Converging from Branching to Linear Metrics on Markov Chains

Bacci, Giorgio; Bacci, Giovanni; Larsen, Kim Guldstrand; Mardare, Radu

doi:10.1007/978-3-319-25150-9_21

Cited by 9 publications

(24 citation statements)

References 17 publications

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“…Even though for a good evaluation of our heuristics more tests are needed, the current results seem promising. Moreover, in the light of [18,30], relating the probabilistic bisimilarity distance to the LTL-model checking problem as δ 1 (M, N ) ≥ |P M (ϕ) − P N (ϕ)|, for all ϕ ∈ LTL, our results might be used to lead saving in the overall model checking time. A deeper study of this topic will be the focus of future work.…”

Section: Discussionmentioning

confidence: 98%

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On the metric-based approximate minimization of Markov Chains

Bacci

Larsen

et al. 2018

Journal of Logical and Algebraic Methods in Programming

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In this paper we address the approximate minimization problem of Markov Chains (MCs) from a behavioral metric-based perspective. Specifically, given a finite MC and a positive integer k, we are looking for an MC with at most k states having minimal distance to the original. The metric considered in this work is the bisimilarity distance of Desharnais et al.. For this metric we show that (1) optimal approximations always exist; (2) the problem has a bilinear program characterization; and (3) prove that its threshold problem is in PSPACE and NP-hard.In addition to the bilinear program solution, we present an approach inspired by expectation maximization techniques for computing suboptimal solutions to the problem. Experiments suggest that our method gives a practical approach that outperforms the bilinear program implementation run on state-of-the-art bilinear solvers. 1 1 Figure 1: An MC M (left) with initial state m 0 ; generic 5-states approximant of M aggregating m 1 and m 2 (for x, y ∈ [0, 1] such that x + y ≤ 1) (middle); an optimal 5-states approximant of M (right). Labels are represented by different colors.related to it. 1. We characterize CBA as a bilinear optimization problem, proving the existence of optimal solutions. As a consequence of this result, approximations of optimal solutions can be obtained by checking the feasibility of bilinear matrix inequalities (BMIs) [20,21].2. We provide upper-and lower-bound complexity results for the threshold problem of CBA, called Bounded Approximant problem (BA), that asks whether there exists a k-state approximant with distance from the original MC bounded by a given rational threshold. We show that BA is in PSPACE and NP-hard.3. We introduce the Minimum Significant Approximant Bound (MSAB) problem, that asks what is the minimum size k for an approximant to have some significant similarity to the original MC (i.e., at distance strictly less than 1). We show that this problem is NP-complete when one considers the undiscounted bisimilarity distance.4. Finally, we present an algorithm for finding suboptimal solutions of CBA that is inspired by Expectation Maximization (EM) techniques [22,23]. Experiments suggest that our method gives a practical approach that outperforms the bilinear program implementation -state-of-the-art bilinear solvers [21] fail to handle MCs with more than 5 states!

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Section: Discussionmentioning

confidence: 98%

“…It is worth noting that Algorithm 1 runs in polynomial time in the size of its input. Computing the coupling structure C in line 4, can be performed in polynomial time in the size of M ⊕ N 0 [18,30]…”

Section: Consider the Following Inequalitiesmentioning

confidence: 99%

On the metric-based approximate minimization of Markov Chains

Bacci

Larsen

et al. 2018

Journal of Logical and Algebraic Methods in Programming

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“…Instead of stating whether the behavior of two processes is exactly the same or not, behavioral metrics measure the disparities in their behavior. Since, moreover, for verification purposes, the desired properties (and observable behavior) of processes are usually expressed in terms of modal formulae, logical characterizations of behavioral metrics have been thoroughly investigated [2,11,18,25,27]. As d X -privacy and weak anonymity are measures over privacy protection guarantees of mechanisms, we aim at providing logical characterizations of them by exploiting the characterizations of the behavioral metrics on the LMCs induced by those mechanisms.…”

Section: Our Contributionmentioning

confidence: 99%

“…Interestingly, we also show how it is possible to define a real-valued semantics for formulae in L starting from their syntactic distance. From this, we obtain a logical characterization of weak anonymity in the classic sense of [2,27] (see Section 5 for a detailed description).…”

Section: Our Contributionmentioning

confidence: 99%

“…Moreover, we show that we can exploit the logical distance to obtain bounds on d X -privacy. Notice that dealing with bisimulation semantics instead of traces would allow us to develop efficient algorithms for the evaluation of the logical distance (following, e.g., [2]), and thus of approximations on d X -privacy. Furthermore, we could exploit the non-expansiveness results obtained in [15] to favor compositional reasoning over d X -privacy.…”

Section: Our Contributionmentioning

confidence: 99%

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A logical characterization of differential privacy

Castiglioni

Chatzikokolakis

Palamidessi

2020

Science of Computer Programming

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Differential privacy is a formal definition of privacy ensuring that sensitive information relative to individuals cannot be inferred by querying a database. In this paper, we exploit a modeling of this framework via labeled Markov Chains (LMCs) to provide a logical characterization of differential privacy: we consider a probabilistic variant of the Hennessy-Milner logic and we define a syntactic distance on formulae in it measuring their syntactic disparities. Then, we define a trace distance on LMCs in terms of the syntactic distance between the sets of formulae satisfied by them. We prove that such distance corresponds to the level of privacy of the LMCs. Moreover, we use the distance on formulae to define a real-valued semantics for them, from which we obtain a logical characterization of weak anonymity: the level of anonymity is measured in terms of the formulae distinguishing the considered LMCs. Then, we focus on bisimulation semantics on nondeterministic probabilistic processes and we provide a logical characterization of generalized bisimulation metrics, namely those defined via the generalized Kantorovich lifting. Our characterization is based on the notion of mimicking formula of a process and the syntactic distance on formulae, where the former captures the observable behavior of the corresponding process and allows us to characterize bisimilarity. We show that the generalized bisimulation distance on processes is equal to the syntactic distance on their mimicking formulae. Moreover, we use the distance on mimicking formulae to obtain bounds on differential privacy.

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A Logical Characterization of Differential Privacy via Behavioral Metrics

Castiglioni

Chatzikokolakis

Palamidessi

2018

Formal Aspects of Component Software

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Differential privacy is a formal definition of privacy ensuring that sensitive information relative to individuals cannot be inferred by querying a database. In this paper, we exploit a modeling of this framework via labeled Markov Chains (LMCs) to provide a logical characterization of differential privacy: we consider a probabilistic variant of the Hennessy-Milner logic and we define a syntactical distance on formulae in it measuring their syntactic disparities. Then, we define a trace distance on LMCs in terms of the syntactic distance between the sets of formulae satisfied by them. We prove that such distance corresponds to the level of privacy of the LMCs. Moreover, we use the distance on formulae to define a real-valued semantics for them, from which we obtain a logical characterization of weak anonymity: the level of anonymity is measured in terms of the smallest formula distinguishing the considered LMCs. Then, we focus on bisimulation semantics on nondeterministic probabilistic processes and we provide a logical characterization of generalized bisimulation metrics, namely those defined via the generalized Kantorovich lifting. Our characterization is based on the notion of mimicking formula of a process and the syntactic distance on formulae, where the former captures the observable behavior of the corresponding process and allows us to characterize bisimilarity. We show that the generalized bisimulation distance on processes is equal to the syntactic distance on their mimicking formulae. Moreover, we use the distance on mimicking formulae to obtain bounds on differential privacy.

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Converging from Branching to Linear Metrics on Markov Chains

Cited by 9 publications

References 17 publications

On the metric-based approximate minimization of Markov Chains

On the metric-based approximate minimization of Markov Chains

A logical characterization of differential privacy

A Logical Characterization of Differential Privacy via Behavioral Metrics

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