implies σ 2 P 1 β e σ 2 P 1, where β = σ 2 α Consequently, since σ 2 P 1 = σ 1 P, we have σ 1 P β e P, where β = σ 2 α and P = σ 2 P 1 (4) Similarly, lemma 4.4 tells us that the α-transition Q 1 e Q 1 in (3) implies that β σ 2 Q 1 e σ 2 Q 1, where β = σ 2 α and since σ 2 Q 1 = σ 1 Q we have that σ 1 Q β e Q, where β = σ 2 α and Q = σ 2 Q 1 (5) By applying (4) and (5) to (3) we get if σ 1 P β e P, there exists a Q, such that σ 1 Q β e Q and P e Q where β = σ 2 α, P = σ 2 P 1 og Q = σ 2 Q 1. The existence of Q follows from (3) and Q = σ 2 Q 1. We must also show that P O Q ; this amounts to showing that (P, Q ) R u. By (3) P 1 e Q 1, i.e. (P 1, Q 1 ) R u. Since R u contains all substitution instances of strong bisimilar process pairs, (P, Q ) R u, da P = σ 2 P 1 og Q = σ 2 Q 1. In conclusion, R u is an open bisimulation. α Corollary 4.8 For the π-calculus without match we have that e = e O 10

In the extended semantics, barbed bisimilarity and barbed congruence are defined as usual, with an extended notion of barbs. Definition 4.9 The set of extended barbs of a process is defined by P m if P my P for some y and P P m if P m y P for some y and P P (x=y)τ if P (x=y)τ P for some P In the extended semantics, barbed congruence still coincides with early congruence. Theorem 4.10 e = e E Proof:Similar to that for Theorem 3.15 in [8]: We show that if P E Q then for some finite summation M and some fresh name s we have that (ν z)(p (M + s)) (ν z)(q (M + s) where z fn(p, Q). We must now also consider the transition label (x = y)τ; this extension is straightforward. Corollary 4.11 For the π-calculus without match we have that [e] = e L = e E = e = e O. Proof:Since the intermediate inclusions [e] e L E e also hold in the extended semantics and since Corollary 4.8 holds, the result follows. 5 An equational theory for the replication-free fragment Corollary 4.11 immediately gives us a sound and complete axiom system for open bisimulation in the replication-free fragment of the π-calculus with the extended semantics the one for strong bisimulation. The new feature of the axiom system is that we must take the conditional τ-actions into account. For this reason, we also extend to Act ext = Act {(x = y)τ x, y N }. The proof system is shown in Table 3. Definition 5.1 In the extended semantics of the π-calculus without match, P = Q if P = Q is provable within the axiom system of Table 3. Theorem 5.2 In the π-calculus without match P e O Q iff P = Q 11

(Struct) P Q P = Q (Prefix) P = Q α.p = α.q α Act ext (Par) P = Q P R = Q R (New) P = Q (νp ) = (νq) (Sum) P = Q P + R = Q + R (Expand-1) x(y).p + w z.q = x(y).(p w z.q) + w z.(x(y).p Q) + (x = w)τ.({z/y}p Q) (Expand-2) x(y).p + x z.q = Table 3: Proof rules for e O x(y).(p x z.q) + x z.(x(y).p Q) + τ.({z/y}p Q) without replication Proof:The soundness part of the theorem follows from the fact that all axioms preserve e. The completeness part of the theorem amounts to establishing a head normal form (hnf) and establishing that For any P P π there exists a P 1 in hnf such that P = P 1 For any P, Q in hnf, if P Q then P = Q. 6 Conclusion and discussion In this paper we have presented an alternative semantics for the π-calculus without match. The new semantics adds an extra action, the conditional τ, to the labelled transition relation. The result is that strong bisimilarity and open congruence coincide. Since early and barbed congruence still coincide, all four notions of bisimilarity coincide for this semantics, giving us a simple labelled characterization of barbed congruence. An obvious question to ask is whether the result can be extended to a larger π-calculus, i.e a calculus with match and possible also mismatch. The answer to this is not simple; if we add conditional transitions for arbitrary actions, we arrive at Sangiorgi s symbolic semantics of the π-calculus [9]. 12

References [1] Michele Boreale and Davide Sangiorgi. Bisimulation in name-passing calculi without matching. In Proceedings of LICS 98. IEEE, Computer Society Press, July 1998. [2] Gérard Boudol. Asynchrony and the π-calculus (note). Rapport de Recherche 1702, INRIA Sofia-Antipolis, May 1992. [3] Martin Hansen, Hans Hüttel, and Josva Kleist. Bisimulations for asynchronous mobile processes. In Insup Lee and Scott A. Smolka, editors, Proceedings of CONCUR 95, volume 962 of LNCS. Springer, 1995. [4] Kohei Honda and Mario Tokoro. An object calculus for asynchronous communication. In Pièrre America, editor, Proceedings of ECOOP 91, volume 512 of LNCS, pages 133 147. Springer, July 1991. [5] Robin Milner, Joachim Parrow, and David Walker. A calculus of mobile processes, part I/II. Journal of Information and Computation, 100:1 77, September 1992. [6] Robin Milner and Davide Sangiorgi. Barbed bisimulation. In W. Kuich, editor, Proceedings of ICALP 92, volume 623 of LNCS, pages 685 695. Springer, 1992. [7] Benjamin C. Pierce and David N. Turner. Pict: A programming language based on the pi-calculus. In Gordon Plotkin, Colin Stirling, and Mads Tofte, editors, Proof, Language and Interaction: Essays in Honour of Robin Milner, 1998. To appear. [8] Davide Sangiorgi. Expressing Mobility in Process Algebras: First-Order and Higher-Order Paradigms. PhD thesis, LFCS, University of Edinburgh, 1993. [9] Davide Sangiorgi. A theory of bisimulation for the π-calculus. Acta Informatica, 33:69 97, 1996. Earlier version published as Report ECS- LFCS-93-270, University of Edinburgh. An extended abstract appeared in the Proceedings of CONCUR 93, LNCS 715. 13