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74
Approximation metrics for discrete and continuous systems
 IEEE Transactions on Automatic Control
, 2005
"... Established system relationships for discrete systems, such as language inclusion, simulation, and bisimulation, require system observations to be identical. When interacting with the physical world, modeled by continuous or hybrid systems, exact relationships are restrictive and not robust. In thi ..."
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Cited by 105 (16 self)
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Established system relationships for discrete systems, such as language inclusion, simulation, and bisimulation, require system observations to be identical. When interacting with the physical world, modeled by continuous or hybrid systems, exact relationships are restrictive and not robust. In this paper, we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion, approximate simulation, and approximate bisimulation relations. We define a hierarchy of approximation pseudometrics between two systems that quantify the quality of the approximation, and capture the established exact relationships as zero sections. Our approximation framework is compositional for a synchronous composition operator. Algorithms are developed for computing the proposed pseudometrics, both exactly and approximately. The exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations, or dually, the solution of a static game whose cost is the socalled branching distance between the systems. Approximations for the pseudometrics can be obtained by considering Lyapunovlike functions called simulation and bisimulation functions. We illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems.
The Metric Analogue of Weak Bisimulation for Probabilistic Processes
, 2002
"... We observe that equivalence is not a robust concept in the presence of numerical information  such as probabilities  in the model. We develop a metric analogue of weak bisimulation in the spirit of our earlier work on metric analogues for strong bisimulation. We give a fixed point characterization ..."
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Cited by 68 (2 self)
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We observe that equivalence is not a robust concept in the presence of numerical information  such as probabilities  in the model. We develop a metric analogue of weak bisimulation in the spirit of our earlier work on metric analogues for strong bisimulation. We give a fixed point characterization of the metric. This makes available coinductive reasoning principles and allows us to prove metric analogues of the usual algebraic laws for process combinators. We also show that quantitative properties of interest are continuous with respect to the metric, which says that if two processes are close in the metric then observable quantitative properties of interest are indeed close. As an important example of this we show that nearby processes have nearby channel capacities  a quantitative measure of their propensity to leak information.
Metrics for Labelled Markov Systems
, 2001
"... The notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions. Thus, a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent. This instability is due to the quantitative nature of ..."
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Cited by 50 (10 self)
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The notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions. Thus, a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent. This instability is due to the quantitative nature of probabilistic processes. In a situation where the process behaviour has a quantitative aspect there should be a more robust approach to process equivalence. This paper studies a metric between labelled Markov processes. This metric has the property that processes are at zero distance if and only if they are bisimilar. The metric is inspired by earlier work on logics for characterizing bisimulation and is related, in spirit, to the Hutchinson metric.
Approximate simulation relations for hybrid systems
, 2006
"... Approximate simulation relations have recently been introduced as a powerful tool for the approximation of discrete and continuous systems. In this paper, we extend this notion to hybrid systems. Using the socalled simulation functions, we develop a computationally effective characterization of ap ..."
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Cited by 34 (1 self)
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Approximate simulation relations have recently been introduced as a powerful tool for the approximation of discrete and continuous systems. In this paper, we extend this notion to hybrid systems. Using the socalled simulation functions, we develop a computationally effective characterization of approximate simulation relations which can be used for hybrid systems approximation. An example of application in the context of safety verification is shown.
Process Equivalence: Comparing Two Process Models Based on Observed Behavior
 International Conference on Business Process Management (BPM 2006), volume 4102 of Lecture Notes in Computer Science
, 2006
"... Abstract. In various application domains there is a desire to compare process models, e.g., to relate an organizationspecific process model to a reference model, to find a web service matching some desired service description, or to compare some normative process model with a process model discover ..."
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Cited by 33 (7 self)
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Abstract. In various application domains there is a desire to compare process models, e.g., to relate an organizationspecific process model to a reference model, to find a web service matching some desired service description, or to compare some normative process model with a process model discovered using process mining techniques. Although many researchers have worked on different notions of equivalence (e.g., trace equivalence, bisimulation, branching bisimulation, etc.), most of the existing notions are not very useful in this context. First of all, most equivalence notions result in a binary answer (i.e., two processes are equivalent or not). This is not very helpful, because, in reallife applications, one needs to differentiate between slightly different models and completely different models. Second, not all parts of a process model are equally important. There may be parts of the process model that are rarely activated while other parts are executed for most process instances. Clearly, these should be considered differently. To address these problems, this paper proposes a completely new way of comparing process models. Rather than directly comparing two models, the process models are compared with respect to some typical behavior. This way we are able to avoid the two problems. Although the results are presented in the context of Petri nets, the approach can be applied to any process modeling language with executable semantics. Keywords: Process Equivalence, Petri Nets, Process Mining. 1
Approximate Bisimulations for Nonlinear Dynamical Systems
, 2005
"... The notion of exact bisimulation equivalence for nondeterministic discrete systems has recently resulted in notions of exact bisimulation equivalence for continuous and hybrid systems. In this paper, we establish the more robust notion of approximate bisimulation equivalence for nondeterministic n ..."
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Cited by 31 (6 self)
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The notion of exact bisimulation equivalence for nondeterministic discrete systems has recently resulted in notions of exact bisimulation equivalence for continuous and hybrid systems. In this paper, we establish the more robust notion of approximate bisimulation equivalence for nondeterministic nonlinear systems. This is achieved by requiring that a distance between system observations starts and remains, close, in the presence of nondeterministic system evolution. We show that approximate bisimulation relations can be characterized using a class of functions called bisimulation functions. For nondeterministic nonlinear systems, we show that conditions for the existence of bisimulation functions can be expressed in terms of Lyapunovlike inequalities, which for deterministic systems can be computed using recent sumofsquares techniques. Our framework is illustrated on a safety verification example.
Weak Bisimulation is Sound and Complete for PCTL
, 2002
"... We investigate weak bisimulation of probabilistic systems in the presence of nondeterminism, i.e. labelled concurrent Markov chains (LCMC) with silent transitions. We build on the work of Philippou, Lee and Sokolsky [1] for finite state LCMCs. Their denition of weak bisimulation destroys the additiv ..."
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Cited by 26 (0 self)
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We investigate weak bisimulation of probabilistic systems in the presence of nondeterminism, i.e. labelled concurrent Markov chains (LCMC) with silent transitions. We build on the work of Philippou, Lee and Sokolsky [1] for finite state LCMCs. Their denition of weak bisimulation destroys the additivity property of the probability distributions, yielding instead capacities. The mathematics behind capacities naturally captures the intuition that when we deal with nondeterminism we must work with estimates on the possible probabilities. Our analysis leads to three...
Quantifying Process Equivalence Based on Observed Behavior
"... Abstract. In various application domains there is a desire to compare process models, e.g., to relate an organizationspecific process model to a reference model, to find a web service matching some desired service description, or to compare some normative process model with a process model discover ..."
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Cited by 24 (0 self)
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Abstract. In various application domains there is a desire to compare process models, e.g., to relate an organizationspecific process model to a reference model, to find a web service matching some desired service description, or to compare some normative process model with a process model discovered using process mining techniques. Although many researchers have worked on different notions of equivalence (e.g., trace equivalence, bisimulation, branching bisimulation, etc.), most of the existing notions are not very useful in this context. First of all, most equivalence notions result in a binary answer (i.e., two processes are equivalent or not). This is not very helpful, because, in reallife applications, one needs to differentiate between slightly different models and completely different models. Second, not all parts of a process model are equally important. There may be parts of the process model that are rarely activated (i.e., “process veins”) while other parts are executed for most process instances (i.e., the “process arteries”). Clearly, differences in some veins of a process are less important than differences in the main arteries of a process. To address the problem, this paper proposes a completely new way of comparing process models. Rather than directly comparing two models, the process models are compared with respect to some typical behavior. This way, we are able to avoid the two problems just mentioned. The approach has been implemented and has been used in the context of genetic process mining. Although the results are presented in the context of Petri nets, the approach can be applied to any process modeling language with executable semantics.
Logical Relations for Monadic Types
, 2002
"... Logical relations and their generalizations are a fundamental tool in proving properties of lambdacalculi, e.g., yielding sound principles for observational equivalence. We propose a natural notion of logical relations able to deal with the monadic types of Moggi's computational lambdacal ..."
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Cited by 23 (6 self)
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Logical relations and their generalizations are a fundamental tool in proving properties of lambdacalculi, e.g., yielding sound principles for observational equivalence. We propose a natural notion of logical relations able to deal with the monadic types of Moggi's computational lambdacalculus. The treatment is categorical, and is based on notions of subsconing and distributivity laws for monads. Our approach has a number of interesting applications, including cases for lambdacalculi with nondeterminism (where being in logical relation means being bisimilar), dynamic name creation, and probabilistic systems.
Measuring anonymity with relative entropy
 In Proceedings of the 4th International Workshop on Formal Aspects in Security and Trust, volume 4691 of LNCS
, 2007
"... Abstract. Anonymity is the property of maintaining secret the identity of users performing a certain action. Anonymity protocols often use random mechanisms which can be described probabilistically. In this paper, we propose a probabilistic process calculus to describe protocols for ensuring anonymi ..."
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Cited by 21 (2 self)
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Abstract. Anonymity is the property of maintaining secret the identity of users performing a certain action. Anonymity protocols often use random mechanisms which can be described probabilistically. In this paper, we propose a probabilistic process calculus to describe protocols for ensuring anonymity, and we use the notion of relative entropy from information theory to measure the degree of anonymity these protocols can guarantee. Furthermore, we prove that the operators in the probabilistic process calculus are nonexpansive, with respect to this measuring method. We illustrate our approach by using the example of the Dining Cryptographers Problem. 1