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Denotational fixed-point semantics for constructive scheduling of synchronous concurrency

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Abstract

The synchronous model of concurrent computation (SMoCC) is well established for programming languages in the domain of safety-critical reactive and embedded systems. Translated into mainstream C/Java programming, the SMoCC corresponds to a cyclic execution model in which concurrent threads are synchronised on a logical clock that cuts system computation into a sequence of macro-steps. A causality analysis verifies the existence of a schedule on memory accesses to ensure each macro-step is deadlock-free and determinate. We introduce an abstract semantic domain \(I(\mathbb {D}, \mathbb {P})\) and an associated denotational fixed-point semantics for reasoning about concurrent and sequential variable accesses within a synchronous cycle-based model of computation. We use this domain for a new and extended behavioural definition of Berry’s causality analysis in terms of approximation intervals. The domain \(I(\mathbb {D}, \mathbb {P})\) extends the domain \(I(\mathbb {D})\) from our previous work and fixes a mistake in the treatment of initialisations. Based on this fixed-point semantics we propose the notion of Input Berry-constructiveness (IBC) for synchronous programs. This new IBC class lies properly between strong (SBC) and normal Berry-constructiveness (BC) defined in previous work. SBC and BC are two ways to interpret the standard constructive semantics of synchronous programming, as exemplified by imperative SMoCC languages such as Esterel or Quartz. SBC is often too restrictive as it requires all variables to be initialised by the program. BC can be too permissive because it initialises all variables to a fixed value, by default. Where the initialisation happens through the memory, e.g., when carrying values from one synchronous tick to the next, then IBC is more appropriate. IBC links two levels of execution, the macro-step level and the micro-step level. We prove that the denotational fixed-point analysis for IBC, and hence Berry’s causality analysis, is sound with respect to operational micro-level scheduling. The denotational model can thus be viewed as a compositional presentation of a synchronous scheduling strategy that ensures reactiveness and determinacy for imperative concurrent programming.

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Notes

  1. This is sometimes referred to as the ‘LAGS’ model and not to be confused with the well-known but orthogonal ‘GALS’ model which features globally asynchronous and locally synchronous computations.

  2. There is an informal sketch of a micro-step semantics in [12], Sect. 4.3] which is not developed further or formally related with the fixed-point semantics for macro-steps.

  3. This stands here for “pure Synchronous Constructive Language” indicating not only that signal variables in pSCL carry Boolean status but also that pSCL is a minimalistic version of control-flow synchronous languages in an abstract algebraic syntax.

  4. This extra bit for indicating predicted resets has been missing in our publication [4] where this fixed-point analysis was introduced for the first time.

  5. In other words, the free set-theoretic “collection semantics”, which defines the completion code of a program as the set of all it possible completions (under a given choice of environments), would produce exactly the sets in \(I(\mathbb {C})\). We could have defined \(I(\mathbb {C})\) more generously as the set of subsets of completion codes \(\mathcal{P}\{ \bot , 0, 1 \}\). However, our explicit description reveals more of the algebraic properties of \(I(\mathbb {C})\) than \(\mathcal{P}\{ \bot , 0, 1 \}\). For instance, it makes clear that the internal logic of \(I(\mathbb {C})\) is not a Boolean algebra.

  6. The oversampling feature of Signal may be seen as an implicit form of reaction to absence in the asynchronous data-flow part of [77]. Synchronous reaction to absence would map \(\bot \) to \(\top \) and \(\top \) to \(\bot \) in the domain \(\mathcal {D}\) which does not seem to be expressible by the control-flow operators considered in [77].

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Acknowledgments

This work has been supported by the German National Research Council DFG as part of the PRETSY Project (HA 4407/6-1, ME 1427/6-1). The authors would also like to thank the anonymous reviewers for their trenchant yet constructive criticisms and their useful suggestions regarding related work and expository improvements of the paper.

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  1. Faculty of Information Systems and Applied Computer Sciences, Bamberg University, Bamberg, Germany

    Joaquín Aguado & Michael Mendler

  2. Department of Computer Science, Kiel University, Kiel, Germany

    Reinhard von Hanxleden & Insa Fuhrmann

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  1. Joaquín Aguado
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Aguado, J., Mendler, M., von Hanxleden, R. et al. Denotational fixed-point semantics for constructive scheduling of synchronous concurrency. Acta Informatica 52, 393–442 (2015). https://doi.org/10.1007/s00236-015-0238-x

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