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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 42 (13 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.
Approximate bisimulation relations for constrained linear systems
 AUTOMATICA
, 2007
"... In this paper, we define the notion of approximate bisimulation relation between two systems, extending the well established exact bisimulation relations for discrete and continuous systems. Exact bisimulation requires that the observations of two systems are and remain identical, approximate bisi ..."
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Cited by 10 (4 self)
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In this paper, we define the notion of approximate bisimulation relation between two systems, extending the well established exact bisimulation relations for discrete and continuous systems. Exact bisimulation requires that the observations of two systems are and remain identical, approximate bisimulation allows the observation to be different provided they are and remain arbitrarily close. Approximate bisimulation relations are conveniently defined as level sets of a function called bisimulation function. For the class of linear systems with constrained initial states and constrained inputs, we develop effective characterizations for bisimulation functions that can be interpreted in terms of linear matrix inequalities, set inclusion and games. We derive a computationally effective algorithm to evaluate the precision of the approximate bisimulation between a constrained linear system and its projection. This algorithm has been implemented in a MATLAB toolbox: MATISSE. Two examples of use of the toolbox in the context of safety verification are shown.
Approximate simulations for taskstructured probabilistic I/O automata
 In LICS workshop on Probabilistic Automata and Logics (PAul06
, 2006
"... A Probabilistic I/O Automaton (PIOA) is a countablestate automaton model that allows nondeterministic and probabilistic choices in state transitions. A taskPIOA adds a task structure on the locally controlled actions of a PIOA as a means for restricting the nondeterminism in the model. The taskPI ..."
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Cited by 6 (4 self)
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A Probabilistic I/O Automaton (PIOA) is a countablestate automaton model that allows nondeterministic and probabilistic choices in state transitions. A taskPIOA adds a task structure on the locally controlled actions of a PIOA as a means for restricting the nondeterminism in the model. The taskPIOA framework defines exact implementation relations based on inclusion of sets of trace distributions. In this paper we develop the theory of approximate implementations and equivalences for taskPIOAs. We propose a new kind of approximate simulation between taskPIOAs and prove that it is sound with respect to approximate implementations. Our notion of similarity of traces is based on a metric on trace distributions and therefore, we do not require the state spaces nor the space of external actions (output alphabet) of the underlying automata to be metric spaces. We discuss applications of approximate implementations to probabilistic safety verification.
Approximate Bisimulation: A Bridge Between Computer Science and Control Theory
 EUROPEAN JOURNAL OF CONTROL (2011)56:568–578
, 2011
"... Fifty years ago, control and computing were part of a broader system science. After a long period of separate development within each discipline, embedded and hybrid systems have challenged us to reunite the, now sophisticated theories of continuous control and discrete computing on a broader syste ..."
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Cited by 2 (0 self)
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Fifty years ago, control and computing were part of a broader system science. After a long period of separate development within each discipline, embedded and hybrid systems have challenged us to reunite the, now sophisticated theories of continuous control and discrete computing on a broader system theoretic basis. In this paper, we present a framework of system approximation that applies to both discrete and continuous systems. We define a hierarchy of approximation metrics between two systems that quantify the quality of the approximation, and capture the established notions in computer science as zero sections. The central notions in this framework are that of approximate simulation and bisimulation relations and their functional characterizations called simulation and bisimulation functions and defined by Lyapunovtype inequalities. In particular, these functions can provide computable upperbounds on the approximation metrics by solving a static game. Our approximation framework will be illustrated by showing some of its applications in various problems such as reachability analysis of continuous systems and hybrid systems, approximation of continuous and hybrid systems by discrete systems, hierarchical control design, and simulationbased approaches to verification of continuous and hybrid systems.
Verifying Statistical Zero Knowledge with Approximate Implementations ⋆
"... Abstract. Statistical zeroknowledge (SZK) properties play an important role in designing cryptographic protocols that enforce honest behavior while maintaining privacy. This paper presents a novel approach for verifying SZK properties, using recently developed techniques based on approximate simula ..."
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Cited by 1 (1 self)
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Abstract. Statistical zeroknowledge (SZK) properties play an important role in designing cryptographic protocols that enforce honest behavior while maintaining privacy. This paper presents a novel approach for verifying SZK properties, using recently developed techniques based on approximate simulation relations. We formulate statistical indistinguishability as an implementation relation in the TaskPIOA framework, which allows us to express computational restrictions. The implementation relation is then proven using approximate simulation relations. This technique separates proof obligations into two categories: those requiring probabilistic reasoning, as well as those that do not. The latter is a good candidate for mechanization. We illustrate the general method by verifying the SZK property of the wellknown identification protocol proposed by Girault, Poupard and Stern.
Testing for Simulation and Bisimulation in Labelled Markov Processes
, 2003
"... This paper presents a fundamental study of similarity and bisimilarity for labelled Markov processes: a particular class of probabilistic labelled transition systems. The main results characterize similarity as a testing preorder and bisimilarity as a testing equivalence. ..."
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This paper presents a fundamental study of similarity and bisimilarity for labelled Markov processes: a particular class of probabilistic labelled transition systems. The main results characterize similarity as a testing preorder and bisimilarity as a testing equivalence.
Authors
, 2007
"... Project cofunded by the European Commission within the Sixth Framework Programme (20022006) ..."
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Project cofunded by the European Commission within the Sixth Framework Programme (20022006)
On The Theory of Stochastic Processors
"... Abstract—Traditional architecture design approaches hide hardware uncertainties from the software stack through overdesign, which is often expensive in terms of power consumption. The recently proposed quantitative alternative of stochastic computing requires circuits and processors to be correct on ..."
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Abstract—Traditional architecture design approaches hide hardware uncertainties from the software stack through overdesign, which is often expensive in terms of power consumption. The recently proposed quantitative alternative of stochastic computing requires circuits and processors to be correct only probabilistically and use less power. In this paper, we present the first step towards a theory of stochastic computing. Specifically, a formal model of a device which computes a deterministic function with stochastic delays is presented; the semantics of a stochastic circuit is obtained by composing such devices; finally, a quantitative notion of stochastic correctness, called correctness factor (CF), is introduced. For random data sources, a closed form expression is derived for CF of devices, which shows that there are two probabilities that contribute positively, namely, the probability of being timely with current inputs and the probability of being lucky with past inputs. Finally, we show the characteristic graphs obtained from the analytical expressions for the variation of correctness factor with clock period, for several simple circuits and sources. Index Terms—probabilistic computing; probabilistic circuits; formal models of computation; I.