J. W. de Bakker

Processes and the denotational semantics of concurrency

A framework allowing a unified and rigorous definition of the semantics of concurrency is proposed. The mathematical model introduces processes as elements of process domains which are obtained as solutions of domain equations in the sense of Scott and Plotkin. Techniques of metric topology as proposed, e.g., by Nivat are used to solve such equations. Processes are then used as meanings of statements in languages with concurrency. Three main concepts are treated, viz. parallellism (arbitrary interleaving of sequences of elementary actions), synchronization, and communication. These notions are embedded in languages which also feature classical sequential concepts such as assignment, tests, iteration or recursion, and guarded commands. In the definitions, a sequence of process domains of increasing complexity is used. The languages discussed include Milners calculus for communicating systems and Hoares communicating sequential processes. The paper concludes with a section with brief remarks on miscellaneous notions in concurrency, and two appendices with mathematical details.

on Linear time, branching time and partial order in logics and models for concurrency

Time, logic and computation.- Process theory based on bisimulation semantics.- Branching time temporal logic.- Observing processes.- The anchored version of the temporal framework.- Basic notions of trace theory.- An introduction to event structures.- A logic for the description of behaviours and properties of concurrent systems.- Permutation of transitions: An event structure semantics for CCS and SCCS.- Expressibility results for linear-time and branching-time logics.- Partial orderings descriptions and observations of nondeterministic concurrent processes.- Modeling concurrency by partial orders and nonlinear transition systems.- An efficient verification method for parallel and distributed programs.- A logic for distributed transition systems.- Fully abstract models for a process language with refinement.- Strong bisimilarity on nets: A new concept for comparing net semantics.- Nets of processes and data flow.- Towards a temporal logic for causality and choice in distributed systems.- Correctness and full abstraction of metric semantics for concurrency.- Temporal logics for CCS.- Behavioural presentations.- Computation tree logic and regular ?-languages.

Denotational semantics of concurrency

A general framework for the denotational treatment of concurrency is introduced. The key idea is the notion of process which is element of a domain obtained as solution of a domain equation in the style as considered previously by Plotkin. We use tools from metric topology as advocated by Nivat to solve this equation, show how operations upon processes can be defined conveniently, and illustrate the approach with the definition of a variety of concepts as encountered in the study of concurrency. Only few proofs of the supporting mathematical theory are given; full proofs will appear in the final version of the paper.

Linear time and branching time semantics for recursion with merge

Abstract We consider two ways of assigning semantics to a class of statements built from a set of atomic actions (the ‘alphabet’), by means of sequential composition, nondeterministic choice, recursion and merge (arbitrary interleaving). The first is linear time semantics (LT), stated in terms of trace theory; the semantic domain is the collection of all closed sets of finite and infinite words. The second is branching time semantics (BT), as introduced by De Bakker and Zucker; here the semantic domain is the metric completion of the collection of finite processes. For LT we prove the continuity of the operations (merge, sequential composition) in a direct, combinatorial way. Next, a connection between LT and BT is established by means of the operation trace which assigns to a process its set of traces. We show that the trace set of a process is closed and that trace is continuous. This requires the compactness of the semantic domains, ensured by the finiteness of the alphabet. Using trace , we then can carry over BT into LT.

Contrasting themes in the semantics of imperative concurrency

A survey is given of work performed by the authors in recent years concerning the semantics of imperative concurrency. Four sample languages are presented for which a number of operational and denotational semantic models are developed. All languages have parallel execution through interleaving, and the last three have as well a form of synchronization. Three languages are uniform, i.e., they have uninterpreted elementary actions; the fourth is nonuniform and has assignment, tests and value-passing communication. The operational models build on Hennessy-Plotkin transition systems; as denotational structures both metric spaces and cpo domains are employed. Two forms of nondeterminacy are distinguished, viz. the local and global variety. As associated model-theoretic distinction that of linear time versus branching time is investigated. In the former we use streams, i.e. finite or infinite sequences of actions; in the latter the (metrically based) notion of process is introduced. We furthermore study a model with only finite observations. Ready sets also appear, used as technical tool to compare various semantics. Altogether, ten models for the four languages are described, and precise statements on (the majority of) their interrelationships are made. The paper supplies no proofs; for these references to technical papers by the authors are provided.

Metric semantics for concurrency

An overview is given of work we have done in recent years on the semantics of concurrency, concentrating on semantic models built on metric structures. Three contrasting themes are discussed, viz. (i) uniform or schematic versus nonuniform or interpreted languages; (ii) operational versus denotational semantics, and (iii) linear time versus branching time models. The operational models are based on Plotkins transition systems. Language constructs which receive particular attention are recursion and merge, synchronization and global nondeterminacy, process creation, and communication with value passing. Various semantic equivalence results are established. Both in the definitions and in the derivation of these equivalences, essential use is made of Banachs theorem for contracting functions.

Transition systems metric spaces and ready sets in the semantics of uniform co currency

Abstract Transition systems as proposed by Hennessy and Plotkin are defined for a series of three languages featuring concurrency. The first has shuffle and local nondeterminacy, the second synchronization merge and local nondeterminacy, and the third synchronization merge and global nondeterminacy. The languages are all uniform in the sense that the elementary actions are uninterpreted. Throughout, infinite behaviour is taken into account and modelled with infinitary languages in the sense of Nivat. A comparison with denotational semantics is provided. For the first two languages, a linear time model suffices; for the third language a branching time model with processes in the sense of de Bakker and Zucker is described. In the comparison an important role is played by an intermediate semantics in the style of Hoare and Olderogs specification oriented semantics. A variant on the notion of ready set is employed here. Precise statements are given relating the various semantics terms of a number of abstraction operators.

Comparative metric semantics for concurrent Prolog

Abstract This paper shows the equivalence of two semantics for a version of Concurrent Prolog with non-flat guards: an operational semantics based on a transition system and a denotational semantics which is a metric semantics (the domains are metric spaces). We do this in the following manner. First a uniform language L is considered, that is a language where the atomic actions have arbitrary interpretations. For this language we define an operational and a denotational semantics, and we prove that the denotational semantics is correct with respect to the operational semantics. This result relies on Banachs fixed point theorem. Techniques stemming from imperative languages are used. Then we show how to translate a Concurrent Prolog program to a program in L by selecting certain basic sets for L and then instantiating the interpretation function for the atomic actions. In this way we induce the two semantics for Concurrent Prolog and the equivalence between the two semantics.

Infinite streams and finite observations in the semantics of uniform concurrency

Two ways of assigning meaning to a language with uniform concurrency are presented and compared. The language has uninterpreted elementary actions from which statements are composed using sequential composition, nondeterministic choice, parallel composition with communication, and recursion. The first semantics uses infinite streams in the sense which is a refinement of the linear time semantics of De Bakker et al. The second semantics uses the finite observations of Hoare et al., situated “in between” the divergence and readiness semantics of Olderog & Hoare. It is shown that the two models are isomorphic and that this isomorphism induces an equivalence result between the two semantics.

On infinite computations in denotational semantics

Abstract Finite and, especially, infinite computations in languages with iteration or recursion are studied in the framework of denotational semantics, and a theorem is proved which relates their syntactic and semantic characterizations. A general proof method is presented to establish this type of relations, and it is shown how—in an induction on the structure of the syntactic constructs of the language—the recursive case follows from the non-recursive one by applying a general definitional scheme. The method is applicable to a variety of other problems concerning recursive constructs such as, for example, fixed point characterizations of several notions of weakest precondition. Also, the connections with the theory of languages with infinite words are discussed, in particular with a substitution theorem due to Nivat (1978).