A Partial Ordering Semantics for CCS

A new operatilcrnql semantics for *‘pure” CCS is proposed that considers the parallel operator as a first class one, and permits a description of the calculus in terms of partial orderings. The new semantics (also for unguarded agents) is given in the SOS style via the partial ordering derivation relation. CCS agenis are decomposed into sets of sequential subagents. The new derivations relate sets of subagents, and describe their actions and the causal Dependencies among them. The computations obtained by composing partial ordering derivations are “observed” either as interleaving or partial orderings of events. Interleavings coincide with Miiner’s many step derivations, and “linearizations” of partial orderings are all and only interleavings. Abstract semantics are obtained by introducing two relations of observational equivalence and congruence that preserve concurrency. These relations are finer than Milner’s iz that they distinguish interleaving of sequential nondeterministic agents from their concurrent execution. 1. 2. 3. 4. 5. Introduction.. .................................................................... CCS and its interleaving semantics .................................................. Defining the partial ordering derivation relation ................................. ... Partial ordering many step derivations and equivalences ................................ Conclusions and related work ....................................................... Appendix A ..................................................................... Appendix B ...................................................................... References ....................................................................... 223 229 231 237 248 251 256 261 Q304-3975/90/$03.50 Q 1990-Elsevier Science Publishers B.V. (North-Holland) P. Degano, R. De Nicola, U. Montanari compared by examini g the way in which they describe the fact that events (atomic actions, synchronizations, communications) can be performed concurrently by subparts of a system, i.e., independently from one to another. If we take t the various models of CO rtncy can be divided into two broad based on inre&dcing and se based on true concurrencg?. on interleaving express concurrency among events by saying that in any order. Thus, a t states are assu 1,3,4, 15,30, 22], stress the simplicity of the reason to advocate this a systems and proving mo e [27, 28, 32, 25, its as a sufficient bout concurrent On the other hand, models based on true concurrency use partial ordering of events where concurrency bsence of ordering. Wit framework, 110 g/&al c!ock i ehaviour of a system is e in terms of the causal relat e events performed by subparts of its (see fo xample [26, 24, 36, 40, 3 1, 39, 16, 20, 5, 38, 2, 17, 181) claim that th a more faithful picture of reality, and that certain liveness properties of concurrent systems can be better understood and . . . studied -within this frame-work. A classical representative of models based on interleaving is Milner’s Calculus of Communicating Systems (CCS) 11273. It relier; on a small number of operators which are used z.o build terms. These are considered as agents which, by performing certain actions, will become other agents. The operational semantics of the calculus is given through labelled transition systems, and the fact that agent E0 evolves to El by performing an action pi is rendered by E,-p E,. The technique used (Structured Operational Semantics or SOS [37]) relies on the well-known idea of describing the behaviour of systems by sequences of transitions between configurations. Transitions of compound systems are defined in a syntax-driven way, via axioms and inference rules. Since the original version of CCS was geared towards the interleaving approach, oes not consider the operator for parallel composition of processes “I” as primitive: given any finite process containing 1, there will a!ways exist another process without 1 which exhibits the same behaviour. This paper proposes a new operational semantics for CCS that considers the parallel operator as a first class operator, and offers a partial ordering semantics for the calculus. The operational semantics is still given in the SOS style, but a different transitirz relation, called the partid ordering deritlation rehtion, is defined. This relates sub arts of CCS agents, rather than their whole global state, and carries information about causal dependencies. CCS agents are decomposed into sets of recesses, called g es, and the new t nsitions not only describe the actions agents ma a given state, relatio n the global sta Partid ordering semantics *for CCS 225 derivation relation is defined via inference rules which directly correspond to those of [27-J. Conf!ec;uently, the deduction of both transitions follows the same pattern. The new transitions have the form I, +rP*.*‘Al I2 where I, and I-, represent sets of grapes, and 92 is a relation providing additional information about the causal relations among agents. The grapes in I, perform the action p and evolve to those in I?. We thus say that the grapes of I1 cause those in I?, through p. Information about other grapes caused by grapes in I,, but not through p, is recorded in 3. The intended dynamic meaning is that, after showing an event labelled by k, the set of grapes II, occurring in the current state, can be replaced by the grapes in I2 and by those related to I, via 3, thus obtaining the new state. As an example, consider the CCS agent (a.NILIjXNIL)+y.NIL, which may evolve to NILI&NIL after resolving the nondeterministic choice (expressed by +) in favour of cy. In the interleaving approach, this will be rendered as (a.NILI/&NIL)+ y.NIL: NILI&NIL. (*) We will write it as {(a.NILIP.NIL)+ y.NIL} [tr.(~rr.Nlt~~~.NIL~+y.NILidf&NiL}] {NILlid) where (ui.NILl/3.NILf+ y.NIL, NJL[id and id[P.NIL are grapes. In this way, we describe the fact that grape (~u.NILI/~.NIL)+ y.NIL causes both grape id I ~.Iw. and the event labelled by LY which in turn causes grape NIL1 id. Note that the possibility that id l&NIL may have to perform p independently of the occurrence of ti is implied by the absence of any causal relation between cy and id / @VIL. The cu-derivatit= of grape ja.NILIP.NIL)+ y.NIL is shown in Fig. 1. It should be noted that every derivation of the original cslculus can always be recovered from our partial ordering derivation simply by “putting together*’ its initial and final sets of graphs. In the example above, we obtain NILIP.NIL by putting together the two grapes NIL[id and idip.NIL. A transition of the above form may look a bit unnatural. We are used to conceiving labelled transitions as relations between a set of processes and an action, and between that action and ai/ the new processes. Instead, in the transition {**) above, Fig. 1. The transition of the partial ordering operational semantics :(a.NILIP.PJlL)+ y.NIL)} [fr*~(CX Ir;lLb l\;‘L)+y NiLC IJ’BN’I-)l { NiLi id}. Grapes are represented bq i&elkd boxes, events by labelled circles and the causal relation is expressed tiliough its asse diagram growing downwards. grape idlP.NIL is directly related to grape (as.NILIP.NIL) + y.NIL. This happens because the evolution of this type of nondeterministic processes requires that first one of the alternatives is chosen, and an action of the chosen grapes is performed. A possible way of describi g the above cu-transition is illustrated in Fig. 2a. First, a choice-event causes two concurrent grapes a.NILIid and idlfi NIL; the former then performs an cy. It is however important to kzote that, in order to be faithful to tb,e n trr;ncli ramant-& Vr+Ic+I 8 be decision .Jrl*rurrrrrr,, i* ” and t+ action ~53~ il_, --_11 only be ccQsidered as a. single indivisible action. Since CCS has no mechanisms for defining atomic actions from sequences, we are left with two alternatives. The first requires hiding inside the source grape the decision to obtain transitions such as those of Fig. 1. We would like to stress that this discussion is just for the sake of clarity and does not imply at all introducing any invisible action whatsoever in our semantics. The 3econd alternative is to incorporate the decision into the action itself to obtain the usual transitions (Fig. 2b). In [8, 101, we have followed the latter approach, but it results in an operational semantics that does not take the possible parallelism oi CCS agents fully into account. For example, independencies are lost between some concurrent actions in +-context; in the case of the agent (a.NILIP.NIL)+ y.NIL, a causal relation between a! and p is enforced, thus identifying this agent with a.fl.NIL+P.a.NIL+ y.NIL. A third approach, i%llowed in [ 1 l] and [34] introduces a new decomposition according to which the agent (cx.NIL~/~.NIL) + y.NIL originates two grapes, namely (cw.NILlid)+ y.NIL and (idIP.NIL)+ y.NIL. These papers will be further discussed later in this section and in the concluding one. a) Fig. 2. Alternative descriptions of the a-transition of agent (a.NILj @NIL) + y.NIL. A computation is a sequence of sets of grapes (i.e., system states corresponding to CCS agents), and of partial ordering derivations (i.e., system transitions). A computation of agent (a.NIL(P.NIL) + y.NIL is Partial ordering semantics for CCS 227