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Opacity for linear constraint Markov chains

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Abstract

On a partially observed system, a secret φ is opaque if an observer cannot ascertain that its trace belongs to φ. We consider specifications given as Constraint Markov Chains (CMC), which are underspecified Markov chains where probabilities on edges are required to belong to some set. The nondeterminism is resolved by a scheduler, and opacity on this model is defined as a worst case measure over all implementations obtained by scheduling. This measures the information obtained by a passive observer when the system is controlled by the smartest scheduler in coalition with the observer. When restricting to the subclass of Linear CMC, we compute (or approximate) this measure and prove that refinement of a specification can only improve opacity.

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Acknowledgments

Partially supported by a grant from Coopération France-Québec, Service Coopération et Action Culturelle 2012/26/SCAC (French Government), the NSERC Discovery Individual grant No. 13321 (Government of Canada), the FQRNT Team grant No. 167440 (Quebec’s Government) and the CFQCU France-Quebec Cooperative grant No. 167671 (Quebec’s Government). This research has been partially performed while the third author was visiting the LIP6, Université Pierre & Marie Curie.

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Authors and Affiliations

  1. Sorbonne Université, UPMC Paris 06, LIP6, CNRS UMR 7606, Paris, France

    Béatrice Bérard

  2. INRIA and LSV, CNRS and ENS Cachan, Université Paris-Saclay, Saclay, France

    Béatrice Bérard

  3. Université Bourgogne Franche-ComtéFEMTO-ST, CNRS UMR 6174, Besançon, France

    Olga Kouchnarenko

  4. Department of Computer & Software Engineering, École Polytechnique de Montréal, Montreal, Quebec, Canada

    John Mullins

  5. Université Paris-Est, LACL, Créteil, France

    Mathieu Sassolas

Authors
  1. Béatrice Bérard
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  2. Olga Kouchnarenko
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  3. John Mullins
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  4. Mathieu Sassolas
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Correspondence to John Mullins.

Additional information

This article belongs to the Topical Collection: Special Issue on Performance Analysis and Optimization of Discrete Event Systems

Guest Editors: Christos G. Cassandras and Alessandro Giua

Appendix: Proof of Proposition 4

Appendix: Proof of Proposition 4

Assume by induction that the proposition holds for every word of length n. Let \(w \in {\mathit {FTr}}({\mathcal {A}}_1) = {\mathit {FTr}}({\mathcal {A}}_2)\) (recall Proposition 3) of length n + 1 with w = w 0 a for some a ∈ Σ. A run \(\rho ^{\prime }\) of \({\mathcal {A}}_2\) that produces w can be assumed to be of the form \(\rho ^{\prime } = \rho _0^{\prime } s_2^{\prime }\) with \({\text {tr}}(\rho _0^{\prime }) = w_0\) and \(\lambda (s_2^{\prime }) = a\). Then \({\mathbf {P}}_{{\mathcal {A}}_2}(C_{\rho ^{\prime }}) = {\mathbf {P}}_{{\mathcal {A}}_2}(C_{\rho _0^{\prime }}) {\Delta}_2(s_2)(s_2^{\prime })\) where \(s_2 = {\text {lst}}(\rho _0^{\prime })\) and hence

$$\begin{array}{@{}rcl@{}} {\mathbf{P}}_{{\mathcal{A}}_2}(C_w) &=& \sum\limits_{\begin{array}{cccccccc}\rho^{\prime} \in \mathit{FRuns}({\mathcal{A}}_2)\\{\text{tr}}(\rho^{\prime})=w \end{array}} {\mathbf{P}}_{{\mathcal{A}}_2}(C_{\rho^{\prime}}) \\\end{array} $$
$$\begin{array}{@{}rcl@{}} &=& \sum\limits_{\begin{array}{cccccccc}s_2,s_2^{\prime}\in S_2 \end{array}} \sum\limits_{\begin{array}{cccccccc}\rho_0^{\prime} \in \mathit{FRuns}({\mathcal{A}}_2)\\{\text{tr}}(\rho_0^{\prime})=w_0,\\ {\text{lst}}(\rho_0^{\prime})=s_2, {\text{lst}}(\rho^{\prime})=s_2^{\prime} \end{array}} {\mathbf{P}}_{{\mathcal{A}}_2}(C_{\rho_0^{\prime}})\cdot {\Delta}_2(s_2)(s_2^{\prime}) \\ \end{array} $$

Now let \({\mathcal {A}}_1\) s.t. \({\mathcal {A}}_2\) simulates \({\mathcal {A}}_1\) then, as

$$ \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}({\mathcal{A}}_1)\\\rho_0 \sim \rho_0^{\prime}\\{\text{lst}}(\rho_0)=s_1 \end{array}} \mu_{\rho_0^{\prime}}(\rho_0) = \sum\limits_{s_1 \in S_1} \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}(\mathcal{A}_1)\\\rho_0 \sim \rho_0^{\prime}\\\text{lst}(\rho_0)=s_1 \end{array}} \mu_{\rho_0^{\prime}}(\rho_0) = 1 $$

we get

$$\begin{array}{@{}rcl@{}} {\mathbf{P}}_{{\mathcal{A}}_2}(C_w) &=& \sum\limits_{\begin{array}{cccccccc}s_2,s_2^{\prime}\in S_2 \end{array}} \sum\limits_{\begin{array}{cccccccc}\rho_0^{\prime} \in \mathit{FRuns}({\mathcal{A}}_2)\\{\text{tr}}(\rho_0^{\prime})=w_0,\\ {\text{lst}}(\rho_0^{\prime})=s_2, {\text{lst}}(\rho^{\prime})=s_2^{\prime} \end{array}}{\mathbf{P}}_{{\mathcal{A}}_2}(C_{\rho_0^{\prime}}) \cdot {\Delta}_2(s_2)(s_2^{\prime}) \cdot \sum\limits_{s_1 \in S_1} \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}(\mathcal{A}_1)\\\rho_0 \sim \rho_0^{\prime}\\\text{lst}(\rho_0)=s_1 \end{array}}\mu_{\rho_0^{\prime}}(\rho_0) \\ &=& \sum\limits_{\begin{array}{cccccccc}s_1 \in S_1\\s_2,s_2^{\prime}\in S_2 \end{array}}\sum\limits_{\begin{array}{cccccccc}\rho_0^{\prime} \in \mathit{FRuns}({\mathcal{A}}_2)\\{\text{tr}}(\rho_0^{\prime})=w_0,\\ {\text{lst}}(\rho_0^{\prime})=s_2, {\text{lst}}(\rho^{\prime})=s_2^{\prime} \end{array}} {\mathbf{P}}_{{\mathcal{A}}_2}(C_{\rho_0^{\prime}}) \cdot {\Delta}_2(s_2)(s_2^{\prime}) \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}({\mathcal{A}}_1)\\\rho_0 \sim \rho_0^{\prime}\\\text{lst}(\rho_0)=s_1 \end{array}} \mu_{\rho_0^{\prime}}(\rho_0) \\ & & \text{As the terms are null if it is not the case that}\, s_1 {\mathcal{R}} s_2, \text{we have:} \\ &=& \sum\limits_{\begin{array}{cccccccc}s_1 \in S_1\\s_2,s_2^{\prime}\in S_2 \end{array}} \sum\limits_{\begin{array}{cccccccc}\rho_0^{\prime} \in \mathit{FRuns}({\mathcal{A}}_2)\\{\text{tr}}(\rho_0^{\prime})=w_0,\\ {\text{lst}}(\rho_0^{\prime})=s_2, {\text{lst}}(\rho^{\prime})=s_2^{\prime} \end{array}} {\mathbf{P}}_{{\mathcal{A}}_2}(C_{\rho_0^{\prime}}) \cdot\\&&\left( \sum\limits_{s^{\prime}_1 \in S_1} {\Delta}_1(s_1)(s^{\prime}_1)\cdot \delta_{s_1,s_2} (s^{\prime}_1)(s^{\prime}_2) \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}(\mathcal{A}_1)\\\rho_0 \sim \rho_0^{\prime}\\\text{lst}(\rho_0)=s_1 \end{array}} \mu_{\rho_0^{\prime}}(\rho_0)\right)\\ & & \text{And as the terms are null if}\, \lambda(s_1^{\prime}) \neq \lambda(s_2^{\prime}) = \lambda({\text{lst}}(\rho)),\, \text{we have:}\\ &=& \sum\limits_{\begin{array}{cccccccc}s_1 \in S_1\\s_2,s_2^{\prime}\in S_2 \end{array}}\sum\limits_{\begin{array}{cccccccc}\rho_0^{\prime} \in \mathit{FRuns}({\mathcal{A}}_2)\\{\text{tr}}(\rho_0^{\prime})=w_0,\\ {\text{lst}}(\rho_0^{\prime})=s_2, {\text{lst}}(\rho^{\prime})=s_2^{\prime} \end{array}} {\mathbf{P}}_{{\mathcal{A}}_2}(C_{\rho_0^{\prime}}) \sum\limits_{\begin{array}{cccccccc}s^{\prime}_1 \in S_1\\\lambda(s_1) = \lambda(\text{lst}(\rho)) \end{array}} {\Delta}_1(s_1)(s^{\prime}_1)\cdot\\&&\left( \delta_{s_1,s_2} (s^{\prime}_1)(s^{\prime}_2) \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}(\mathcal{A}_1)\\\rho_0 \sim \rho_0^{\prime}\\\text{lst}(\rho_0)=s_1 \end{array}} \mu_{\rho_0^{\prime}}(\rho_0)\right) \end{array} $$
$$\begin{array}{@{}rcl@{}} {\mathbf{P}}_{{\mathcal{A}}_2}(C_w) &=& \sum\limits_{s_1 \in S_1} \sum\limits_{\begin{array}{cccccccc}s^{\prime}_1 \in S_1\\\lambda(s_1) = \lambda({\text{lst}}(\rho)) \end{array}} \sum\limits_{\begin{array}{cccccccc}s_2\in S_2\\s_2^{\prime}\in S_2 \end{array}} \delta_{s_1,s_2} (s^{\prime}_1)(s^{\prime}_2) \cdot \\&&\left( \sum\limits_{\begin{array}{cccccccc}\rho_0^{\prime} \in \mathit{FRuns}(\mathcal{A}_2)\\\text{tr}(\rho_0^{\prime})=w_0,\\ \text{lst}(\rho_0^{\prime})=s_2, \text{lst}(\rho^{\prime})=s_2^{\prime} \end{array}} \mathbf{P}_{\mathcal{A}_2}(C_{\rho_0^{\prime}}) {\Delta}_1(s_1)(s^{\prime}_1) \right. \cdot\left.\sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}(\mathcal{A}_1)\\\rho_0 \sim \rho_0^{\prime}\\\text{lst}(\rho_0)=s_1 \end{array}} \mu_{\rho_0^{\prime}}(\rho_0)\right) \\ & & \text{And since}\, \sum\limits_{s_2^{\prime}\in S_2} \delta_{s_1,s_2} (s^{\prime}_1)(s^{\prime}_2) = 1: \\ &=& \sum\limits_{s_1 \in S_1} \sum\limits_{\begin{array}{cccccccc}s^{\prime}_1 \in S_1\\\lambda(s_1) = \lambda(\text{lst}(\rho)) \end{array}} \sum\limits_{s_2\in S_2} \sum\limits_{\begin{array}{cccccccc}\rho_0^{\prime} \in \mathit{FRuns}(\mathcal{A}_2)\\\text{tr}(\rho_0^{\prime})=w_0,\\ \text{lst}(\rho_0^{\prime})=s_2 \end{array}} \mathbf{P}_{\mathcal{A}_2}(C_{\rho_0^{\prime}}) \cdot {\Delta}_1(s_1)(s^{\prime}_1) \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}(\mathcal{A}_1)\\\rho_0 \sim \rho_0^{\prime}\\\text{lst}(\rho_0)=s_1 \end{array}} \mu_{\rho_0^{\prime}}(\rho_0) \\ &=& \sum\limits_{s_1 \in S_1} \sum\limits_{\begin{array}{cccccccc}s^{\prime}_1 \in S_1\\\lambda(s_1) = \lambda(\text{lst}(\rho)) \end{array}} \sum\limits_{\begin{array}{cccccccc}\rho_0^{\prime} \in \mathit{FRuns}(\mathcal{A}_2)\\\text{tr}(\rho_0^{\prime})=w_0 \end{array}} \mathbf{P}_{\mathcal{A}_2}(C_{\rho_0^{\prime}}) \cdot {\Delta}_1(s_1)(s^{\prime}) \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}(\mathcal{A}_1)\\\rho_0 \sim \rho_0^{\prime}\\\text{lst}(\rho_0)=s_1 \end{array}} \mu_{\rho_0^{\prime}}(\rho_0) \\ &=& \sum\limits_{s_1 \in S_1} \sum\limits_{\begin{array}{cccccccc}s^{\prime}_1 \in S_1\\\lambda(s_1) = \lambda(\text{lst}(\rho)) \end{array}} {\Delta}_1(s_1)(s^{\prime}_1) \cdot \sum\limits_{\begin{array}{cccccccc}\rho_0^{\prime} \in \mathit{FRuns}(\mathcal{A}_2)\\\text{tr}(\rho_0^{\prime})=w_0 \end{array}} \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}(\mathcal{A}_1)\\\rho_0 \sim \rho_0^{\prime}\\\text{lst}(\rho_0)=s_1 \end{array}} \mathbf{P}_{\mathcal{A}_2}(C_{\rho_0^{\prime}}) \mu_{\rho_0^{\prime}}(\rho_0) \\ &=& \sum\limits_{s_1 \in S_1} \sum\limits_{\begin{array}{cccccccc}s^{\prime}_1 \in S_1\\\lambda(s_1) = \lambda(\text{lst}(\rho)) \end{array}} {\Delta}_1(s_1)(s^{\prime}_1) \sum\limits_{\begin{array}{cccccccc}\rho_0^{\prime} \in \mathit{FRuns}(\mathcal{A}_2)\\\text{tr}(\rho_0^{\prime})=w_0 \end{array}} \cdot \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}(\mathcal{A}_1)\\\rho_0 \sim \rho_0^{\prime}\\\text{lst}(\rho_0)=s_1 \end{array}} \mathbf{P}_{\mathcal{A}_2}(C_{\rho_0^{\prime}}) \frac{\mathbf{P}_{\mathcal{A}_1}(C_{\rho_0})}{\mathbf{P}_{\mathcal{A}_1}(sim(\rho_0^{\prime}))} \\ &=& \sum\limits_{s_1 \in S_1} \sum\limits_{\begin{array}{cccccccc}s^{\prime}_1 \in S_1\\\lambda(s_1) = \lambda(\text{lst}(\rho)) \end{array}} {\Delta}_1(s_1)(s^{\prime}_1) \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}(\mathcal{A}_1)\\\text{tr}(\rho_0)=w_0\\\text{lst}(\rho_0)=s_1 \end{array}} \frac{\mathbf{P}_{\mathcal{A}_1}(C_{\rho_0})}{\mathbf{P}_{\mathcal{A}_1}(sim(\rho_0^{\prime}))} \sum\limits_{\begin{array}{cccccccc}\rho_0^{\prime} \in \mathit{FRuns}(\mathcal{A}_2)\\\text{tr}(\rho_0^{\prime})=w_0 \end{array}} \mathbf{P}_{\mathcal{A}_2}(C_{\rho_0^{\prime}}) \\ &=& \sum\limits_{s_1 \in S_1} \sum\limits_{\begin{array}{cccccccc}s^{\prime}_1 \in S_1\\\lambda(s_1) = \lambda(\text{lst}(\rho)) \end{array}} {\Delta}_1(s_1)(s^{\prime}_1) \cdot \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}(\mathcal{A}_1)\\\text{tr}(\rho_0)=w_0\\\text{lst}(\rho_0)=s_1 \end{array}} \frac{\mathbf{P}_{\mathcal{A}_1}(C_{\rho_0})}{\mathbf{P}_{\mathcal{A}_1}(sim(\rho_0^{\prime}))} \cdot \mathbf{P}_{\mathcal{A}_2}(C_{w_0}) \\ &&\text{and by induction hypothesis, } {\mathbf{P}}_{{\mathcal{A}}_2}(C_{w_0}) = {\mathbf{P}}_{{\mathcal{A}}_1}(C_{w_0}) \text{, hence:}\\ &=& \sum\limits_{s_1 \in S_1} \sum\limits_{\begin{array}{cccccccc}s^{\prime}_1 \in S_1\\\lambda(s_1) = \lambda(\text{lst}(\rho)) \end{array}} {\Delta}_1(s_1)(s^{\prime}_1) \cdot \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}(\mathcal{A}_1)\\\text{tr}(\rho_0)=w_0\\\text{lst}(\rho_0)=s_1 \end{array}} \frac{\mathbf{P}_{\mathcal{A}_1}(C_{\rho_0})}{\mathbf{P}_{\mathcal{A}_1}(sim(\rho_0^{\prime}))} \cdot \mathbf{P}_{\mathcal{A}_1}(C_{w_0}) \\ \end{array} $$
$$\begin{array}{@{}rcl@{}} &&\text{and since } {\mathbf{P}}_{{\mathcal{A}}_1}(sim(\rho_0^{\prime})) = {\mathbf{P}}_{{\mathcal{A}}_1}(C_{w_0}) \text{:}\\ &=& \sum\limits_{s_1 \in S_1} \sum\limits_{\begin{array}{cccccccc}s^{\prime}_1 \in S_1\\\lambda(s_1) = \lambda(\text{lst}(\rho)) \end{array}} {\Delta}_1(s_1)(s^{\prime}_1) \cdot \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}(\mathcal{A}_1)\\\text{tr}(\rho_0)=w_0\\\text{lst}(\rho_0)=s_1 \end{array}} \mathbf{P}_{\mathcal{A}_1}(C_{\rho_0}) \\ &=& \sum\limits_{s_1 \in S_1} \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}({\mathcal{A}}_1)\\{\text{tr}}(\rho_0)=w_0\\{\text{lst}}(\rho_0)=s_1 \end{array}} \mathbf{P}_{\mathcal{A}_1}(C_{\rho_0}) \cdot\sum\limits_{\begin{array}{cccccccc}s^{\prime}_1 \in S_1\\\lambda(s_1) = \lambda(\text{lst}(\rho)) \end{array}} {\Delta}_1(s_1)(s^{\prime}_1)\\ &=& \sum\limits_{\begin{array}{cccccccc}\rho_0 \in \mathit{FRuns}({\mathcal{A}}_1)\\{\text{tr}}(\rho_0)=w_0 \end{array}} {\mathbf{P}}_{{\mathcal{A}}_1}(C_{\rho_0}) {\Delta}_1({\text{lst}}(\rho_0))({\text{lst}}(\rho)) \ =\ {\mathbf{P}}_{{\mathcal{A}}_1}(C_w) \end{array} $$

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Bérard, B., Kouchnarenko, O., Mullins, J. et al. Opacity for linear constraint Markov chains. Discrete Event Dyn Syst 28, 83–108 (2018). https://doi.org/10.1007/s10626-017-0259-4

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