Abstract
Stochastic automata provide a way to symbolically model systems in which the occurrence time of events may respond to any continuous random variable. We introduce here an input/output variant of stochastic automata that, once the model is closed —i.e., all synchronizations are resolved—, the resulting automaton does not contain non-deterministic choices. This is important since fully probabilistic models are amenable to simulation in the general case and to much more efficient analysis if restricted to Markov models. We present here a theoretical introduction to input/output stochastic automata (IOSA) for which we (i) provide a concrete semantics in terms of non-deterministic labeled Markov processes (NLMP), (ii) prove that bisimulation is a congruence for parallel composition both in NLMP and IOSA, (iii) show that parallel composition commutes in the symbolic and concrete level, and (iv) provide a proof that a closed IOSA is indeed deterministic.
Supported by ANPCyT PICT-2012-1823 and SeCyT-UNC 05/BP12 and 05/B497.
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Notes
- 1.
Strictly speaking, \(\mathcal {P}(\mathcal {I}_1)||\mathcal {P}(\mathcal {I}_2)\) should also contain states of the form \((s,\vec {v}_1)||(\mathsf {init},\vec {v}_2)\) and \((\mathsf {init},\vec {v}_1)||(s,\vec {v}_2)\) with \(s\ne \mathsf {init}\). Nonetheless, these states are not reachable. Thus, we do not consider them since otherwise the result would not be strictly an isomorphism and it would only add irrelevant technical problems to the proof.
- 2.
Note that the domain and image of \(H\!f\) appear apparently inverted. This is necessary in [12] since they only deal with morphisms, and we are following their definitions. In our case, we could have also defined a direct map from \(\varDelta (\mathscr {B}(\mathbf S ))\) to \(\varDelta (\mathscr {B}(\mathbf S '))\) since \(\varDelta f\) is bimeasurable, namely \(H(f^{-1})=(\varDelta (f^{-1}))^{-1}\).
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Acknowledgments
We thank Pedro Sánchez Terraf for the help provided in measure theory, and Carlos E. Budde for early discussions on IOSAs.
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D’Argenio, P.R., Lee, M.D., Monti, R.E. (2016). Input/Output Stochastic Automata. In: Fränzle, M., Markey, N. (eds) Formal Modeling and Analysis of Timed Systems. FORMATS 2016. Lecture Notes in Computer Science(), vol 9884. Springer, Cham. https://doi.org/10.1007/978-3-319-44878-7_4
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