Broadcasting in Time - Semantic Scholar
Broadcasting in Time K. V. S. Prasad? Department of Computing Science Chalmers University of Technology S- 412 96 Gothenburg, Sweden E-mail: [email protected].
December 29, 1995
Abstract. In the calculus of broadcasting systems (CBS), speech is autonomous, but hearing takes place only when the environment speaks. This paper develops a timed CBS (TCBS) where no time may pass if there is speech pending. A process wishing to speak can be forced, by attaching a time-out to it, to rst listen for a speci ed length of time. Those forced to wait for a long time can be seen as having low priority. This reading is consistent with PCBS, which is CBS with priorities. TCBS has an expansion theorem even if time is dense. A delay pre x operator can be derived up to weak bisimulation if time dependent behaviour is allowed. TCBS casts, by contrast, light on handshake communication: on time abstracted bisimulation, and on the relations between time and priority, between expansion theorems and density of time, and between delays and time-outs.
1 Introduction Broadcast vs. handshake. CBS [Pra95], a Calculus of Broadcasting Systems, is
a simple and natural CCS-like calculus [Mil89] where processes speak one at a time and are heard instantaneously by all others. Speech is autonomous, contention between speakers being resolved non-deterministically, but hearing only happens when someone else speaks. CBS di ers from almost all other process calculi, which use handshake (or rendezvous) communication. The change in communication primitive has far reaching consequences. The rst is that there is a natural way to run a CBS process: return a list of values it broadcasts. This allows CBS to be both a process calculus and a coordination language. It provides a small set of process constructors and functions to run these processes, both of which can be implemented on top of a host language, typically a functional language such as Haskell or SML. By contrast, there is no obviously correct way to stand back and let a CCS process run; in a straightforward interpretation, the user must interact with the process at each step. [Pra95] develops the basic theory of CBS and gives several programming examples. ?
Funding: Swedish Government agency TFR and Esprit BRA CONCUR2.
[Pra95] also develops PCBS, which is CBS with priorities. Processes that wish to speak do so at some priority. This resolves contention between utterances of di ering priorities. Hearing has no priority associated with it, since what a process hears is anyway decided by the environment. Things are much harder in CCS with priorities [CH90, CW91]. Here there is only one autonomous action, a completed handshake (written ) which is silent. Priorities are associated with potential (i.e., non-autonomous or controlled) actions. This leads to obvious problems. A process p can do actions a or b and prefers a, say, but what use is this if the environment only allows b? Worse, what if the environment allows both actions, but prefers b? Translated to a timed setting, the topic of the present paper, the di erence in communication primitive can be illustrated as follows. Consider the following classroom situation: the teacher gives the class two minutes to think about a question. The quickest student will answer, unless even this person takes more than two minutes, in which case the teacher will preempt the class. This can be described as a timed broadcast system with maximal progress, because people speak as soon as they are ready to. Handshake communication is illustrated by a marketplace, where buyer and seller must agree before a sale goes ahead. Each must wait for the other to appear; neither can expect in general to perform an action when it is ready. Timed process calculi. There is now a plethora of these. A tutorial, [Hen92],
notes that there is very little di erence between untimed and timed languages. Current research nds little to say about the setting up of timed calculi, and focusses instead on more esoteric issues such as the abstract structure of time domains, the diculties caused by dense time, and the like. But almost all timed calculi are based on handshake communication,even if occasional postulates such as \urgent actions" reveal the beginnings of dissatisfaction with it. As one might guess, setting up a timed calculus based on a di erent communication model turns out to be an instructive exercise. Main results. This paper develops TCBS, a timed extension of CBS. TCBS shares several properties with existing timed calculi such as the Timed CCS of [Yi91] or TEPL, the simple timed calculus introduced in [Hen92]. As foreshadowed, it turns out that TCBS, a calculus with maximal progress and a time-out operator, is easier to set up than than timed handshake calculi with maximal progress [Yi91, HR90, Hen92]. Other main results are: a priority interpretation of TCBS is consistent with PCBS; TCBS has an expansion theorem even if time is dense. Delays and time-outs both t into a more detailed calculus where each process keeps track of time. Yet the simpler TCBS without time dependence is expressive and powerful enough for many purposes. It is easier to read time as priority than the other way around, which suggests that time is more basic than priority, at least in this communication model. [Pra94] shows that strong bisimulation is the largest congruence contained in a bisimulation equivalence that ignores priority. The corresponding result for time is similarly simple for TCBS; it is a main result of [LY93].
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Related work. There have been few comparisons of timed CCS with prioritised CCS. However, [Jef92b] gives a translation of timed algebra into a prioritised algebra where every action is time stamped. Some confusion surrounds the expansion theorem for timed CCS [GL92, Hen92, Yi90, Yi91]. One in uential view is that \in order to have expansion theorems in real-timed calculi, one should focus on calculi with time variables or similar notions" [GL92]. This is misleading. An expansion theorem for CCS is awkward with the delay operator (which is what [GL92] use) regardless of the density of time. To get an expansion theorem with the more powerful time-out operator (in CCS, the delay operator can be derived from the time-out), the density of time is indeed signi cant. Time variables, or some other means of keeping track of time, are needed if time is dense, but not if it is discrete. Readership. Readers are assumed to be familiar with CCS [Mil89]. Familiarity with some timed process calculus and with one of [Pra93, Pra94, Pra95] would help.
2 The Syntax and Semantics of TCBS TCBS is an extension of CBS [Pra95]. The treatment below is largely self contained, but concentrates only on the timed aspects. The syntax and communication actions of TCBS processes are given in Table 1. TCBS is a framework or a coordination language, not a complete programming language. No syntax or computation rules are given for data expressions. The user chooses these for the application at hand. The evaluation of data is not represented, but is assumed to terminate, and closed data expressions merely stand for their values. Types. Given the datatype , the syntax de nes inductively the set Proc . Not every type is permissible here: it must be possible to determine when two elements of are equal. Further, may not itself involve the type Proc for any , that is, this paper is restricted to rst order TCBS. A translator : Proc ! Proc is speci ed by a pair of functions " : ! and # : ! . That is, p Trans h " ; #i p, where Trans is the translation operator, always dropped in writing. Informally, " and # are treated as projections yielding the components " and # of a translator . The time domain. This is denoted T. It can be taken to be either the natural numbers or the positive real numbers (including zero). De ning equations and guardedness. This paper assumes that there is an unmentioned set of guarded constant de nitions available for use in process terms. De nition1 Guardedness. Processes of the form 0, !s, ?f or f &s are guarded. If p and q are guarded, so are p j q and if b then p else q. If p is guarded, so are : p and p. If A z = p then A d is guarded if p[d=z ] is. A de nition A z = p is guarded if p[d=z ] is guarded for all d. def
def
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Let be a datatype. Let be a distinguished value, 2= , and be [ f g. Let x: be a variable and w: an expression. Let be another datatype. Let be speci ed by " : ! and # : ! satisfying " = and # = . Let , " 2 T, the time domain (the natural numbers or the positive reals with zero). Let b be a boolean expression. Then the elements of Proc are given by
p : : = 0 !s ?f f & s : p pjp p if b then p else p A d where f :: = [x]p, s: : = hw; pi, p : Proc , and A ranges over constants, declared in (mutually) recursive guarded de nitions A z = p, parameterised by a datatype ranged over by variable z and expression d. The semantics below also uses pi , p0 , p00 , p0i : Proc , p0 : Proc , and values v: , w : , and u: [ T. def
p ?? ! p
Tau
?
Guarded Sum
!hw; pi ??w! p v! !s v! p[v=x] !s ?? ?[x]p ??
f & hw; pi ??w! p v! p[v=x] [x]p & s ?? p ??! p0 p0 ??"! p00 " p ????? ! p00
!
0 ??v! 0 ?
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0 ??! 0 ?f ??! ?f p ??! p
Time
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( + ):
p ??w! p0 0: p ??w! p0
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p ??v! p0 : p ??v! p0
: p ??! p
Timeout
Compose
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( + ") : p ??! ": p :
! ? : ! ? !? ? ! ?? : ? ?:
u]1 0 u]2 p ??? ! p p ??? ! p0 ] ] 6= ? u ]1 ]2 ! p0 jp0 p jp ?????? 1
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# w ! p0 p ???? w p0 p ??!
w ! p0 p ???
Translate
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p ??! p0 p ??! p0 :
w p ????? ! p0 p ??u]! p0 p ??u]! p0 Conditional if true then p else p ??u]! p0 if false then p else p ??u]! p0 p[d=z ] ??u]! p0 De ne A d ??u]! p0 Az = p ], ] , ] range over f! ; ? ; : g. ? means \unde ned" in the synchronisation algebra. 1
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Table 1. The syntax of TCBS and the semantics of closed processes 4
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Abbreviation Meaning x? p ?[x]p w! p !hw; pi x? p & s [x]p & s f & w! p f & hw; pi x? p & w! p [x]p & hw;pi A=p A () = p where () is the only element of the type \Unit". def
def
This paper often uses f v for the function application f (v). NOTE the abuse of notation in the third, fourth and fth lines. In the context of the operator & the subexpressions x? p and w! p stand for the abstraction [x]p and the pair hw;pi respectively; outside of a & context, they stand for the processes ?[x]p and !hw; pi respectively.
Table 2. Syntactic abbreviations, freely used throughout. Since every de nition in this paper is guarded, so is every process. Examples of unguarded de nitions are p = p, p = p, p = 2! q j p and p = 2: p. In the last example, it is not possible to determine p's response to hearing something from the environment|see Proposition 2 below. Thus pre xing by a timeout is not enough to ensure guardedness. Induction on the structure of processes does not work for constant applications, because to prove the hypothesis for A d, the starting point is p[d=z ], which is not smaller than A d. But the proof of guardedness of p[d=z ] is smaller than that of A d. Induction can therefore be carried out on the depth of proof of guardedness of processes (abbreviated \induction on guardedness"). def
def
def
def
Open and closed processes. Let x: be a (data) variable. Occurrences of x in p become bound in the process abstraction [x]p, and the scope of x in [x]p is p. Bound variables are assumed to be renamed as necessary to avoid clashes under substitution. A process is closed if it has no free variables, and is open if it does. Thus x? x! 0 is closed while x! 0 is open. The set of all (open) processes is denoted P, and the set of closed processes Pcl . Let v: be a (data) value and let p[v=x] denote the result of substituting v for x in p. The user has to supply the functions that substitute values for variables in data expressions. These functions extend from to P in the evident way. For example, (w! p)[v=x] = (w[v=x])! p[v=x] and (if b then p else p ) [v=x] = if b[v=x] then p [v=x] else p [v=x]. Only closed processes can communicate. Open ones cannot, by de nition. 1
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w and ??! w over Communication actions. For each w: , there are relations ??! Proc . Also, for each 2 T there is a relation ??! over Proc . These are the !
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least relations satisfying the axioms and inference rules in Table 1. Let ], ] and ] range over f! ; ? ; : g and u over [ T. 1
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Expansion. Section 4 presents an extended guarded sum f & fsi i 2 I g with
several output branches, needed (only) for an expansion theorem. This form is not needed for programming, which only needs at most one output branch as in the syntax presented here.
2.1 Examples and explanation In (T)CBS, speech is autonomous, while hearing is a response to speech by the environment. TCBS incorporates maximal progress: time passes i no process has anything to say, i.e., i there are no autonomous actions available. Thus time actions in TCBS are like hearing in that they are not autonomous. TCBS is built from CBS by augmenting the syntax by time-outs, and the semantics by the rules in the second box of Table 1, and by time instances of the other rules, notably p ??! p0 p ??! p0 p j p ??! p0 j p0 The processes 0 and ?f have nothing to say, and allow any amount of time to pass. If ?f hears v, it evolves to f v. The process f & (5 + 1) ! p wishes to say 6 and become p. If it hears v immediately, it evolves to f v. It does not let time pass. f & (5 + 1)! p ??! p f & (5 + 1) ! p ??! f 4 The process : (f & v! p) times out after units of time by saying v, and becomes p. Upto this time, it is prepared to hear v0 and become f v0 . : (f & (5 + 1) ! p) ??!??!p : (f & (5 + 1) ! p) ??"!??!f 4 for 0 " Attaching a time-out to ?f achieves little; in fact, : (?f ) ?f . The next two examples show that care is needed when programming with time-outs. First, note the behaviour of : w! p. : (5 + 1) ! p??!??!p : (5 + 1) ! p??"!??! (5 + 1)! p for 0 " Thus : (5 + 1) ! p is not a good representation of a process that takes time to evaluate 5+1, for it does the evaluation immediately upon any interruption by the environment! The following attempt is not much better. :
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w! p = : (x? w! p & w! p) def
This yields the behaviour (5 + 1) ! p??!??!p (5 + 1)! p??"!??! (5 + 1) ! p for 0 " where any interruption causes the evaluation to be restarted. A process that takes to evaluate 5+1 is best represented by a delay operator (see Section 6). Readers familiar with [Pra95] or [Pra94] can skip the rest of this section. :
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If both components of a parallel composition wish to speak, one is chosen arbitrarily. j is associative, and communication is synchronous. Whatever is said is heard instantaneously by all processes in parallel with the speaker. ?f j 5! p j y? y! 0??! f 5j pj5! 0 ?f j 5! p j y? y! 0??! f 4j5! pj4! 0 ?f j : (5! p) ??! ?f j 5! p All processes respond deterministically to what they hear; w! p and f & w! p are deterministic also for speech. Contending speakers are the only source of non-determinism. This is captured by the extended guarded sum used for the expansion theorem (Section 4). Finally, a translator is a pair h "; # i. The process p says "5 if p says 5; its response to 4 depends on p's response to #4. Translators do not a ect the passage of time. Hiding and restriction are achieved by translation to , an aside appended to every data type. Asides are heard but always ignored. Translating functions must map to . 5!
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2.2 Properties of the calculus u] u] 0 Letu]p??! mean \9 p0 such that p ??! p ", and let p??u]! = mean \6 9 p0 such that 0 p ??! p ". The propositions below con rm that TCBS is well behaved. First, some untimed properties. See [Pra95] for proofs and more detail.
w p0 . Proposition2 Input enabling and determinism. 8p, w, 9! p0 such that p ??! w p0 . De nition3. p=w, the image of p under w, is the unique p0 such that p ??! ?
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For example, (?f ) =v = f v, p= = p, and (p) =v = p=v w Proposition4 Finite output branching. 8p, the set fw p??!g is nite.
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u] 0 p g is nite. Proposition5 Image nite. 8p, u, ], the set fp0 p ??! To derive 5! 0 j x? 3! 0 ??! 0 j 3! 0, the premise x? 3! 0 ??! 3! 0 is needed. But to derive any ??! transition, no premises are needed that involve a ??v! for any
5!
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v. The next proposition formulates this. w transitions can be derived independently Proposition6. For all w and w0, ??! 0 w of ???! transitions. ?
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Timing properties. All the propositions below can be proved by induction on
guardedness.
Proposition7 Time determinism. If p ??! p0 and p ??! p00 then p0 p00. :
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If the time domain is restricted to non-zero elements, the proposition can be strengthened to yield p p0. But 0: p ??! p and 0: p ??! 0: p, so processes are time deterministic only upto strong bisimulation equivalence , de ned in the next section. Time determinism holds also for Timed CCS and TEPL. 0:
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" Proposition8 Time additivity. p??!??"!r i p ????? ! r. So if p??! and " < (i.e., 9"0 6= 0 such that " + "0 = ) then also p??"!. :
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Additivity is associated with density (\continuity" in [Yi91]). If time is discrete, then : p can be taken to be an abbreviation for 1 : p, ( + ") : p for : (": p), " and ????? ! for ??!??"!. This makes the two \+" rules of Table 1 unnecessary, and the property of additivity a matter of de nition. The property becomes interesting, and the rules necessary, only if a time interval can be divided in nitely nely. The non-zero naturals make a simple time domain for a calculus where further the rule p ??! p is dropped. Such a calculus is enough to convey all the concepts of TCBS except the handling of dense time. Additivity holds for Timed CCS; TEPL does not need it because it is a discrete timed calculus. ( + ):
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Proposition9 Maximal progress. If p??! for 6= 0 then 8w, p??w!= . w and p?? "!? w0! then = ". Corollary 10. If p??!??! ?? :
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Thus each process can be characterised by the maximum amount of time for which it can wait. This can be read as the \priority" of the process; the longer it can wait, the lower it's priority. This is explored in Section 5.
Proposition11 Patience. If 8w p??w!= then 9 6= 0, p??!. !
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Maximal progress and its inverse, patience, are much debated decisions with handshake communication. They seem natural here, as autonomous actions in (T)CBS are local, not composed of handshakes between possibly geographically separated components. A speaker has all the information it needs, and no reason to wait. Conversely, listening is a matter of waiting. Maximal progress and patience hold for TEPL and Timed CCS. They need = . Similarly for the time-out negative premises. In TEPL, p j q??! only if p j q?! operator. [Yi91] codes negative premises into \timed sorts", used to say when the components of a parallel composition can communicate. In TCBS, the basic dichotomy is between 0 and ?f , which allow time to pass, and !s and f & s, which don't. The rest follows naturally from the way autonomy and communication are captured by parallel composition. Thus, as in :
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PCBS, the communication model yields a simple transition system with no need for negative premises. In passing, note that some handshake calculi postulate \urgent actions" (usually with little physical interpretation). These have maximal progress but not patience. De nition12 Alert processes. A translator alerts if 8v # v 6= . A process p is alert if all translators occurring in p alert. The following two properties hold only for alert processes. v q and p ?? ! r then r ??! v q. Proposition13 Persistence. Let p be alert. If p ??! v q. Proposition14 Readiness. Let p be alert. If p ??! r and r ??v! q then p ??! ?
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These properties are illustrated by the process p = 2: 5! 0. Then p ??! 5! 0, v 5! 0 for any value v. but also p ??! To see why alertness is needed, let Deaf be a translator de ned by Deaf " x = x and Deaf # x = . That is, Deaf lets through all outgoing values, but restricts v Deaf (5! 0), but Deaf p ??! v Deaf p. all incoming ones. Then Deaf p??!??! Persistence applies only to non-autonomous actions. Most timed calculi do not have this property. In TEPL, which adds a time-out operator to CCS, the property holds only for the CCS subcalculus. In Timed CCS, which uses a delay pre x, the property holds for all processes, but in a weaker form, corresponding v and p ?? ! r then r?? v!". This is uninteresting for TCBS because of to \if p??! input enabling. Persistence says that time does not decrease input capability. Readiness says it does not increase it either. Readiness seems to be interesting only in TCBS. Processes pre xed by delays, as in Section 7, are not persistent. Delay therefore cannot be expressed in TCBS using only alert processes. Alert processes are \time independent" in the sense that their responses to input are independent of when the input arives. With time dependence, TCBS can express delays. def
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3 Strong Bisimulation De nition15 Strong bisimulation for closed processes. R Pcl Pcl is a strongu]bisimulation if whenever pRu]q, (i) if p ??! p0 then 9q0 such that q ??! q0 and p0Rq0 , u] 0 (ii) if q ??! q then 9p0 such that p ??u]! p0 and p0 Rq0 The largest strong bisimulation is an equivalence, denoted . To show p q, nd a bisimulation R such that pRq. All the laws below are shown this way. Proposition16 Some untimed laws. 1. (a) x? 0 0 (b) (x? w! p) & (w! p) w! p 9
2. (a) 0 X where X = x?X (b) w! p X where X = x?X & w! p 3. (P= ; j; 0) is a commutative monoid. def
def
See [Pra95] for more laws. CBS has no laws corresponding to the CCS + laws of associativity, commutativity, idempotence, and 0-identity, because the guarded sums have at most one output branch, and their input behaviour is deterministic. is extended to process abstractions: [x]p [x]q i 8v: p[v=x] q[v=x]. From [x]p [x]q, it follows that x? p x? q and that x? p & s x? q & s.
Proposition17. is a congruence for TCBS. Proposition18 Timed laws. 1. 0: p p 2. : 0 0 3. : (?f ) ?f 4. : (": p) ( + ") : p 5. : (p j q) ( : p) j ( : q) 6. If alerts, then : ( p) ( : p) The processes generated by 0, !, ?, & and conditionals are called \ nite guarded sums". [Pra95] shows that Laws 1(a) and 1(b) of Proposition 16 are the only axioms needed to axiomatise strong bisimulation for nite guarded sums. Adding timeout to this syntax yields \timed nite guarded sums". The axiomatisation has not yet been carried out for this version of TCBS; it is a conjecture that the above laws are enough. Suitable normal forms seem to be 0, ?f , : (!s) and : (f & s). The above laws are enough to put any timed nite guarded sum into normal form. w\ Time stamped actions. Let \ range over f! ; ? g. The de nition of matches ??!
w\ and ??! separately. In fact, only combinations of the form ??!??! need be considered. A reading is \w\ at time ". Such a reading is useful when reading w\ time as priority (Section 5). The action ??! is represented as a combination where = 0. The action ??! is represented as a combination where w\ = ?. :
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De nition19 Time stamped bisimulation. R Pcl Pcl is a time stamped bisimulationw\if whenever P RQ, w\ 0 (i) if p??!??! p0 then 9q0 such that q??!??! q and p0 Rq0 , w\ w\ (ii) if q??!??!q0 then 9p0 such that p??!??!p0 and p0Rq0 Let the largest time stamped bisimulation be 0 . Proposition20. =0 :
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Proof. 0 is obvious. For the other direction, use time determinism.
A main result of [LY93] shows that time abstracted bisimulation, where the passage of time is ignored, yields an equivalence that is not a congruence. The largest congruence contained in this equivalence is strong bisimulation. For TCBS, this result is easy, as is the corresponding result for PCBS [Pra94]. Note that the only operator involved is j.
De nition21 Time abstracted bisimulation. R Pcl Pcl is a time abstracted bisimulation if whenever P RQ, w\ 0 w\ 0 (i) if p??!??! p then 9q0 , " such that q??"!??! q and p0Rq0 , w\ w\ (ii) if q??!??!q0 then 9p0, " such that p??"!??!p0 and p0 Rq0 The largest time abstracted bisimulation, denoted ', is an equivalence, but not a congruence. For 3! p ' 3! p, yet 3! p j 5! q??!??!, while for any , 3! p j 5! q??!??!= . Let 'c be the largest congruence contained in '. Proposition22. ='c Proof. It is easy to see that w\ ', and is a congruence. For the other direction, w\ 0 let p 'c q, and let p??!??!p0 . Then q??"!??! q , and p0 'c q0. But = ", otherwise a j context can be found to distinguish p and q. Use Proposition 20. :
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4 Expansion For this section, an extended calculus TCBSe is introduced where guarded sums have an output tree, a nite set of output branches instead of just one as in TCBS. The syntax of TCBSe is given by
p : : = !s f & s : p p j p p if b then p else p A d
where f : : = [x]p and s : : = fhwi; pii i 2 I g, where I is a nite set. The semantics below is extended from Table 1; the common parts are not repeated. wi f & s ??? ! pi hwi ; pii 2 s [x]p & s ??v! p[v=x] wi v !s !s ???! pi hwi ; pii 2 s !s ??! Now !; says nothing and loops on all input. It corresponds to 0 in TCBS. Similarly, f &; corresponds to ?f in TCBS. Writing just hw; pi instead of fhw; pig, it is easily seen that TCBS is a subcalculus of TCBSe . TCBSe processes too have a unique response to each input. Again, p=w denotes the image of p under w (see De nition 3). Bisimulation extends naturally to TCBSe , as do the notions of (timed) nite guarded sums. First, the untimed expansion theorem. The nite number of output branches in a sum are enough for the expansion theorem below, since parallel compositions are nite. For legibility, fhwi; pii i 2 I g is often written fwi! pi i 2 I g.
Guarded Sum
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Proposition23 Expansion theorem. Let p, q be nite guarded sums.Then w p0 and r = 0; 1g p j p x? (p =x j p =x) & fw! (p0r j p ?r =w) pr ??! r 0
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1
For example, the process 2! p j 7! q can be expanded into a guarded sum: 2! p j 7! q x? (2! p j 7! q)&f2! (p j 7! q) ; 7! (2! p j q)g Now, assuming the normal forms proposed earlier for timed nite guarded sums, the expansion theorem can be broken up into special cases. Consider : p j ": q where p, q are nite guarded sums. Assume that ". Then : p j ": q : (p j (" ? ) : q). In the case " = , the composition is of the form p j q where both components are just guarded sums. The case < " results in a composition of the form p j : q. Further, if p is of the form ?f , it can be rewritten p : p, and the composition again reduces to one of just guarded sums. Thus there are only two cases to consider, the untimed theorem above, and
Proposition24 Timed Expansion theorem. Let p f & fwi ! pi i 2 I g, and I = 6 ;. Then p j : q x? (p=x j q=x) & fwi! (pi j ( : q) =wi) i 2 I g. The existence of this theorem for TCBS calls for some comment, in the light of the discussion in the introduction. [Hen92] points out that in TEPL, all agents without time-outs are persistent. Let the delay pre x \:" be de ned by the sole axiom : p ??! p. Then : p is a : p, persistent as well. But the process a: 0 j : p is not persistent, for a: 0 j : p ?! a p, i.e., after the time action, the agent no longer has but a: 0 j : p ??! a: 0 j p?! an action to the same target state. Therefore CCS augmented by a delay pre x cannot yield an expansion theorem, even if time is discrete. [Hen92] gets an expansion theorem using the time-out construct p + q which behaves like p for and then becomes q. This is clearly not persistent. [Yi91] gets around the problem by allowing time dependence: (in adapted notation) the action pre x is now written a: f where f is a function from time to a f 0 and a: f ?! a: (t: f (t + )). The process processes. The axioms are a: f ?! a: f is also not persistent. (It is weakly persistent). Thus far it would appear that TCBS follows a path similar to that of [Hen92]. But [Hen92] only considers discrete time. What happens if time is dense? [GL92] motivate their study of dense time by showing that it is hard to get an expansion theorem using the delay pre x. This is misleading, since the impossibility here is independent of the density of time. However, their result holds. Roughly, the problem with dense time is this. For brevity here, CCS processes are written without trailing 0's, i.e., a: 0 is written just a. Consider the agent (a + 0 q) j c. The plausible expansion is (a: c + c: (a + 0 q)) + 0 (q j c) :
:
+
+
12
+
Consider the action
(a + 0 q) j c (a + 0 q) j c ?! The plausible expansion cannot match it (a: c + c: (a + (a: c + c: (a + 0 q)) + 0 (q j c) ?! 0 q)) +0 (q j c) The clock behind the action pre x does not tick. It is dicult to demonstrate the \corresponding" situation in TCBS, but roughly speaking, the problem does not arise because input received by one component is also received by the other. Consider the expansion : (f & v! p) j ?g : (x? (f x j g x) & v! (p j g v)) When the second component receives input, the rst one also loses its time-out. The fundamental di erences in communication model mean that the result of [GL92] does not apply to TCBS. +
+
+
+
5 Interpreting time as priority . To avoid introducing the notation of PCBS, this section instead provides a priority reading of TCBS, and proves the necessary propositions to show that this reading is consistent. An operational de nition of the priority of a process has already been suggested: it is the longest time the process can wait. Corollary 10 shows that this is well de ned. The highest priority processes, with priority 0, do not wait at all before saying what they want to. Processes with nothing to say have the lowest priority, \1", and are willing to wait inde nitely. Thus time actions can be interpreted as pauses to ensure that higher priority processes speak rst. The priority of 3! p is 1 and that of 5! q is 2. In 3! p j 5! q, the lower priority process is prevented from speaking rst because the higher priority process times out before it is ready to do so. The latter needs a further wait of 1 unit, the former refuses, and therefore so does the composition. Now every TCBS process can be assigned a syntactic priority. The proposition to follow checks that this agrees with the operational de nition. De nition25. The priority (p) of a closed process p is given by the rules (0) =1 (?f ) =1 (!s) =0 (f & s) =0 ( : p) = + (p) (p j q) = min( (p) ; (q)) ( p) = (p) (if b then p else p ) = if b then (p ) else (p ) (A d) = (p[d=z ]) where A z = p 1
1
2
2
1
1
2
def
13
2
Proposition26. p??! i (p). w , the proposition above shows that a process Since p??! implies 8w, p??!??! :
:
:
?
is prepared to hear others of priority at least as high as its own. Next, a syntax for a \PCBS" as a restriction of TCBS. Use timeouts only in the forms : !s and : (f & s). The non-alert processes of TCBS cannot be put into this form, so \PCBS" is strictly a subcalculus of TCBS. The terms of \PCBS" annotate speech requests with priorities. The priority of a process as de ned above is merely that of its most urgent speech (a parallel composition may have more than one), as in the PCBS of p w i (p) 6= 1. [Pra95]: 9w such that p???? !??! Finally, the semantics of TCBS can be cast into prioritised form by using w\ time stamped actions as in Section 3. Read ??!??! as \do w\ at priority ". This completes the interpretation of time as priority. As with PCBS, this appears to suggest that hearing happens at a certain w is interpreted as willingness to accept transmissions of priority. Now ??!??! that priority. Processes that cannot get an audience cannot speak! This explains how parallel composition respects priority. Expansion theorem. The untimed theorem corresponds to the case where the two components have the same priority, so that priority can be ignored. The timed case has one component of higher priority. In PCBS, the component with lower priority cannot speak. All that matters is that it is of lower priority, it does not matter how much lower. This explains why the plays no role on the right hand side of the timed expansion theorem. Comparison of \PCBS" with PCBS. The PCBS of [Pra95] takes priorities to be natural numbers. \PCBS" shows that they can be real numbers as well. PCBS takes translators to be triples, with the rst component specifying deprioritisation. \PCBS" suggests deprioritisation could usefully be separated out. PCBS then suggests that a static (or persistent) timeout is useful|an operator that attaches the timeout to every process the original one evolves to. It turns out that the notion of reversed translators is useful in CBS. The reverse of = h " ; #i is R = h # ; "i, and can be used to describe the environment as seen from inside the scope of . This suggests in turn a static timeout operator that also allows \inverse time". This is yet to be explored fully. Preliminary investigation suggests another, inverse, relation between time and priority. ( ):
!
:
:
?
6 Weak Bisimulation
De nition27 Weak bisimulation for closed processes. R Pcl Pcl is a weak bisimulation if whenever pRq, u] 0 u] (i) if p ??! p then 9q0 such that q ?????? ! q0 and p0 Rq0 , u] u] 0 (ii) if q ??! q then 9p0 such that p ??????! p0 and p0Rq0 The largest weak bisimulation is an equivalence, denoted . Because sums !
!
!
!
are guarded in (T)CBS, it is a congruence. 14
7 Delay pre xes and time-outs Consider a delay pre x, :, de ned by the following rules. v : p ( + ") : p ??! ": p : p ??! p : p ??! Now : p is neither persistent nor ready, and can therefore not be derived from alert processes in TCBS, since these are both persistent and ready. It is interesting to note that things are the other way around in TEPL. Since non-alert processes show time dependence, it is an open question whether delay can be derived (at least upto ) from TCBS. It can if we explicitly allow time dependent behaviour. :
:
?
7.1 Time dependence Now suppose TCBS is extended to allow time dependence. This means rstly that the functions f for input are to be ! T ! Proc . De ne a function add from processes to processes as below.
De nition28.
add (0) add (!s) add (?f ) add (f & w! p) add (": p) add (p j q) add ( p) add (if b then p else p add (A d) 1
2
=0 = !s = ? ([v; t]f v (t + )) = ([v; t]f v (t + )) & w! p = ": (add p) = add p j add q = (add p) ) = if b then add p else add p = add (p[d=z ])where A z = p 1
2
def
Next, the semantic rules are changed as below ( + ") : p ??! add (": p) : p ??! add p ?f ??! add (?f ) v f v0 v f v0 ?f ??! f & w! p ??! Call the resulting calculus TCBS'. In TCBS', processes are neither persistent nor ready. The calculus is otherwise very like TCBS. Now consider the de nition :
:
?
:
?
X = : ([v; t]if = t then p=v else X ( ? t) & ! p) It is easy to see that v X X ( + ") ??! X " X ??! v? (p=v) & ! p X ??! Since p v? (p=v) & ! p, it follows that X : p. Thus delay can be derived upto weak bisimulation using time dependence. def
:
:
?
15
The translation from time to priority in [Jef92a] is similar in spirit to the derivation of : P here. Finally, a curious continuation of the connection between time and priority. This paper would appear to have shown that time is more expressive than priority. But in fact the features of TCBS' can be mimicked by a version of PCBS where processes hear not just the highest priority transmission but also its priority, i.e., the priority annotation is no longer treated as only for deriving transitions, but is made available to receiving processes.
8 Examples First, note that all the examples in [Pra95] of CBS are (degenerate) examples of TCBS. So are the prioritised examples, by reading priority as time. The main new techniqes is that time-outs can be used to detect termination. Several simple protocols have been written and simulated in TCBS. Space does not permit their inclusion. The following brief description should give a taste of what is possible. Example 1 Distributed search: Root of a monotonic function. Let R be the set of real numbers. Given a monotonically increasing function f : R ! R, and a y: R, binary search is a sequential way to nd the x within a given range such that f x = y upto a desired accuracy. (If there is no such x, the program loops). It is easy to parallelise the search by dividing the range into n sections. Let the process pk compute f xk , where xk is the midpoint of the k'th. section. When pk is done computing fxk , it reports this value, and the range is adjusted. This may result in some processes nding that they are no longer in the range. For example, if pk reports a positive value for f xk , then the root is less than xk , and all process pl with l > k abandon their current computation, and start a fresh one. This description assumes computations can be interrupted. Otherwise, each round of computation has all processes ready to report, and all but the two nearest the root will nd that they have completed a useless computation. So processes come up for communication several times during a computation, which is divided into grain sized bits. If a process has nothing to report for that round, it waits. If it does have something to say, it will to do so, unless it hears a better value. The program is ommitted for lack of space. It has been run on a quasi-parallel implementation [RW93] of Haskell, which allows interesting experiments showing the e ects of changing n and the grain.
9 Conclusions This is a working paper, with many loose ends. However, it appears clear that the concepts proposed for TCBS are new and interesting. They are therefore reported here in preliminary form. 16
The main conclusions are as follows. Time is added easily to CBS, and signi cantly extends its power. The transition system for TCBS yields only those transitions that can actually take place. TCBS is simple because only autonomous actions need be considered when deciding which process should act next. These actions are distinct and audible. By contrast, it is precisely the fact that there is only one autonomous action in CCS, and further that this action is the result of communication, that makes it dicult to add time to it. Negative premises are needed more or less explicitly. Strong and weak bisimulations in TCBS are congruences, and yield observationally meaningful equivalences. Time can be read consistently as priority. Delay can be derived from timeout upto weak bisimulation in a calculus with time dependence. TCBS has an expansion theorem even if time is dense. This and other aspects of TCBS cast light, by contrast, on calculi with handshake communication. Further work. Several gaps have been indicated in the paper. Various exten-
sions of the calculus presented here have been tried; a stable version has yet to be reached incorporating all the avenues explored. Apart from this, work is in progress to move TCBS implementations to parallel implementations of Haskell. The simulator for untimed CBS has been proved correct w.r.t. the operational semantics of the language, as part of a project to develop proof methods for CBS programs. See [HPP95, AHP95] for some aspects. This work needs to be extended to TCBS.
References [AHP95] Jorgen Andersen, Ed Harcourt, and K.V.S. Prasad. A machine veri ed sorting algorithm. Preliminary version, 1995. [CH90] Rance Cleaveland and Matthew Hennessy. Priorities in process algebras. Information and Computation, 87, 1990. [CW91] Juanito Camilleri and Glynn Winskel. CCS with priority choice. In Symposium on Logic in Computer Science. IEEE, 1991. [GL92] Jens Chr. Godskesen and Kim G. Larsen. Real-time calculi and expansion theorems. In FST&TCS'92, 1992. Springer Verlag LNCS 652. [Hen92] Matthew Hennessy. Timed process algebras: a tutorial. In Marktoberdorf Summer School on process design calculi, 1992. [HPP95] Ed Harcourt, Pawel Paczkowski, and K.V.S. Prasad. A framework for representing value-passing parametric processes. Preliminary version, 1995. [HR90] Matthew Hennessy and Tim Regan. A temporal process algebra. Computer Science Internal Report 2/90, University of Sussex, 1990. [Jef92a] Alan Je rey. Observation Spaces and Timed Processes. D.phil, Oxford University, 1992. [Jef92b] Alan Je rey. Translating timed process algebra into prioritised process algebra. In Nijmegen Symposium on Real-Time and Fault-Tolerant Systems, 1992.
17
[LY93] Kim Larsen and Wang Yi. Time abstracted bisimulation: Implicit speci cations and decidability. In Mathematical Foundations of Programming Semantics, 1993. [Mil89] Robin Milner. Communication and Concurrency. Prentice Hall, 1989. [Pra93] K. V. S. Prasad. Programming with broadcasts. In CONCUR, August 1993. Springer Verlag LNCS 715. [Pra94] K. V. S. Prasad. Broadcasting with priority. In ESOP, April 1994. Springer Verlag LNCS 788. [Pra95] K. V. S. Prasad. A calculus of broadcasting systems. Science of Computer Programming, 25, 1995. [RW93] C. Runciman and D. Wakeling. Pro ling parallelism. Internal report, Department of Computer Science, University of York, 1993. [Yi90] Wang Yi. Real time behaviour of asynchronous agents. In CONCUR'90, 1990. Springer Verlag LNCS 458. [Yi91] Wang Yi. A Calculus of Real Time Systems. PhD thesis, Chalmers University of Technology, June 1991.
This article was processed using the LaTEX macro package with LLNCS style
18
December 29, 1995
Abstract. In the calculus of broadcasting systems (CBS), speech is autonomous, but hearing takes place only when the environment speaks. This paper develops a timed CBS (TCBS) where no time may pass if there is speech pending. A process wishing to speak can be forced, by attaching a time-out to it, to rst listen for a speci ed length of time. Those forced to wait for a long time can be seen as having low priority. This reading is consistent with PCBS, which is CBS with priorities. TCBS has an expansion theorem even if time is dense. A delay pre x operator can be derived up to weak bisimulation if time dependent behaviour is allowed. TCBS casts, by contrast, light on handshake communication: on time abstracted bisimulation, and on the relations between time and priority, between expansion theorems and density of time, and between delays and time-outs.
1 Introduction Broadcast vs. handshake. CBS [Pra95], a Calculus of Broadcasting Systems, is
a simple and natural CCS-like calculus [Mil89] where processes speak one at a time and are heard instantaneously by all others. Speech is autonomous, contention between speakers being resolved non-deterministically, but hearing only happens when someone else speaks. CBS di ers from almost all other process calculi, which use handshake (or rendezvous) communication. The change in communication primitive has far reaching consequences. The rst is that there is a natural way to run a CBS process: return a list of values it broadcasts. This allows CBS to be both a process calculus and a coordination language. It provides a small set of process constructors and functions to run these processes, both of which can be implemented on top of a host language, typically a functional language such as Haskell or SML. By contrast, there is no obviously correct way to stand back and let a CCS process run; in a straightforward interpretation, the user must interact with the process at each step. [Pra95] develops the basic theory of CBS and gives several programming examples. ?
Funding: Swedish Government agency TFR and Esprit BRA CONCUR2.
[Pra95] also develops PCBS, which is CBS with priorities. Processes that wish to speak do so at some priority. This resolves contention between utterances of di ering priorities. Hearing has no priority associated with it, since what a process hears is anyway decided by the environment. Things are much harder in CCS with priorities [CH90, CW91]. Here there is only one autonomous action, a completed handshake (written ) which is silent. Priorities are associated with potential (i.e., non-autonomous or controlled) actions. This leads to obvious problems. A process p can do actions a or b and prefers a, say, but what use is this if the environment only allows b? Worse, what if the environment allows both actions, but prefers b? Translated to a timed setting, the topic of the present paper, the di erence in communication primitive can be illustrated as follows. Consider the following classroom situation: the teacher gives the class two minutes to think about a question. The quickest student will answer, unless even this person takes more than two minutes, in which case the teacher will preempt the class. This can be described as a timed broadcast system with maximal progress, because people speak as soon as they are ready to. Handshake communication is illustrated by a marketplace, where buyer and seller must agree before a sale goes ahead. Each must wait for the other to appear; neither can expect in general to perform an action when it is ready. Timed process calculi. There is now a plethora of these. A tutorial, [Hen92],
notes that there is very little di erence between untimed and timed languages. Current research nds little to say about the setting up of timed calculi, and focusses instead on more esoteric issues such as the abstract structure of time domains, the diculties caused by dense time, and the like. But almost all timed calculi are based on handshake communication,even if occasional postulates such as \urgent actions" reveal the beginnings of dissatisfaction with it. As one might guess, setting up a timed calculus based on a di erent communication model turns out to be an instructive exercise. Main results. This paper develops TCBS, a timed extension of CBS. TCBS shares several properties with existing timed calculi such as the Timed CCS of [Yi91] or TEPL, the simple timed calculus introduced in [Hen92]. As foreshadowed, it turns out that TCBS, a calculus with maximal progress and a time-out operator, is easier to set up than than timed handshake calculi with maximal progress [Yi91, HR90, Hen92]. Other main results are: a priority interpretation of TCBS is consistent with PCBS; TCBS has an expansion theorem even if time is dense. Delays and time-outs both t into a more detailed calculus where each process keeps track of time. Yet the simpler TCBS without time dependence is expressive and powerful enough for many purposes. It is easier to read time as priority than the other way around, which suggests that time is more basic than priority, at least in this communication model. [Pra94] shows that strong bisimulation is the largest congruence contained in a bisimulation equivalence that ignores priority. The corresponding result for time is similarly simple for TCBS; it is a main result of [LY93].
2
Related work. There have been few comparisons of timed CCS with prioritised CCS. However, [Jef92b] gives a translation of timed algebra into a prioritised algebra where every action is time stamped. Some confusion surrounds the expansion theorem for timed CCS [GL92, Hen92, Yi90, Yi91]. One in uential view is that \in order to have expansion theorems in real-timed calculi, one should focus on calculi with time variables or similar notions" [GL92]. This is misleading. An expansion theorem for CCS is awkward with the delay operator (which is what [GL92] use) regardless of the density of time. To get an expansion theorem with the more powerful time-out operator (in CCS, the delay operator can be derived from the time-out), the density of time is indeed signi cant. Time variables, or some other means of keeping track of time, are needed if time is dense, but not if it is discrete. Readership. Readers are assumed to be familiar with CCS [Mil89]. Familiarity with some timed process calculus and with one of [Pra93, Pra94, Pra95] would help.
2 The Syntax and Semantics of TCBS TCBS is an extension of CBS [Pra95]. The treatment below is largely self contained, but concentrates only on the timed aspects. The syntax and communication actions of TCBS processes are given in Table 1. TCBS is a framework or a coordination language, not a complete programming language. No syntax or computation rules are given for data expressions. The user chooses these for the application at hand. The evaluation of data is not represented, but is assumed to terminate, and closed data expressions merely stand for their values. Types. Given the datatype , the syntax de nes inductively the set Proc . Not every type is permissible here: it must be possible to determine when two elements of are equal. Further, may not itself involve the type Proc for any , that is, this paper is restricted to rst order TCBS. A translator : Proc ! Proc is speci ed by a pair of functions " : ! and # : ! . That is, p Trans h " ; #i p, where Trans is the translation operator, always dropped in writing. Informally, " and # are treated as projections yielding the components " and # of a translator . The time domain. This is denoted T. It can be taken to be either the natural numbers or the positive real numbers (including zero). De ning equations and guardedness. This paper assumes that there is an unmentioned set of guarded constant de nitions available for use in process terms. De nition1 Guardedness. Processes of the form 0, !s, ?f or f &s are guarded. If p and q are guarded, so are p j q and if b then p else q. If p is guarded, so are : p and p. If A z = p then A d is guarded if p[d=z ] is. A de nition A z = p is guarded if p[d=z ] is guarded for all d. def
def
3
Let be a datatype. Let be a distinguished value, 2= , and be [ f g. Let x: be a variable and w: an expression. Let be another datatype. Let be speci ed by " : ! and # : ! satisfying " = and # = . Let , " 2 T, the time domain (the natural numbers or the positive reals with zero). Let b be a boolean expression. Then the elements of Proc are given by
p : : = 0 !s ?f f & s : p pjp p if b then p else p A d where f :: = [x]p, s: : = hw; pi, p : Proc , and A ranges over constants, declared in (mutually) recursive guarded de nitions A z = p, parameterised by a datatype ranged over by variable z and expression d. The semantics below also uses pi , p0 , p00 , p0i : Proc , p0 : Proc , and values v: , w : , and u: [ T. def
p ?? ! p
Tau
?
Guarded Sum
!hw; pi ??w! p v! !s v! p[v=x] !s ?? ?[x]p ??
f & hw; pi ??w! p v! p[v=x] [x]p & s ?? p ??! p0 p0 ??"! p00 " p ????? ! p00
!
0 ??v! 0 ?
?
?
0 ??! 0 ?f ??! ?f p ??! p
Time
:
!
?
:
( + ):
p ??w! p0 0: p ??w! p0
?
:
?
:
0:
p ??v! p0 : p ??v! p0
: p ??! p
Timeout
Compose
:
!
!
( + ") : p ??! ": p :
! ? : ! ? !? ? ! ?? : ? ?:
u]1 0 u]2 p ??? ! p p ??? ! p0 ] ] 6= ? u ]1 ]2 ! p0 jp0 p jp ?????? 1
1
2
1
(
1
2
2
1
?
!
"
2
2
# w ! p0 p ???? w p0 p ??!
w ! p0 p ???
Translate
1
)
p ??! p0 p ??! p0 :
w p ????? ! p0 p ??u]! p0 p ??u]! p0 Conditional if true then p else p ??u]! p0 if false then p else p ??u]! p0 p[d=z ] ??u]! p0 De ne A d ??u]! p0 Az = p ], ] , ] range over f! ; ? ; : g. ? means \unde ned" in the synchronisation algebra. 1
?
!
1
2
1
1
:
2
1
2
1
1
def
1
1
2
Table 1. The syntax of TCBS and the semantics of closed processes 4
2
2
Abbreviation Meaning x? p ?[x]p w! p !hw; pi x? p & s [x]p & s f & w! p f & hw; pi x? p & w! p [x]p & hw;pi A=p A () = p where () is the only element of the type \Unit". def
def
This paper often uses f v for the function application f (v). NOTE the abuse of notation in the third, fourth and fth lines. In the context of the operator & the subexpressions x? p and w! p stand for the abstraction [x]p and the pair hw;pi respectively; outside of a & context, they stand for the processes ?[x]p and !hw; pi respectively.
Table 2. Syntactic abbreviations, freely used throughout. Since every de nition in this paper is guarded, so is every process. Examples of unguarded de nitions are p = p, p = p, p = 2! q j p and p = 2: p. In the last example, it is not possible to determine p's response to hearing something from the environment|see Proposition 2 below. Thus pre xing by a timeout is not enough to ensure guardedness. Induction on the structure of processes does not work for constant applications, because to prove the hypothesis for A d, the starting point is p[d=z ], which is not smaller than A d. But the proof of guardedness of p[d=z ] is smaller than that of A d. Induction can therefore be carried out on the depth of proof of guardedness of processes (abbreviated \induction on guardedness"). def
def
def
def
Open and closed processes. Let x: be a (data) variable. Occurrences of x in p become bound in the process abstraction [x]p, and the scope of x in [x]p is p. Bound variables are assumed to be renamed as necessary to avoid clashes under substitution. A process is closed if it has no free variables, and is open if it does. Thus x? x! 0 is closed while x! 0 is open. The set of all (open) processes is denoted P, and the set of closed processes Pcl . Let v: be a (data) value and let p[v=x] denote the result of substituting v for x in p. The user has to supply the functions that substitute values for variables in data expressions. These functions extend from to P in the evident way. For example, (w! p)[v=x] = (w[v=x])! p[v=x] and (if b then p else p ) [v=x] = if b[v=x] then p [v=x] else p [v=x]. Only closed processes can communicate. Open ones cannot, by de nition. 1
1
2
2
w and ??! w over Communication actions. For each w: , there are relations ??! Proc . Also, for each 2 T there is a relation ??! over Proc . These are the !
?
:
least relations satisfying the axioms and inference rules in Table 1. Let ], ] and ] range over f! ; ? ; : g and u over [ T. 1
2
5
Expansion. Section 4 presents an extended guarded sum f & fsi i 2 I g with
several output branches, needed (only) for an expansion theorem. This form is not needed for programming, which only needs at most one output branch as in the syntax presented here.
2.1 Examples and explanation In (T)CBS, speech is autonomous, while hearing is a response to speech by the environment. TCBS incorporates maximal progress: time passes i no process has anything to say, i.e., i there are no autonomous actions available. Thus time actions in TCBS are like hearing in that they are not autonomous. TCBS is built from CBS by augmenting the syntax by time-outs, and the semantics by the rules in the second box of Table 1, and by time instances of the other rules, notably p ??! p0 p ??! p0 p j p ??! p0 j p0 The processes 0 and ?f have nothing to say, and allow any amount of time to pass. If ?f hears v, it evolves to f v. The process f & (5 + 1) ! p wishes to say 6 and become p. If it hears v immediately, it evolves to f v. It does not let time pass. f & (5 + 1)! p ??! p f & (5 + 1) ! p ??! f 4 The process : (f & v! p) times out after units of time by saying v, and becomes p. Upto this time, it is prepared to hear v0 and become f v0 . : (f & (5 + 1) ! p) ??!??!p : (f & (5 + 1) ! p) ??"!??!f 4 for 0 " Attaching a time-out to ?f achieves little; in fact, : (?f ) ?f . The next two examples show that care is needed when programming with time-outs. First, note the behaviour of : w! p. : (5 + 1) ! p??!??!p : (5 + 1) ! p??"!??! (5 + 1)! p for 0 " Thus : (5 + 1) ! p is not a good representation of a process that takes time to evaluate 5+1, for it does the evaluation immediately upon any interruption by the environment! The following attempt is not much better. :
1
1
2
:
2
1
2
:
1
2
6!
:
:
4?
6!
:
6!
:
4?
4?
w! p = : (x? w! p & w! p) def
This yields the behaviour (5 + 1) ! p??!??!p (5 + 1)! p??"!??! (5 + 1) ! p for 0 " where any interruption causes the evaluation to be restarted. A process that takes to evaluate 5+1 is best represented by a delay operator (see Section 6). Readers familiar with [Pra95] or [Pra94] can skip the rest of this section. :
6!
:
6
4?
If both components of a parallel composition wish to speak, one is chosen arbitrarily. j is associative, and communication is synchronous. Whatever is said is heard instantaneously by all processes in parallel with the speaker. ?f j 5! p j y? y! 0??! f 5j pj5! 0 ?f j 5! p j y? y! 0??! f 4j5! pj4! 0 ?f j : (5! p) ??! ?f j 5! p All processes respond deterministically to what they hear; w! p and f & w! p are deterministic also for speech. Contending speakers are the only source of non-determinism. This is captured by the extended guarded sum used for the expansion theorem (Section 4). Finally, a translator is a pair h "; # i. The process p says "5 if p says 5; its response to 4 depends on p's response to #4. Translators do not a ect the passage of time. Hiding and restriction are achieved by translation to , an aside appended to every data type. Asides are heard but always ignored. Translating functions must map to . 5!
4?
:
2.2 Properties of the calculus u] u] 0 Letu]p??! mean \9 p0 such that p ??! p ", and let p??u]! = mean \6 9 p0 such that 0 p ??! p ". The propositions below con rm that TCBS is well behaved. First, some untimed properties. See [Pra95] for proofs and more detail.
w p0 . Proposition2 Input enabling and determinism. 8p, w, 9! p0 such that p ??! w p0 . De nition3. p=w, the image of p under w, is the unique p0 such that p ??! ?
?
For example, (?f ) =v = f v, p= = p, and (p) =v = p=v w Proposition4 Finite output branching. 8p, the set fw p??!g is nite.
!
u] 0 p g is nite. Proposition5 Image nite. 8p, u, ], the set fp0 p ??! To derive 5! 0 j x? 3! 0 ??! 0 j 3! 0, the premise x? 3! 0 ??! 3! 0 is needed. But to derive any ??! transition, no premises are needed that involve a ??v! for any
5!
5?
5?
!
v. The next proposition formulates this. w transitions can be derived independently Proposition6. For all w and w0, ??! 0 w of ???! transitions. ?
!
7
Timing properties. All the propositions below can be proved by induction on
guardedness.
Proposition7 Time determinism. If p ??! p0 and p ??! p00 then p0 p00. :
:
If the time domain is restricted to non-zero elements, the proposition can be strengthened to yield p p0. But 0: p ??! p and 0: p ??! 0: p, so processes are time deterministic only upto strong bisimulation equivalence , de ned in the next section. Time determinism holds also for Timed CCS and TEPL. 0:
0:
" Proposition8 Time additivity. p??!??"!r i p ????? ! r. So if p??! and " < (i.e., 9"0 6= 0 such that " + "0 = ) then also p??"!. :
:
( + ):
:
:
Additivity is associated with density (\continuity" in [Yi91]). If time is discrete, then : p can be taken to be an abbreviation for 1 : p, ( + ") : p for : (": p), " and ????? ! for ??!??"!. This makes the two \+" rules of Table 1 unnecessary, and the property of additivity a matter of de nition. The property becomes interesting, and the rules necessary, only if a time interval can be divided in nitely nely. The non-zero naturals make a simple time domain for a calculus where further the rule p ??! p is dropped. Such a calculus is enough to convey all the concepts of TCBS except the handling of dense time. Additivity holds for Timed CCS; TEPL does not need it because it is a discrete timed calculus. ( + ):
:
:
0:
Proposition9 Maximal progress. If p??! for 6= 0 then 8w, p??w!= . w and p?? "!? w0! then = ". Corollary 10. If p??!??! ?? :
:
!
:
!
!
Thus each process can be characterised by the maximum amount of time for which it can wait. This can be read as the \priority" of the process; the longer it can wait, the lower it's priority. This is explored in Section 5.
Proposition11 Patience. If 8w p??w!= then 9 6= 0, p??!. !
:
Maximal progress and its inverse, patience, are much debated decisions with handshake communication. They seem natural here, as autonomous actions in (T)CBS are local, not composed of handshakes between possibly geographically separated components. A speaker has all the information it needs, and no reason to wait. Conversely, listening is a matter of waiting. Maximal progress and patience hold for TEPL and Timed CCS. They need = . Similarly for the time-out negative premises. In TEPL, p j q??! only if p j q?! operator. [Yi91] codes negative premises into \timed sorts", used to say when the components of a parallel composition can communicate. In TCBS, the basic dichotomy is between 0 and ?f , which allow time to pass, and !s and f & s, which don't. The rest follows naturally from the way autonomy and communication are captured by parallel composition. Thus, as in :
8
PCBS, the communication model yields a simple transition system with no need for negative premises. In passing, note that some handshake calculi postulate \urgent actions" (usually with little physical interpretation). These have maximal progress but not patience. De nition12 Alert processes. A translator alerts if 8v # v 6= . A process p is alert if all translators occurring in p alert. The following two properties hold only for alert processes. v q and p ?? ! r then r ??! v q. Proposition13 Persistence. Let p be alert. If p ??! v q. Proposition14 Readiness. Let p be alert. If p ??! r and r ??v! q then p ??! ?
:
:
?
?
?
These properties are illustrated by the process p = 2: 5! 0. Then p ??! 5! 0, v 5! 0 for any value v. but also p ??! To see why alertness is needed, let Deaf be a translator de ned by Deaf " x = x and Deaf # x = . That is, Deaf lets through all outgoing values, but restricts v Deaf (5! 0), but Deaf p ??! v Deaf p. all incoming ones. Then Deaf p??!??! Persistence applies only to non-autonomous actions. Most timed calculi do not have this property. In TEPL, which adds a time-out operator to CCS, the property holds only for the CCS subcalculus. In Timed CCS, which uses a delay pre x, the property holds for all processes, but in a weaker form, corresponding v and p ?? ! r then r?? v!". This is uninteresting for TCBS because of to \if p??! input enabling. Persistence says that time does not decrease input capability. Readiness says it does not increase it either. Readiness seems to be interesting only in TCBS. Processes pre xed by delays, as in Section 7, are not persistent. Delay therefore cannot be expressed in TCBS using only alert processes. Alert processes are \time independent" in the sense that their responses to input are independent of when the input arives. With time dependence, TCBS can express delays. def
2:
?
2:
?
:
?
?
?
3 Strong Bisimulation De nition15 Strong bisimulation for closed processes. R Pcl Pcl is a strongu]bisimulation if whenever pRu]q, (i) if p ??! p0 then 9q0 such that q ??! q0 and p0Rq0 , u] 0 (ii) if q ??! q then 9p0 such that p ??u]! p0 and p0 Rq0 The largest strong bisimulation is an equivalence, denoted . To show p q, nd a bisimulation R such that pRq. All the laws below are shown this way. Proposition16 Some untimed laws. 1. (a) x? 0 0 (b) (x? w! p) & (w! p) w! p 9
2. (a) 0 X where X = x?X (b) w! p X where X = x?X & w! p 3. (P= ; j; 0) is a commutative monoid. def
def
See [Pra95] for more laws. CBS has no laws corresponding to the CCS + laws of associativity, commutativity, idempotence, and 0-identity, because the guarded sums have at most one output branch, and their input behaviour is deterministic. is extended to process abstractions: [x]p [x]q i 8v: p[v=x] q[v=x]. From [x]p [x]q, it follows that x? p x? q and that x? p & s x? q & s.
Proposition17. is a congruence for TCBS. Proposition18 Timed laws. 1. 0: p p 2. : 0 0 3. : (?f ) ?f 4. : (": p) ( + ") : p 5. : (p j q) ( : p) j ( : q) 6. If alerts, then : ( p) ( : p) The processes generated by 0, !, ?, & and conditionals are called \ nite guarded sums". [Pra95] shows that Laws 1(a) and 1(b) of Proposition 16 are the only axioms needed to axiomatise strong bisimulation for nite guarded sums. Adding timeout to this syntax yields \timed nite guarded sums". The axiomatisation has not yet been carried out for this version of TCBS; it is a conjecture that the above laws are enough. Suitable normal forms seem to be 0, ?f , : (!s) and : (f & s). The above laws are enough to put any timed nite guarded sum into normal form. w\ Time stamped actions. Let \ range over f! ; ? g. The de nition of matches ??!
w\ and ??! separately. In fact, only combinations of the form ??!??! need be considered. A reading is \w\ at time ". Such a reading is useful when reading w\ time as priority (Section 5). The action ??! is represented as a combination where = 0. The action ??! is represented as a combination where w\ = ?. :
:
:
De nition19 Time stamped bisimulation. R Pcl Pcl is a time stamped bisimulationw\if whenever P RQ, w\ 0 (i) if p??!??! p0 then 9q0 such that q??!??! q and p0 Rq0 , w\ w\ (ii) if q??!??!q0 then 9p0 such that p??!??!p0 and p0Rq0 Let the largest time stamped bisimulation be 0 . Proposition20. =0 :
:
:
:
10
Proof. 0 is obvious. For the other direction, use time determinism.
A main result of [LY93] shows that time abstracted bisimulation, where the passage of time is ignored, yields an equivalence that is not a congruence. The largest congruence contained in this equivalence is strong bisimulation. For TCBS, this result is easy, as is the corresponding result for PCBS [Pra94]. Note that the only operator involved is j.
De nition21 Time abstracted bisimulation. R Pcl Pcl is a time abstracted bisimulation if whenever P RQ, w\ 0 w\ 0 (i) if p??!??! p then 9q0 , " such that q??"!??! q and p0Rq0 , w\ w\ (ii) if q??!??!q0 then 9p0, " such that p??"!??!p0 and p0 Rq0 The largest time abstracted bisimulation, denoted ', is an equivalence, but not a congruence. For 3! p ' 3! p, yet 3! p j 5! q??!??!, while for any , 3! p j 5! q??!??!= . Let 'c be the largest congruence contained in '. Proposition22. ='c Proof. It is easy to see that w\ ', and is a congruence. For the other direction, w\ 0 let p 'c q, and let p??!??!p0 . Then q??"!??! q , and p0 'c q0. But = ", otherwise a j context can be found to distinguish p and q. Use Proposition 20. :
:
:
:
1
2
1
:
2
1
1
1:
3!
3!
:
:
4 Expansion For this section, an extended calculus TCBSe is introduced where guarded sums have an output tree, a nite set of output branches instead of just one as in TCBS. The syntax of TCBSe is given by
p : : = !s f & s : p p j p p if b then p else p A d
where f : : = [x]p and s : : = fhwi; pii i 2 I g, where I is a nite set. The semantics below is extended from Table 1; the common parts are not repeated. wi f & s ??? ! pi hwi ; pii 2 s [x]p & s ??v! p[v=x] wi v !s !s ???! pi hwi ; pii 2 s !s ??! Now !; says nothing and loops on all input. It corresponds to 0 in TCBS. Similarly, f &; corresponds to ?f in TCBS. Writing just hw; pi instead of fhw; pig, it is easily seen that TCBS is a subcalculus of TCBSe . TCBSe processes too have a unique response to each input. Again, p=w denotes the image of p under w (see De nition 3). Bisimulation extends naturally to TCBSe , as do the notions of (timed) nite guarded sums. First, the untimed expansion theorem. The nite number of output branches in a sum are enough for the expansion theorem below, since parallel compositions are nite. For legibility, fhwi; pii i 2 I g is often written fwi! pi i 2 I g.
Guarded Sum
!
?
!
?
11
Proposition23 Expansion theorem. Let p, q be nite guarded sums.Then w p0 and r = 0; 1g p j p x? (p =x j p =x) & fw! (p0r j p ?r =w) pr ??! r 0
1
0
1
!
1
For example, the process 2! p j 7! q can be expanded into a guarded sum: 2! p j 7! q x? (2! p j 7! q)&f2! (p j 7! q) ; 7! (2! p j q)g Now, assuming the normal forms proposed earlier for timed nite guarded sums, the expansion theorem can be broken up into special cases. Consider : p j ": q where p, q are nite guarded sums. Assume that ". Then : p j ": q : (p j (" ? ) : q). In the case " = , the composition is of the form p j q where both components are just guarded sums. The case < " results in a composition of the form p j : q. Further, if p is of the form ?f , it can be rewritten p : p, and the composition again reduces to one of just guarded sums. Thus there are only two cases to consider, the untimed theorem above, and
Proposition24 Timed Expansion theorem. Let p f & fwi ! pi i 2 I g, and I = 6 ;. Then p j : q x? (p=x j q=x) & fwi! (pi j ( : q) =wi) i 2 I g. The existence of this theorem for TCBS calls for some comment, in the light of the discussion in the introduction. [Hen92] points out that in TEPL, all agents without time-outs are persistent. Let the delay pre x \:" be de ned by the sole axiom : p ??! p. Then : p is a : p, persistent as well. But the process a: 0 j : p is not persistent, for a: 0 j : p ?! a p, i.e., after the time action, the agent no longer has but a: 0 j : p ??! a: 0 j p?! an action to the same target state. Therefore CCS augmented by a delay pre x cannot yield an expansion theorem, even if time is discrete. [Hen92] gets an expansion theorem using the time-out construct p + q which behaves like p for and then becomes q. This is clearly not persistent. [Yi91] gets around the problem by allowing time dependence: (in adapted notation) the action pre x is now written a: f where f is a function from time to a f 0 and a: f ?! a: (t: f (t + )). The process processes. The axioms are a: f ?! a: f is also not persistent. (It is weakly persistent). Thus far it would appear that TCBS follows a path similar to that of [Hen92]. But [Hen92] only considers discrete time. What happens if time is dense? [GL92] motivate their study of dense time by showing that it is hard to get an expansion theorem using the delay pre x. This is misleading, since the impossibility here is independent of the density of time. However, their result holds. Roughly, the problem with dense time is this. For brevity here, CCS processes are written without trailing 0's, i.e., a: 0 is written just a. Consider the agent (a + 0 q) j c. The plausible expansion is (a: c + c: (a + 0 q)) + 0 (q j c) :
:
+
+
12
+
Consider the action
(a + 0 q) j c (a + 0 q) j c ?! The plausible expansion cannot match it (a: c + c: (a + (a: c + c: (a + 0 q)) + 0 (q j c) ?! 0 q)) +0 (q j c) The clock behind the action pre x does not tick. It is dicult to demonstrate the \corresponding" situation in TCBS, but roughly speaking, the problem does not arise because input received by one component is also received by the other. Consider the expansion : (f & v! p) j ?g : (x? (f x j g x) & v! (p j g v)) When the second component receives input, the rst one also loses its time-out. The fundamental di erences in communication model mean that the result of [GL92] does not apply to TCBS. +
+
+
+
5 Interpreting time as priority . To avoid introducing the notation of PCBS, this section instead provides a priority reading of TCBS, and proves the necessary propositions to show that this reading is consistent. An operational de nition of the priority of a process has already been suggested: it is the longest time the process can wait. Corollary 10 shows that this is well de ned. The highest priority processes, with priority 0, do not wait at all before saying what they want to. Processes with nothing to say have the lowest priority, \1", and are willing to wait inde nitely. Thus time actions can be interpreted as pauses to ensure that higher priority processes speak rst. The priority of 3! p is 1 and that of 5! q is 2. In 3! p j 5! q, the lower priority process is prevented from speaking rst because the higher priority process times out before it is ready to do so. The latter needs a further wait of 1 unit, the former refuses, and therefore so does the composition. Now every TCBS process can be assigned a syntactic priority. The proposition to follow checks that this agrees with the operational de nition. De nition25. The priority (p) of a closed process p is given by the rules (0) =1 (?f ) =1 (!s) =0 (f & s) =0 ( : p) = + (p) (p j q) = min( (p) ; (q)) ( p) = (p) (if b then p else p ) = if b then (p ) else (p ) (A d) = (p[d=z ]) where A z = p 1
1
2
2
1
1
2
def
13
2
Proposition26. p??! i (p). w , the proposition above shows that a process Since p??! implies 8w, p??!??! :
:
:
?
is prepared to hear others of priority at least as high as its own. Next, a syntax for a \PCBS" as a restriction of TCBS. Use timeouts only in the forms : !s and : (f & s). The non-alert processes of TCBS cannot be put into this form, so \PCBS" is strictly a subcalculus of TCBS. The terms of \PCBS" annotate speech requests with priorities. The priority of a process as de ned above is merely that of its most urgent speech (a parallel composition may have more than one), as in the PCBS of p w i (p) 6= 1. [Pra95]: 9w such that p???? !??! Finally, the semantics of TCBS can be cast into prioritised form by using w\ time stamped actions as in Section 3. Read ??!??! as \do w\ at priority ". This completes the interpretation of time as priority. As with PCBS, this appears to suggest that hearing happens at a certain w is interpreted as willingness to accept transmissions of priority. Now ??!??! that priority. Processes that cannot get an audience cannot speak! This explains how parallel composition respects priority. Expansion theorem. The untimed theorem corresponds to the case where the two components have the same priority, so that priority can be ignored. The timed case has one component of higher priority. In PCBS, the component with lower priority cannot speak. All that matters is that it is of lower priority, it does not matter how much lower. This explains why the plays no role on the right hand side of the timed expansion theorem. Comparison of \PCBS" with PCBS. The PCBS of [Pra95] takes priorities to be natural numbers. \PCBS" shows that they can be real numbers as well. PCBS takes translators to be triples, with the rst component specifying deprioritisation. \PCBS" suggests deprioritisation could usefully be separated out. PCBS then suggests that a static (or persistent) timeout is useful|an operator that attaches the timeout to every process the original one evolves to. It turns out that the notion of reversed translators is useful in CBS. The reverse of = h " ; #i is R = h # ; "i, and can be used to describe the environment as seen from inside the scope of . This suggests in turn a static timeout operator that also allows \inverse time". This is yet to be explored fully. Preliminary investigation suggests another, inverse, relation between time and priority. ( ):
!
:
:
?
6 Weak Bisimulation
De nition27 Weak bisimulation for closed processes. R Pcl Pcl is a weak bisimulation if whenever pRq, u] 0 u] (i) if p ??! p then 9q0 such that q ?????? ! q0 and p0 Rq0 , u] u] 0 (ii) if q ??! q then 9p0 such that p ??????! p0 and p0Rq0 The largest weak bisimulation is an equivalence, denoted . Because sums !
!
!
!
are guarded in (T)CBS, it is a congruence. 14
7 Delay pre xes and time-outs Consider a delay pre x, :, de ned by the following rules. v : p ( + ") : p ??! ": p : p ??! p : p ??! Now : p is neither persistent nor ready, and can therefore not be derived from alert processes in TCBS, since these are both persistent and ready. It is interesting to note that things are the other way around in TEPL. Since non-alert processes show time dependence, it is an open question whether delay can be derived (at least upto ) from TCBS. It can if we explicitly allow time dependent behaviour. :
:
?
7.1 Time dependence Now suppose TCBS is extended to allow time dependence. This means rstly that the functions f for input are to be ! T ! Proc . De ne a function add from processes to processes as below.
De nition28.
add (0) add (!s) add (?f ) add (f & w! p) add (": p) add (p j q) add ( p) add (if b then p else p add (A d) 1
2
=0 = !s = ? ([v; t]f v (t + )) = ([v; t]f v (t + )) & w! p = ": (add p) = add p j add q = (add p) ) = if b then add p else add p = add (p[d=z ])where A z = p 1
2
def
Next, the semantic rules are changed as below ( + ") : p ??! add (": p) : p ??! add p ?f ??! add (?f ) v f v0 v f v0 ?f ??! f & w! p ??! Call the resulting calculus TCBS'. In TCBS', processes are neither persistent nor ready. The calculus is otherwise very like TCBS. Now consider the de nition :
:
?
:
?
X = : ([v; t]if = t then p=v else X ( ? t) & ! p) It is easy to see that v X X ( + ") ??! X " X ??! v? (p=v) & ! p X ??! Since p v? (p=v) & ! p, it follows that X : p. Thus delay can be derived upto weak bisimulation using time dependence. def
:
:
?
15
The translation from time to priority in [Jef92a] is similar in spirit to the derivation of : P here. Finally, a curious continuation of the connection between time and priority. This paper would appear to have shown that time is more expressive than priority. But in fact the features of TCBS' can be mimicked by a version of PCBS where processes hear not just the highest priority transmission but also its priority, i.e., the priority annotation is no longer treated as only for deriving transitions, but is made available to receiving processes.
8 Examples First, note that all the examples in [Pra95] of CBS are (degenerate) examples of TCBS. So are the prioritised examples, by reading priority as time. The main new techniqes is that time-outs can be used to detect termination. Several simple protocols have been written and simulated in TCBS. Space does not permit their inclusion. The following brief description should give a taste of what is possible. Example 1 Distributed search: Root of a monotonic function. Let R be the set of real numbers. Given a monotonically increasing function f : R ! R, and a y: R, binary search is a sequential way to nd the x within a given range such that f x = y upto a desired accuracy. (If there is no such x, the program loops). It is easy to parallelise the search by dividing the range into n sections. Let the process pk compute f xk , where xk is the midpoint of the k'th. section. When pk is done computing fxk , it reports this value, and the range is adjusted. This may result in some processes nding that they are no longer in the range. For example, if pk reports a positive value for f xk , then the root is less than xk , and all process pl with l > k abandon their current computation, and start a fresh one. This description assumes computations can be interrupted. Otherwise, each round of computation has all processes ready to report, and all but the two nearest the root will nd that they have completed a useless computation. So processes come up for communication several times during a computation, which is divided into grain sized bits. If a process has nothing to report for that round, it waits. If it does have something to say, it will to do so, unless it hears a better value. The program is ommitted for lack of space. It has been run on a quasi-parallel implementation [RW93] of Haskell, which allows interesting experiments showing the e ects of changing n and the grain.
9 Conclusions This is a working paper, with many loose ends. However, it appears clear that the concepts proposed for TCBS are new and interesting. They are therefore reported here in preliminary form. 16
The main conclusions are as follows. Time is added easily to CBS, and signi cantly extends its power. The transition system for TCBS yields only those transitions that can actually take place. TCBS is simple because only autonomous actions need be considered when deciding which process should act next. These actions are distinct and audible. By contrast, it is precisely the fact that there is only one autonomous action in CCS, and further that this action is the result of communication, that makes it dicult to add time to it. Negative premises are needed more or less explicitly. Strong and weak bisimulations in TCBS are congruences, and yield observationally meaningful equivalences. Time can be read consistently as priority. Delay can be derived from timeout upto weak bisimulation in a calculus with time dependence. TCBS has an expansion theorem even if time is dense. This and other aspects of TCBS cast light, by contrast, on calculi with handshake communication. Further work. Several gaps have been indicated in the paper. Various exten-
sions of the calculus presented here have been tried; a stable version has yet to be reached incorporating all the avenues explored. Apart from this, work is in progress to move TCBS implementations to parallel implementations of Haskell. The simulator for untimed CBS has been proved correct w.r.t. the operational semantics of the language, as part of a project to develop proof methods for CBS programs. See [HPP95, AHP95] for some aspects. This work needs to be extended to TCBS.
References [AHP95] Jorgen Andersen, Ed Harcourt, and K.V.S. Prasad. A machine veri ed sorting algorithm. Preliminary version, 1995. [CH90] Rance Cleaveland and Matthew Hennessy. Priorities in process algebras. Information and Computation, 87, 1990. [CW91] Juanito Camilleri and Glynn Winskel. CCS with priority choice. In Symposium on Logic in Computer Science. IEEE, 1991. [GL92] Jens Chr. Godskesen and Kim G. Larsen. Real-time calculi and expansion theorems. In FST&TCS'92, 1992. Springer Verlag LNCS 652. [Hen92] Matthew Hennessy. Timed process algebras: a tutorial. In Marktoberdorf Summer School on process design calculi, 1992. [HPP95] Ed Harcourt, Pawel Paczkowski, and K.V.S. Prasad. A framework for representing value-passing parametric processes. Preliminary version, 1995. [HR90] Matthew Hennessy and Tim Regan. A temporal process algebra. Computer Science Internal Report 2/90, University of Sussex, 1990. [Jef92a] Alan Je rey. Observation Spaces and Timed Processes. D.phil, Oxford University, 1992. [Jef92b] Alan Je rey. Translating timed process algebra into prioritised process algebra. In Nijmegen Symposium on Real-Time and Fault-Tolerant Systems, 1992.
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[LY93] Kim Larsen and Wang Yi. Time abstracted bisimulation: Implicit speci cations and decidability. In Mathematical Foundations of Programming Semantics, 1993. [Mil89] Robin Milner. Communication and Concurrency. Prentice Hall, 1989. [Pra93] K. V. S. Prasad. Programming with broadcasts. In CONCUR, August 1993. Springer Verlag LNCS 715. [Pra94] K. V. S. Prasad. Broadcasting with priority. In ESOP, April 1994. Springer Verlag LNCS 788. [Pra95] K. V. S. Prasad. A calculus of broadcasting systems. Science of Computer Programming, 25, 1995. [RW93] C. Runciman and D. Wakeling. Pro ling parallelism. Internal report, Department of Computer Science, University of York, 1993. [Yi90] Wang Yi. Real time behaviour of asynchronous agents. In CONCUR'90, 1990. Springer Verlag LNCS 458. [Yi91] Wang Yi. A Calculus of Real Time Systems. PhD thesis, Chalmers University of Technology, June 1991.
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