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Event algebra for transition systems composition application to timed automata

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Abstract

Formal specification languages have a lot of notions in common. They all introduce entities usually called processes, offer similar operators, and most importantly define their operational semantics based on labelled transition systems (LTS). However, each language defines specific synchronizing and/or memory structures. For instance, in CSP, the synchronization is defined between identical events, while in CCS and in synchronization vectors-based views it is defined respectively between complementary events or between possibly different events. In this paper, we aim at capturing some similarities of specification languages by defining a label-based formal framework for reasoning on LTS, their semantics and related properties. Firstly, we define a high-level synchronization mechanism in the form of an abstract label structure and identify some properties. Then, we introduce operators for composing and transforming label structures, study their intrinsic properties and explore how label structure properties propagate. Secondly, we introduce a LTS-based behavioral framework. We then lift the label structure composition and transformation operators to the LTS level and establish LTS-related properties derived from those of their underlying labelled structures. Thirdly, we consider extended transition systems, more specifically timed automata, as LTS built on top of specific labelled structures. Their semantics is reconstructed by applying operators of our framework on the syntactic LTS, which allows the direct proof of some semantic properties such as compositionality.

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Notes

  1. not their union.

  2. The set projection could also be defined as \(S\downarrow _1 = \mathbf {fst}[\bowtie ^{-1}[S]]\) where \(\mathbf {fst}(a,b) = a\).

  3. \(l \le l'\) is defined by \(l \le l' \triangleq (l,l') \in \mathbf {dom}(\bowtie ) \wedge l \bowtie l' = l'\) in Sect. 2.4.

  4. not as a sub-expression of the right-hand side of the set-difference operator.

  5. We do not consider here the global variables and clocks can only be reset to 0.

  6. This is verified for CCS and CSP.

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Acknowledgements

We would like to thank the anonymous reviewers for their careful reading of our manuscript and their helpful suggestions and comments.

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Correspondence to Jean-Paul Bodeveix.

Appendices

Coq definition of a label structure, associativity and commutativity

figure t

Coq definition for the product of label structures

The \(\bowtie \) operator and its domain of definition are specified in Coq as follows:

figure u

Coq definition for the sum of label structures

figure v

Coq statement of the general associativity lemma for LTS product

figure w

Coq script for compositionality

This Coq proof script illustrates how the various results established in the paper can be reused to prove the compositionality of the product of two automata.

figure x

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Fares, E., Bodeveix, JP. & Filali, M. Event algebra for transition systems composition application to timed automata. Acta Informatica 55, 363–400 (2018). https://doi.org/10.1007/s00236-017-0302-9

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  • DOI: https://doi.org/10.1007/s00236-017-0302-9

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