Abstract
Dense-Time Petri Nets (TPNs) are a widely-accepted model suitable for qualitative and quantitative modeling and verifying safety-critical, computer-controlled, and real-time systems. Testing equivalences used to compare the behavior (processes) of systems and reduce their structure are defined in terms of tests that processes may or must pass. The intention of the paper is to present a framework for developing, studying and comparing testing equivalences with extended tests (extended future) in interleaving, partial order, and causal tree semantics in the context of safe TPNs (transitions are labeled with time firing intervals, can fire only if their lower time bounds are attained and must fire when their upper time bounds are reached). Additionally, we establish, for the whole class of TPNs and their various subclasses, relationships between testing equivalences and other equivalences from the interleaving—partial order and linear time—branching time spectra. This allows studying in complete detail the timing behavior, in addition to the degrees of relative concurrency and nondeterminism of processes.
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Notes
- 1.
The alternative definition of (ii): for all \(1\le i < j\le k\) it holds: \(\bigcup _{1\le l \le i} e_{l}\) is a downward-closed and timely sound subset of \(\bigcup _{1\le m \le j} e_{l}\).
- 2.
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 \(l:X\rightarrow Act\); and a timing function \(\tau : X \rightarrow \mathbb {T}\) such that \(x\preceq x'\Rightarrow \tau (x)\le \tau (x')\).
- 3.
Notice that in \(\mathcal{C}\mathcal{T}({\mathcal{T}\mathcal{N}})\), for any vertex \(\sigma \in V\), there is a unique path starting from the root and finishing in \(\sigma \).
- 4.
\(\mathcal {F}(\mathcal {L}^*_{po}({\mathcal{T}\mathcal{N}})\), \(\mathcal {L}^*_{po}({\mathcal{T}\mathcal{N}}'))= \{f:TP\rightarrow TP'\mid \) f is a mapping, \([TP]_{\simeq } \in \mathcal {L}^*_{po}({\mathcal{T}\mathcal{N}})\), \([TP']_{\simeq } \in \mathcal {L}^*_{po}({\mathcal{T}\mathcal{N}}') \}\).
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Bozhenkova, E., Virbitskaite, I. (2023). Extended Future in Testing Semantics for Time Petri Nets. In: Schlingloff, BH., Vogel, T., Skowron, A. (eds) Concurrency, Specification and Programming. Studies in Computational Intelligence, vol 1091. Springer, Cham. https://doi.org/10.1007/978-3-031-26651-5_4
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