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).
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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 \(x \in X\).
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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
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