Abstract
The intention of the paper is towards a causality-based framework for developing, studying, and comparing testing equivalences with causal net and causal tree semantics in the setting of time Petri nets (elementary net systems whose transitions are labeled with time firing intervals, can fire only if their lower time bounds are attained, and are forced to fire when their upper time bounds are reached). We establish the relationships between the equivalences showing the similarity of the semantics under consideration. This allows studying in detail the timing behaviour in addition to the degrees of relative concurrency of processes generated during the functioning of time Petri nets.
This work is supported in part by DFG (project CAVER, grant BE 1267/14-1).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For technical convenience, we do not use the classical definition: a transition \(t\in T\) is enabled at a marking M if \(^{\bullet }t\subseteq M\) and \(M \cap t^{\bullet }=\emptyset \). We will require the second part in the definition of the contact-free property.
- 2.
Clearly, if the underlying Petri net of \(\mathcal{TN}\) is contact-free, then \(\mathcal{TN}\) must be contact-free as well, but not vice versa.
- 3.
The time-progressive property shall guarantee the correctness of the modified definition of the contact-free property, for our purposes.
- 4.
A (labeled over Act) time poset (partially ordered set) is a tuple \(\eta =(X,\preceq ,\lambda ,\tau )\) consisting of a finite set X of elements; a reflexive, antisymmetric and transitive relation \(\preceq \); a labeling function \(\lambda :X\rightarrow Act\); and a timing function \(\tau : X \rightarrow \mathbb {T}\) such that \(e\preceq e'\Rightarrow \tau (e)\le \tau (e')\). Let \(\tau (\eta )=\max \{\tau (x)\mid x\in X\}\).
- 5.
We assume \(path(\epsilon )=\epsilon \). Notice that in \(TCT(\mathcal{TN})\), for any node \(\sigma \in \mathcal {FS}(\mathcal{TN})\), there is a path starting from the root and finishing in \(\sigma \).
- 6.
A time poset \(\eta \) is a direct prefix of a time poset \(\eta '\) (denoted \({\eta }\prec \cdot \;{\eta '}\)) iff \(X\subseteq X'\), \(X'\setminus X = \{x\}\), \(\preceq =\preceq '\cap (X\times X)\), \(\lambda =\lambda '\mid _{X}\), \(\tau =\tau '\mid _{X}\), and x is a maximal w.r.t. \(\preceq '\) element of \(X'\).
- 7.
Time posets, \(\eta =(X,\preceq ,\lambda ,\tau )\) and \(\eta '=(X',\preceq ',\lambda ',\tau ')\), are isomorphic (denoted \(\eta \simeq \eta '\)) iff there is a bijective mapping \(\beta :X \rightarrow X'\) such that (i) \(x \preceq y\ \iff \ \beta (x) \preceq ' \beta (y)\), for all \(x,y\in X\); (ii) \(\lambda (x)=\lambda '(\beta (x))\) and \(\tau (x)=\tau '(\beta (x))\), for all \(x \in X\).
References
Aceto, L.: History preserving, causal and mixed-ordering equivalence over stable event structures. Fundamenta Informaticae 17(4), 319–331 (1992)
Aceto, L., De Nicola, R., Fantechi, A.: Testing equivalences for event structures. Lect. Notes Comput. Sci. 280, 1–20 (1987)
Aura, T., Lilius, J.: A causal semantics for time Petri nets. Theor. Comput. Sci. 243, 409–447 (2000)
Andreeva, M., Bozhenkova, E., Virbitskaite, I.: Analysis of timed concurrent models based on testing equivalenc. Fundamenta Informaticae 43, 1–20 (2000)
Andreeva, M., Virbitskaite, I.: Observational equivalences for timed stable event structures. Fundamenta Informaticae 72(1–3), 1–19 (2006)
Bihler, E., Vogler, W.: Timed Petri nets: efficiency of asynchronous systems. Lect. Notes Comput. Sci. 3185, 25–58 (2004)
Cleaveland, R., Hennessy, M.: Testing equivalence as a bisimulation equivalence. Lect. Notes Comput. Sci. 407, 11–23 (1989)
Cleaveland, R., Zwarico, A.E.: A theory of testing for real-time. In: Proceedings of 6th IEEE Symposium on Logic in Computer Science ( LICS 1991), Amsterdam, The Netherlands, pp. 110–119 (1991)
Corradini, F., Vogler, W., Jenner, L.: Comparing the worst-case efficiency of asynchronous systems with PAFAS. Technical report, N 2000–6, Inst. fur Informatik of Univ. of Augsburg (2000)
Darondeau, P., Degano, P.: Refinement of actions in event structures and causal trees. Theor. Comput. Sci. 118, 21–48 (1993)
De Nicola, R.: Extensional equivalences for transition systems. Acta Informatica 24, 211–237 (1987)
De Nicola, R., Hennessy, M.: Testing equivalence for processes. Theor. Comput. Sci. 34, 83–133 (1984)
Rozenberg, G., Engelfriet, J.: Elementary net systems. Lect. Notes Comput. Sci. 1491, 12–121 (1998)
Goltz, U., Wehrheim, H.: Causal testing. Lect. Notes Comput. Sci. 1113, 394–406 (1996)
Hennessy, M., Regan, T.: A process algebra for timed systems. Inf. Comput. 117, 221–239 (1995)
Llana, L., de Frutos, D.: Denotational semantics for timed testing. Lect. Notes Comput. Sci. 1233, 368–382 (1997)
Murphy, D.: Time and duration in noninterleaving concurrency. Fundamenta Informaticae 19, 403–416 (1993)
Steffen, B., Weise, C.: Deciding testing equivalence for real-time processes with dense time. Lect. Notes Comput. Sci. 711, 703–713 (1993)
Virbitskaite, I., Bushin, D., Best, E.: True concurrent equivalences in time Petri nets. Fundamenta Informaticae 149(4), 401–418 (2016)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Bozhenkova, E., Virbitskaite, I., Popova-Zeugmann, L. (2019). Causality-Based Testing in Time Petri Nets. In: Bjørner, N., Virbitskaite, I., Voronkov, A. (eds) Perspectives of System Informatics. PSI 2019. Lecture Notes in Computer Science(), vol 11964. Springer, Cham. https://doi.org/10.1007/978-3-030-37487-7_22
Download citation
DOI: https://doi.org/10.1007/978-3-030-37487-7_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-37486-0
Online ISBN: 978-3-030-37487-7
eBook Packages: Computer ScienceComputer Science (R0)