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‘True Concurrency’ Semantics for Time Petri Nets with Weak Time and Persistent Atomic Policies

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Abstract

Dense-Time Petri nets (TPNs), where time intervals for transition firings are assigned, are now a well-established model, which is used to describe and study safety-critical systems. Generally, the state space of the TPN is uncountably infinite, which increases the complexity of model analysis. ‘True concurrency’ semantics represents the behavior of the TPN as a set of actions, the causality relation between which is modeled by a partial order, and the absence of causality implies concurrency. This representation is more adequate for verifying properties of concurrent systems, e.g., deadlock-freedom, fairness, maximum parallelism, etc. This paper introduces and investigates step semantics (based on sets of concurrent actions) and partial order semantics (based on sets of causally ordered and concurrent actions) for TPNs the behavior of which is governed by the weak time policy (i.e., the passage of model time is not constrained by transition firing) and persistent atomic memory technique (i.e., transition firing is regarded as an atomic action when clocks are reset).

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Notes

  1. Two transitions are in conflict if they have a common input place and, when they fire, one transition consumes input tokens of the other transition, thereby making it disabled.

  2. Recall that, in [8, 9], it was shown that the weak and strong time semantics are incomparable and persistent atomic policy is preferable for modeling and verifying real-world systems.

  3. In the case of the intermediate memory policy, this predicate is defined as \( \uparrow enabled(t,M,U)\) = \((t \notin En(M{{{{\backslash }}}^{ \bullet }}U) \vee t \in U) \wedge t\) ∈ \(En((M{{{{\backslash }}}^{ \bullet }}U) \cup {{U}^{ \bullet }})\); in the case of the atomic memory policy, it is defined as \( \uparrow enabled(t,M,U)\) = \((t \notin En(M) \vee t \in U) \wedge t\) ∈ \(En((M{{{{\backslash }}}^{ \bullet }}U) \cup {{U}^{ \bullet }})\).

  4. Note that cuts \(C\) and \(C' = (C{{{{\backslash }}}^{ \bullet }}e) \cup {{e}^{ \bullet }}\) (\(C' = (C{{{{\backslash }}}^{ \bullet }}V) \cup {{V}^{ \bullet }}\)) uniquely define event e (step V) the occurrence of which in C leads to \(C'\), which is guaranteed by the structure of the CN.

  5. Thus, it is forbidden for the cut obtained as a result of firing an enabled event (step) to have an indefinite temporal characteristic (i.e., \( \bot \)), because this means that the firing time of this event (step) has not yet come.

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Authors and Affiliations

  1. Ershov Institute of Informatics Systems, Siberian Branch, Russian Academy of Sciences, pr. Akademika Lavrent’eva 6, 630090, Novosibirsk, Russia

    I. B. Virbitskaite & A. Yu. Zubarev

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  1. I. B. Virbitskaite
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  2. A. Yu. Zubarev
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Correspondence to I. B. Virbitskaite or A. Yu. Zubarev.

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Translated by Yu. Kornienko

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Virbitskaite, I.B., Zubarev, A.Y. ‘True Concurrency’ Semantics for Time Petri Nets with Weak Time and Persistent Atomic Policies. Program Comput Soft 47, 389–401 (2021). https://doi.org/10.1134/S0361768821050078

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  • DOI: https://doi.org/10.1134/S0361768821050078

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