Results 1  10
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18
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 104 (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.
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 19 (5 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 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 12 (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.
Panangaden Taking it to the limit: Approximate reasoning for Markov processes
 In Proceedings of MFCS’12, LNCS 7464
, 2012
"... Abstract. We develop a fusion of logical and metrical principles for reasoning about Markov processes. More precisely, we lift metrics from processes to sets of processes satisfying a formula and explore how the satisfaction relation behaves as sequences of processes and sequences of formulas approa ..."
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Cited by 8 (5 self)
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Abstract. We develop a fusion of logical and metrical principles for reasoning about Markov processes. More precisely, we lift metrics from processes to sets of processes satisfying a formula and explore how the satisfaction relation behaves as sequences of processes and sequences of formulas approach limits. A key new concept is dynamicallycontinuous metric bisimulation which is a property of (pseudo)metrics. We prove theorems about satisfaction in the limit, robustness theorems as well as giving a topological characterization of various classes of formulas. This work is aimed at providing approximate reasoning principles for Markov processes. 1
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.
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.
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.
Parameterized Metatheory for Continuous Markovian Logic
"... Abstract—This paper shows that a classic metalogical framework, including all Boolean operators, can be used to support the development of a metric behavioural theory for Markov processes. Previously, only intuitionistic frameworks or frameworks without negation and logical implication have been d ..."
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Abstract—This paper shows that a classic metalogical framework, including all Boolean operators, can be used to support the development of a metric behavioural theory for Markov processes. Previously, only intuitionistic frameworks or frameworks without negation and logical implication have been developed to fulfill this task. The focus of this paper is on continuous Markovian logic (CML), a logic that characterizes stochastic bisimulation of Markov processes with arbitrary measurable state space and continuoustime transitions. For a parameter ε> 0 interpreted as observational error, we introduce an εparametrized metatheory for CML: we define the concepts of εsatisfiability and εprovability related by a sound and complete axiomatization and prove a series of ”parametrized” metatheorems including decidability, weak completeness and finite model property. We also prove results regarding the relations between metalogical concepts defined for different parameters. Using this framework, we can characterize both the stochastic bisimulation relation and various observational preorders based on behavioural pseudometrics. The main contribution of this paper is proving that all these analyses can actually be done using a unified complete Boolean framework. And this extends the state of the art in this field, since the related works only propose intuitionistic contexts that limit, for instance, the use of the Boolean logical implication. I.
Approximate Bisimulations for Constrained Linear Systems
"... Abstract — In this paper, inspired by exact notions of bisimulation equivalence for discreteevent and continuoustime systems, we establish approximate bisimulation equivalence for linear systems with internal but bounded disturbances. This is achieved by developing a theory of approximation for t ..."
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Abstract — In this paper, inspired by exact notions of bisimulation equivalence for discreteevent and continuoustime systems, we establish approximate bisimulation equivalence for linear systems with internal but bounded disturbances. This is achieved by developing a theory of approximation for transition systems with observation metrics, which require that the distance between system observations is and remains arbitrarily close in the presence of nondeterministic evolution. Our notion of approximate bisimulation naturally reduces to exact bisimulation when the distance between the observations is zero. Approximate bisimulation relations are then characterized by a class of Lyapunovlike functions which are called bisimulation functions. For the class of linear systems with constrained disturbances, we obtain computable characterizations of bisimulation functions in terms of linear matrix inequalities, set inclusions, and optimal values of static games. We illustrate our framework in the context of safety verification. I.