Preface 1. Introduction According to the Oxford English Dictionary (OED II CD-ROM), a process is a series of actions or events, and an algebra is a calculus of symbols combining according to certain defined laws. Completing the picture, a calculus is a system or method of calculation. Despite going back as far as the 13th Century, collectively, these definitions do a good job of accurately conveying the meaning of this Handbook's subject: process algebra. A process algebra is a formal description technique for complex computer systems, especially those with communicating, concurrently executing components. A number of different process algebras have been developed - ACP [1], CCS [6], and TCSP [2] being perhaps the best-known - but all share the following key ingredients. • Compositional modeling. Process algebras provide a small number of constructs for building larger systems up from smaller ones. CCS, for example, contains six operators in total, including ones for composing systems in parallel and others for choice and scoping. • Operational semantics. Process algebras are typically equipped with a Plotkin-style [7] structural operational semantics (SOS) that describes the single-step execution capabilities of systems. Using SOS, systems represented as terms in the algebra can be "compiled" into labeled transition systems. • Behavioral reasoning via equivalences and preorders. Process algebras also feature the use of behavioral relations as a means for relating different systems given in the algebra. These relations are usually equivalences, which capture a notion of ''same behavior", or preorders, which capture notions of ''refinement". In a process-algebraic approach to system verification, one typically writes two specifications. One, call it SYS, captures the design of the actual system and the other, call it SPEC, describes the system's desired "high-level" behavior. One may then establish the correctness of SYS with respect to SPEC by showing that SYS behaves the "same as" SPEC (if using an equivalence) or by showing that it refines SPEC (if using a preorder). Establishing the correctness of SYS with respect to SPEC can be done in a syntaxoriented manner or in a semantics-oriented manner. In the former case, an axiomatization of the behavioral relation of choice is used to show that one expression can be transformed into the other via syntactic manipulations. In the latter case, one can appeal directly to the definition of the behavioral relation, and to the operational semantics of the two expressions, to show that they are related. In certain cases, e.g., when SYS and SPEC are "finite-state", verification, be it syntax-based or semantics-based, can be carried out automatically.
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The advantages to an algebraic approach are the following. • System designers need learn only one language for specifications and designs. • Related processes may be substituted for one another inside other processes. This makes process algebras particularly suitable for the modular analysis of complex systems, since a specification and a design adhering to this specification may be used interchangeably inside larger systems. • Processes may be minimized with respect to the equivalence relation before being analyzed; this sometimes leads to orders of magnitude improvement in the performance of verification routines. Process-algebraic system descriptions can also be verified using model checking [3], a technique for ascertaining if a labeled transition system satisfies a correctness property given as a temporal-logic formula. Model checking has enjoyed considerable success in application to hardware designs. Progress is now being seen in other application domains such as software and protocol verification. 2. Classical roots Process algebra can be viewed as a generalization of the classical theory of formal languages and automata [4], focusing on system specification and behavior rather than language recognition and generation. Process algebra also embodies the principles of cellular automata [5] - cells receiving inputs from neighboring cells and then taking appropriate action - while adding a notion of programmability: nondeterminism, dynamic topologies, evolving cell behavior, etc. Process algebra lays the groundwork for a rigorous system-design ideology, providing support for specification, verification, implementation, testing and other life-cycle-critical activities. Interest in process algebra, however, extends beyond the system-design arena, to areas such as programming language design and semantics, complexity theory, real-time programming, and performance modeling and analysis.
3. About this Handbook This Handbook documents the fate of process algebra from its modem inception in the late 1970's to the present. It is intended to serve as a reference source for researchers, students, and system designers and engineers interested in either the theory of process algebra or in learning what process algebra brings to the table as a formal system description and verification technique. The Handbook is divided into six parts, the first five of which cover various theoretical and foundational aspects of process algebra. Part 6, the final part, is devoted to tools for applying process algebra and to some of the applications themselves. Each part contains between two and four chapters. Chapters are self-contained and can be read independently of each other. In total, there are 19 chapters spanning roughly 1300 pages. Collectively, the Handbook chapters give a comprehensive, albeit necessarily incomplete, view of the field. Part 1, consisting of four chapters, covers a broad swath of the basic theory of process algebra. In Chapter 1, The Linear Time - Branching Time Spectrum /, van Glabbeek gives
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a useful structure to, and an encyclopedic account of, the many behavioral relations that have been proposed in the process-algebra literature. Chapter 2, Trace-Oriented Models of Concurrency by Broy and Olderog, provides an in-depth presentation of trace-oriented models of process behavior, where a trace is a communication sequence that a process can perform with its environment. Aceto, Fokkink and Verhoef present a thorough account of Structural Operational Semantics in Chapter 3. Part 1 concludes with Chapter 4, Modal Logics and Mu-Calculi: An Introduction by Bradfield and Stirling. Modal logics, which extend classical logic with operators for possibility and necessity, play an important role in filling out the semantic picture of process algebra. Part 2 is devoted to the sub-specialization of process algebra known as finite-state processes. This class of processes holds a strong practical appeal as finite-state systems can be verified in an automatic, push-button style. The two chapters in Part 2 address finitestate processes from an axiomatic perspective: Chapter 5, Process Algebra with Recursive Operations by Bergstra, Fokkink and Ponse; and from an algorithmic one: Chapter 6, Equivalence and Preorder Checking for Finite-State Systems by Cleaveland and Sokolsky. Infinite-state processes, the subject of Part 3, capture process algebra at its most expressive. Chapter 7, the first of the three chapters in this part, A Symbolic Approach to Value-Passing Processes by Ingolfsdottir and Lin, systematically examines the class of infinite-state processes arising from the ability to transmit data from an arbitrary domain of values. Symbolic techniques are proposed as a method for analyzing such systems. Chapter 8, by Parrow, is titled An Introduction to the n-Calculus. This chapter investigates the area of mobile processes, an enriched form of value-passing process that is capable of transmitting communication channels and even processes themselves from one process to another. Finally, Burkhart, Caucal, Moller and Steffen consider the equivalence-checking and model-checking problems for a large variety of infinite-state processes in Chapter 9, Verification on Infinite Structures. The three chapters of Part 4 explore several extensions to process algebra that make it easier to model the kinds of systems that arise in practice. Chapter 10 focuses on real-time systems. Process Algebra with Timing: Real Time and Discrete Time by Middelburg and Baeten, presents a real-time extension of the process algebra ACP that extends ACP in a natural way. The final two chapters of Part 4 study the impact on process algebra of replacing the standard notion of "nondeterministically choose the next transition to execute" with one in which probability or priority information play pivotal roles. Chapter 11, Probabilistic Extensions of Process Algebras by Jonsson, Larsen and Yi, targets the probabilistic case, which is especially useful for modeling system failure, reliability, and performance. Chapter 12, Priority in Process Algebra by Cleaveland, Luttgen and Natarajan, considers the case of priority, and shows how a process algebra with priority can be used to model interrupts, prioritized choice and real-time behavior. Process algebra was originally conceived with the view that concurrency equals interleaving. That is, the concurrent execution of a collection of events can be modeled as their interleaved execution, in any order. More recent versions of process algebra known as non-interleaving process algebras, aim to model concurrency directly, for example, as embodied in Petri nets. The four chapters of Part 5 address this subject. Chapter 13, Partial-Order Process Algebra by Baeten and Basten, thoroughly considers the impact of a non-interleaving semantics on ACP. Chapter 14, A Unified Model for Nets and Process
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Algebras by Best, Devillers and Koutny, examines a range of issues that arise when process algebra and Petri nets are combined together. Another kind of non-interleaving treatment of concurrency is put forth in Chapter 15, Castellani's Process Algebras with Localities. In this approach, "locations" are assigned to parallel components, resulting in what Castellani calls a "distributed semantics" for process algebra. Finally, in Chapter 16, Gorrieri and Rensink's Action Refinement gives a thorough treatment of process algebra with action refinement, the operation of replacing a high-level atomic action with a low-level process. The interplay between action refinement and non-interleaving semantics is carefully considered. Part 6, the final part of the Handbook, contains three chapters dealing with tools and applications of process algebra. The first of these. Chapter 17, Algebraic Process Verification by Groote and Reniers, gives a close-up account of verification techniques for distributed algorithms and protocols, using process algebra extended with data (/xCRL). Chapter 18, Discrete Time Process Algebra and the Semantics of SDL by Bergstra, Middelburg and Usenko, introduces a discrete-time process algebra that is used to provide a formal semantics for SDL, a widely used formal description technique for teleconmiunications protocols. Finally, Chapter 19, A Process Algebra for Interworkings by Mauw and Reniers, devises a process-algebra-based semantics for Interworkings, a graphical design language of Philips Kommunikations Industrie.
Acknowledgements The editors gratefully acknowledge the constant support of Arjen Sevenster, our manager at Elsevier; without his efforts, this Handbook would not have seen the light of day. We are equally grateful to all the authors; their diligence, talent, and patience are greatly appreciated. We would also Uke to thank the referees, whose reports significantly enhanced the final contents of the Handbook. They are: Luca Aceto, Jos Baeten, Wan Fokkink, Rob Goldblatt, Hardi Hungar, Joost-Pieter Katoen, Alexander Letichevsky, Bas Luttik, Faron MoUer, Uwe Nestmann, Nikolaj Nikitchenko, Benjamin Pierce, Piet Rodenburg, Marielle Stoehnga, PS. Thiagarajan, and Yaroslav Usenko. Finally, we would like to thank Ranee Cleaveland for his help in writing this preface. Autumn 2000 Jan A. Bergstra (Amsterdam), Alban Ponse (Amsterdam), Scott A. Smolka (Stony Brook, New York)
References [1] J. A. Bergstra and J.W. Klop, Process algebra for synchronous communication. Inform, and Control 60 (1/3) (1984), 109-137. [2] S.D. Brookes, C.A.R. Hoare and A.W. Roscoe, A theory of communicating sequential processes, J. ACM 31 (3) (1984), 560-599.
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[3] E.M. Clarke, E.A. Emerson and A.P. Sistla, Automatic verification of finite-state concurrent systems using temporal logic specifications, ACM TOPLAS 8 (2) (1986). [4] J.E. Hopcroft and J.D. UUman, Introduction to Automata Theory, Languages, and Computation, AddisonWesley (1979). [5] J. von Neumann, Theory of self-reproducing automata, A.W. Burks, ed., Urbana, University of Illinois Press (1966). [6] R. Milner, A Calculus of Communicating Systems, Lecture Notes in Comput. Sci. 92, Springer-Verlag (1980). [7] G.D. Plotkin, A structural approach to operational semantics. Report DAIMI FN-19, Computer Science Department, Aarhus University (1981).
Jan A. Bergstra^'^, Alban Ponse^'^, Scott A. Smolka"* ^CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands http://www. cwi. nU ^ University of Amsterdam, Programming Research Group, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands http://www. science, uva. nl/research/prog/ ^Utrecht University, Department of Philosophy, Heidelberglaan 8, 3584 CS Utrecht, The Netherlands http://www.phil. uu. nl/eng/home. htmlE-mail: State University of New York at Stony Brook, Department of Computer Science Stony Brook, NY 11794-4400, USA http://www. CS. sunysb. eduJ E-mails: janb@science, uva.nl, alban @science, uva.nl, sas @cs.sunysb. edu
List of Contributors Aceto, L., Aalborg University, Aalhorg (Ch. 3). Baeten, J.C.M. Eindhoven University of Technology, Eindhoven (Chs. 10, 13). Basten, T., Eindhoven University of Technology, Eindhoven (Ch. 13). Bergstra, J.A., University of Amsterdam, Amsterdam and Utrecht University, Utrecht (Chs. 5, 18). Best, E., Carl von Ossietzky Universitdt, Oldenburg (Ch. 14). Bradfield, J.C., University of Edinburgh, Edinburgh, UK (Ch. 4). Broy, M., Technische Universitdt MUnchen, MUnchen (Ch. 2). Burkart, O., Universitdt Dortmund, Dortmund (Ch. 9). Castellani, I., INRIA, Sophia-Antipolis (Ch. 15). Caucal, D., IRISA, Rennes (Ch. 9). Cleaveland, R., SUNYat Stony Brook, Stony Brook, NY (Chs. 6, 12). Devillers, R., Universite Libre de Bruxelles, Bruxelles (Ch. 14). Fokkink, W.J., CWI, Amsterdam (Chs. 3, 5). Glabbeek, R.J. van, Stanford University, Stanford, CA (Ch. 1). Gorrieri, R., Universitd di Bologna, Bologna (Ch. 16). Groote, J.F., Eindhoven University of Technology, Eindhoven (Ch. 17). Ingolfsdottir, k., Aalborg University, Aalborg (Ch. 7). Jonsson, B., Uppsala University, Uppsala (Ch. 11). Koutny, M., University of Newcastle, Newcastle upon Tyne, UK (Ch. 14). Larsen, K.G., Aalborg University, Aalborg (Ch. 11). Lin, H., Institute of Software, Chinese Academy of Sciences, Republic of China (Ch. 7). Liittgen, G., NASA Langley Research Center, Hampton, VA (Ch. 12). Mauw, S., Eindhoven University of Technology, Eindhoven (Ch. 19). Middelburg, C.A., Eindhoven University of Technology, Eindhoven and Utrecht University, Utrecht {Ch^. 10, 18). MoUer, R, University of Wales Swansea, Swansea, UK (Ch. 9). Natarajan, V., IBM Corporation, Research Triangle Park, NC (Ch. 12). Olderog, E.-R., Universitdt Oldenburg, Oldenburg (Ch. 2). Parrow, J., Royal Institute of Technology, Stockholm (Ch. 8). Ponse, A., University of Amsterdam and CWI, Amsterdam (Ch. 5). Renters, M.A., Eindhoven University of Technology, Eindhoven (Chs. 17, 19). Rensink, A., University ofTwente, Enschede (Ch. 16). Sokolsky, O., University of Pennsylvania, Philadelphia, PA (Ch. 6). Steffen, B., Universitdt Dortmund, Dortmund (Ch. 9). Stirling, C , University of Edinburgh, Edinburgh, UK (Ch. 4).
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Usenko, Y.S., CWl Amsterdam (Ch. 18). Verhoef, C , Free University of Amsterdam, Amsterdam (Ch. 3). Wang Yi, Uppsala University, Uppsala (Ch. 11).
CHAPTER 1
The Linear Time - Branching Time Spectrum I.* The Semantics of Concrete, Sequential Processes RJ. van Glabbeek Computer Science Department, Stanford University, Stanford, CA 94305-9045, USA E-mail:
[email protected] Contents Introduction 1. Labelled transition systems and process graphs 1.1. Labelled transition systems 1.2. Process graphs 1.3. Embedding labelled transition systems in G 1.4. Equivalences relations and preorders on labelled transition systems 1.5. Initial nondeterminism 2. Trace semantics 3. Completed trace semantics 4. Failures semantics 5. Failure trace semantics 6. Ready trace semantics 7. Readiness semantics and possible-futures semantics 8. Simulation semantics 9. Ready simulation semantics 10. Reactive versus generative testing scenarios 11. 2-nested simulation semantics 12. Bisimulation semantics 13. Tree semantics 14. Possible worlds semantics 15. Summary 16. Deterministic and saturated processes 17. Complete axiomatizations 17.1. A language for finite, concrete, sequential processes
5 9 9 10 11 12 13 13 16 18 23 27 30 35 39 43 45 47 55 56 59 64 70 70
*This is an extension of [20]. The research reported in this paper has been initiated at CWI in Amsterdam, continued at the Technical University of Munich, and finalized at Stanford University. It has been supported by Sonderforschungsbereich 342 of the TU Munchen and by ONR under grant number N00014-92-J-1974. Part of it was carried out in the preparation of a course Comparative Concurrency Semantics, given at the University of Amsterdam, Spring 1988. A coloured version of this paper is available at http://boole.stanford.edu/pub/spectruml.ps.gz. HANDBOOK OF PROCESS ALGEBRA Edited by Jan A. Bergstra, Alban Ponse and Scott A. Smolka © 2001 Elsevier Science B.V. All rights reserved
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\12. Axiomatizing the equivalences 17.3. Axiomatizing the preorders 17.4. A language forfinite,concrete, sequential processes with internal choice 18. Criteria for selecting a semantics for particular applications 19. Distinguishing deadlock and successful termination Concluding remarks Acknowledgement References Subject index
Abstract In this paper various semantics in the linear time - branching time spectrum are presented in a uniform, model-independent way. Restricted to the class of finitely branching, concrete, sequential processes, only fifteen of them turn out to be different, and most semantics found in the literature that can be defined uniformly in terms of action relations coincide with one of these fifteen. Several testing scenarios, motivating these semantics, are presented, phrased in terms of 'button pushing experiments' on generative and reactive machines. Finally twelve of these semantics are applied to a simple language for finite, concrete, sequential, nondeterministic processes, and for each of them a complete axiomatization is provided.
72 78 81 85 91 94 95 95 97
The linear time - branching time spectrum I
5
Introduction Process theory. A process is the behaviour of a system. The system can be a machine, an elementary particle, a communication protocol, a network of falling dominoes, a chess player, or any other system. Process theory is the study of processes. Two main activities of process theory are modelling and verification. Modelling is the activity of representing processes, mostly by mathematical structures or by expressions in a system description language. Verification is the activity of proving statements about processes, for instance that the actual behaviour of a system is equal to its intended behaviour. Of course, this is only possible if a criterion has been defined, determining whether or not two processes are equal, i.e., two systems behave similarly. Such a criterion constitutes the semantics of a process theory. (To be precise, it constitutes the semantics of the equality concept employed in a process theory.) Which aspects of the behaviour of a system are of importance to a certain user depends on the environment in which the system will be running, and on the interests of the particular user. Therefore it is not a task of process theory to find the 'true' semantics of processes, but rather to determine which process semantics is suitable for which applications. Comparative concurrency semantics. This paper aims at the classification of process semantics.^ The set of possible process semantics can be partially ordered by the relation 'makes strictly more identifications on processes than', thereby becoming a complete lattice.-^ Now the classification of some useful process semantics can be facilitated by drawing parts of this lattice and locating the positions of some interesting process semantics, found in the literature. Furthermore the ideas involved in the construction of these semantics can be unravelled and combined in new compositions, thereby creating an abundance of new process semantics. These semantics will, by their intermediate positions in the semantic lattice, shed light on the differences and similarities of the established ones. Sometimes they also turn out to be interesting in their own right. Finally the semantic lattice serves as a map on which it can be indicated which semantics satisfy certain desirable properties, and are suited for a particular class of applications. Most semantic notions encountered in contemporary process theory can be classified along four different fines, corresponding with four different kinds of identifications. First there is the dichotomy of linear time versus branching time: to what extent should one identify processes differing only in the branching structure of their execution paths? Secondly there is the dichotomy of interleaving semantics versus partial order semantics: to what extent should one identify processes differing only in the causal dependencies between their actions (while agreeing on the possible orders of execution)? Thirdly one encounters This field of research is called comparative concurrency- semantics, a terminology first used by Meyer in [36]. Here concurrency is taken to be synonymous with process theory, although strictly speaking it is only the study of parallel (as opposed to sequential) processes. These are the behaviours of systems capable of performing different actions at the same time. In this paper the term concurrency is considered to include sequential process theory. This may be justified since much work on sequential processes is intended to facilitate later studies involving parallehsm. ^ The supremum of a set of process semantics is the semantics identifying two processes whenever they are identified by every semantics in this set.
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different treatments of abstraction from internal actions in a process: to what extent should one identify processes differing only in their internal or silent actions? And fourthly there are different approaches to infinity: to what extent should one identify processes differing only in their infinite behaviour? These considerations give rise to a four-dimensional representation of the proposed semantic lattice. However, at least three more dimensions can be distinguished. In this paper, stochastic and real-time aspects of processes are completely neglected. Furthermore it deals with uniform concurrency"^ only. This means that processes are studied, performing actions^ a,b,c,... which are not subject to further investigations. So it remains unspecified if these actions are in fact assignments to variables or the falling of dominoes or other actions, ff also the options are considered of modelling (to a certain degree) the stochastic and realtime aspects of processes and the operational behaviour of the elementary actions, three more parameters in the classification emerge. Process domains. In order to be able to reason about processes in a mathematical way, it is common practice to represent processes as elements of a mathematical domain.^ Such a domain is called a process domain. The relation between the domain and the world of real processes is mostly stated informally. The semantics of a process theory can be modelled as an equivalence on a process domain, called a semantic equivalence. In the literature one finds among others: • graph domains, in which a process is represented as a process graph, or state transition diagram, • net domains, in which a process is represented as a (labelled) Petri net, • event structure domains, in which a process is represented as a (labelled) event structure, • explicit domains, where a process is represented as a mathematically coded set of its properties, • projective limit domains, which are obtained as projective limits of series of finite term domains, • and term domains, in which a process is represented as a term in a system description language. Action relations. Write p - ^ q if the process p can evolve into the process q, while performing the action a. The binary predicates —^ are called action relations. The semantic equivalences which are treated in this paper will be defined entirely in terms of action relations. Hence these definitions apply to any process domain on which action relations are defined. Such a domain is called a labelled transition system. Furthermore they will be defined uniformly in terms of action relations, meaning that all actions are treated in the same way. For reasons of convenience, even the usual distinction between internal and external actions is dropped in this paper. '^ The term uniform concurrency is employed by De Bakker et al. [8]. ^ Strictly speaking processes do not perform actions, but systems do. However, for reasons of convenience, this paper sometimes uses the word process, when actually referring to a system of which the process is the behaviour. I use the word domain in the sense of universal algebra; it can be any class of mathematical objects - typically the first component of an algebra; the other component being a collection of operators defined on this domain. Without further adjectives I do not refer to the more restrictive domains employed in domain theory.
The linear time - branching time spectrum I Finitely branching, concrete, sequential processes. Being a first step, this paper limits itself to a very simple class of processes. First of all only sequential processes are investigated: processes capable of performing at most one action at a time. Furthermore, instead of dropping the usual distinction between internal and external actions, one can equivalently maintain to study concrete processes: processes in which no internal actions occur. For this simple class of processes the announced semantic lattice collapses in two out of four dimensions and covers only the infinitary linear time - branching time spectrum. Moreover, the main interest is infinitely branching processes: processes having in each state only finitely many possible ways to proceed. The material pertaining to infinitely branching processes - coloured brown in the electronic version of this paper - can easily be omitted in first reading. Literature. In the literature on uniform concurrency 12 semantics can be found which are uniformly definable in terms of action relations and different on the domain of finitely branching, sequential processes (see Figure 1). The coarsest one (i.e., the semantics making the most identifications) is trace semantics, as presented in Hoare [30]. In trace semantics only partial traces are employed. The finest one (making less identifications than any of the others) is bisimulation semantics, as presented in Milner [39]. Bisimulation semantics is the standard semantics for the system description language CCS (Milner [37]). The
{frae semantics)
bisimulation
semantics
2-nested simulation
ready simulation
semantics
semantics
I
possible-fiLtiLres semantics
possible worlds se^mantics
ready trace semantics
failure trace semantics simulation
ir.adincss semantics
semantics failures semantics
completed trace semantics
trace sem.antics
Fig. 1. The linear time - branching time spectrum.
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notion of bisimulation was introduced in Park [41]. Bisimulation equivalence is a refinement of observational equivalence, as introduced by Hennessy and Milner in [27]. On the domain of finitely branching, concrete, sequential processes, both equivalences coincide. Also the semantics of De Bakker and Zucker, presented in [9], coincides with bisimulation semantics on this domain. Then there are ten semantics in between. First of all a variant of trace semantics can be obtained by using complete traces besides partial ones. In this paper it is called completed trace semantics. Failures semantics is introduced in Brookes, Hoare and Roscoe [13], and used in the construction of a model for the system description language CSP (Hoare [29,31]). It is finer than completed trace semantics. The semantics based on testing equivalences, as developed in De Nicola and Hennessy [17], coincides with failures semantics on the domain of finitely branching, concrete, sequential processes, as do the semantics of Kennaway [34] and Darondeau [15]. This has been established in De Nicola [16]. In Olderog and Hoare [40] readiness semantics is presented, which is shghtly finer than failures semantics. Between readiness and bisimulation semantics one finds ready trace semantics, as introduced independently in Pnueli [43] (there called barbed semantics), Baeten, Bergstra and Klop [6] and Pomello [44] (under the name exhibited behaviour semantics). The natural completion of the square, suggested by failures, readiness and ready trace semantics y'loXds failure trace semantics. For finitely branching processes this is the same as refusal semantics, introduced in Phillips [42]. Simulation semantics, based on the classical notion of simulation (see, e.g.. Park [41]), is independent of the last five semantics. Ready simulation semantics was introduced in Bloom, Istrail and Meyer [12] under the name GSOS trace congruence. It is finer than ready trace as well as simulation semantics. In Larsen and Skou [35] a more operational characterization of this equivalence was given under the name |-Z?/5/mw/ar/oAz equivalence. The (denotational) notion of possible worlds semantics of Veglioni and De Nicola [49] fits between ready trace and ready simulation semantics. Finally 2-nested simulation semantics, introduced in Groote and Vaandrager [25], is located between ready simulation and bisimulation semantics, SLud possible-futures semantics, as proposed in Rounds and Brookes [46], can be positioned between 2-nested simulation and readiness semantics. Tree semantics, employed in Winskel [50], is even finer than bisimulation semantics. However, a proper treatment requires more than mere action relations. About the contents. The first section of this paper introduces labelled transition systems and process graphs. A labelled transition system is any process domain that is equipped with action relations. The domain of process graphs or state transition diagrams is one of the most popular labelled transition systems. In Sections 2-14 all semantic equivalences mentioned above are defined on arbitrary labelled transition systems. In particular these definitions apply to the domain of process graphs. Most of the equivalences can be motivated by the observable behaviour of processes, according to some testing scenario. (Two processes are equivalent if they allow the same set of possible observations, possibly in response to certain experiments.) I will try to capture these motivations in terms of button pushing experiments (cf. Milner [37], pp. 10-12). Furthermore the semantics will be partially ordered by the relation 'makes at least as many identifications as'. This yields the linear time - branching time spectrum. Counterexamples are provided, showing that on the graph domain this ordering cannot be further expanded. However, for deterministic
The linear time - branching time spectrum I
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processes the spectrum collapses, as was first observed by Park [41]. Secfion 6 describes various other classes of processes on which parts of the spectrum collapse. In Section 17, the semantics are applied to a simple language for finite, concrete, sequential, nondeterministic processes, and for twelve of them a complete axiomatization is provided. Section 18 applies a few criteria indicating which semantics are suitable for which applications. Finally, in Section 19 the work of this paper is extended to labelled transition systems that distinguish between deadlock and successful termination. With each of the semantic equivalences treated in this paper (except for tree semantics) a preorder is associated that may serve as an implementation relation between processes. The results obtained for the equivalences are extended to the associated preorders as well. 1. Labelled transition systems and process graphs 1.1. Labelled transition systems In this paper processes will be investigated that are capable of performing actions from a given set Act. By an action any activity is understood that is considered as a conceptual entity on a chosen level of abstraction. Actions may be instantaneous or durational and are not required to terminate, but in a finite time only finitely many actions can be carried out. Any activity of an investigated process should be part of some action a e Act performed by the process. Different activities that are indistinguishable on the chosen level of abstraction are interpreted as occurrences of the same action a e Act. A process is sequential if it can perform at most one action at the same time. In this paper only sequential processes will be considered. A class of sequential processes can often be conveniently represented as a labelled transition system. This is a domain P on which infix written binary predicates - ^ are defined for each action a e Act. The elements of P represent processes, and p —^ q means that p can start performing the action a and after completion of this action reach a state where q is its remaining behaviour. In a labelled transition system it may happen that p - % q and p -^ r for different actions a and b or different processes q and r. This phenomenon is called branching. It need not be specified how the choice between the alternatives is made, or whether a probability distribution can be attached to it. Certain actions may be synchronizations of a process with its environment, or the receipt of a signal sent by the environment. Naturally, these actions can only occur if the environment cooperates. In the labelled transition system representation of processes all these potential actions are included, so p —^ q merely means that there is an environment in which the action a can occur. Notation. For any alphabet iJ, let iJ* be the set of finite sequences and E^ the set of infinite sequences over T. Z"^ := iT* U T ^ . Write e for the empty sequence, op for the concatenation of a G i7* and p e X"^, and a for the sequence consisting of the single symbol a e U. DenNlTlON 1.1. A labelled transition system is a pair (P, ->) with P a class and -> C P X Ac? X P, such that for /? € P and a e Act the class {^ G P | (/?,«, ^) € ->} is a set.
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Most of this paper should be read in the context of a given labelled transition system (P, ->), ranged over by p,q,r, Write p - ^ q for (/?,a,q) £-^. The binary predicates - ^ are called action relations. DEFINITION 1.2 (Remark that the following, concepts are defined in terms of action relations only). • The generalized action relations - ^ for a eAct* are defined recursively by: (1) /? —^ /7, for any process p. (2) (p,a,q) e^- with a e Act implies p -^ q with a 6 Act*. (3) p - ^ q - ^ r implies p - ^ r. In words: the generalized action relations - ^ are the reflexive and transitive closure of the ordinary action relations - % . p - ^ q means that p can evolve into q, while performing the sequence a of actions. Remark that the overloading of the notion p - % q is quite harmless. • A process ^ G P is reachable from p eFif p - ^ q for some a e Act*. • The set of initial actions of a process p is defined by: I(p) = {a £ Act \3q: p —^ q). • A process p e P infinite if the set {(a,
