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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 18 (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 bisimulations for constrained linear systems
, 2005
"... 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 ..."
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Cited by 14 (5 self)
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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.
Duality for Labelled Markov Processes
"... Labelled Markov processes (LMPs) are automata whose transitions are given by probability distributions. In this paper we present a `universal' LMP as the spectrum of a commutative C # algebra consisting of formal linear combinations of labelled trees. We characterize the state space of the ..."
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Cited by 13 (1 self)
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Labelled Markov processes (LMPs) are automata whose transitions are given by probability distributions. In this paper we present a `universal' LMP as the spectrum of a commutative C # algebra consisting of formal linear combinations of labelled trees. We characterize the state space of the universal LMP as the set of homomorphims from an ordered commutative monoid of labelled trees into the multiplicative unit interval. This yields a simple semantics for LMPs which is fully abstract with respect to probabilistic bisimilarity. We also consider LMPs with entry points and exit points in the setting of iteration theories. We define an iteration theory of LMPs by specifying its categorical dual: a certain category of C*algebras. We find that the basic operations for composing LMPs have simple definitions in the dual category.
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.
Labeled Markov processes: stronger and faster approximations
, 2004
"... This paper reports on and discusses three notions of approximation for Labelled Markov Processes that have been developed last year. The three schemes are improvements over former constructions [11,12] in the sense that they define approximants that capture more properties than before and that conve ..."
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Cited by 11 (3 self)
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This paper reports on and discusses three notions of approximation for Labelled Markov Processes that have been developed last year. The three schemes are improvements over former constructions [11,12] in the sense that they define approximants that capture more properties than before and that converge faster to the approximated process. One scheme is constructive and the two others are driven by properties on which one wants to focus. All three constructions involve quotienting the statespace in some way and the last two are quotients with respect to sets of temporal properties expressed in a simple logic with a greatest fixed point operator. This gives the possibility of customizing approximants with respect to properties of interest and is thus an important step towards using automated techniques intended for finite state systems, e.g., model checking, for continuous state systems. Another difference between the schemes is how they relate approximants with the approximated process. The requirement that approximants should be simulated by
Logical, Metric, and Algorithmic Characterisations of Probabilistic Bisimulation
, 2011
"... Many behavioural equivalences or preorders for probabilistic processes involve a lifting operation that turns a relation on states into a relation on distributions of states. We show that several existing proposals for lifting relations can be reconciled to be different presentations of essentially ..."
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Many behavioural equivalences or preorders for probabilistic processes involve a lifting operation that turns a relation on states into a relation on distributions of states. We show that several existing proposals for lifting relations can be reconciled to be different presentations of essentially the same lifting operation. More interestingly, this lifting operation nicely corresponds to the Kantorovich metric, a fundamental concept used in mathematics to lift a metric on states to a metric on distributions of states, besides the fact the lifting operation is related to the maximum flow problem in optimisation theory. The lifting operation yields a neat notion of probabilistic bisimulation, for which we provide logical, metric, and algorithmic characterisations. Specifically, we extend the HennessyMilner logic and the modal mucalculus with a new modality, resulting in an adequate and an expressive logic for probabilistic bisimilarity, respectively. The correspondence of the lifting operation and the Kantorovich metric leads to a natural characterisation of bisimulations as pseudometrics which are postfixed points of a monotone function. We also present an “on the fly ” algorithm to check if two states in a finitary system are related by probabilistic bisimilarity, exploiting the close relationship
Approximating Continuous Markov Processes
, 2000
"... Markov processes with continuous state spaces arise in the analysis of stochastic physical systems or stochastic hybrid systems. The standard logical and algorithmic tools for reasoning about discrete (finitestate) systems are, of course, inadequate for reasoning about such systems. In this work we ..."
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Cited by 10 (3 self)
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Markov processes with continuous state spaces arise in the analysis of stochastic physical systems or stochastic hybrid systems. The standard logical and algorithmic tools for reasoning about discrete (finitestate) systems are, of course, inadequate for reasoning about such systems. In this work we develop three related ideas for making such reasoning principles applicable to continuous systems. ffl We show how to approximate continuous systems by a countable family of finitestate probabilistic systems, we can reconstruct the full system from these finite approximants, ffl we define a metric between processes and show that the approximants converge in this metric to the full process, ffl we show that reasoning about properties definable in a rich logic can be carried out in terms of the approximants. The systems that we consider are Markov processes where the state space is continuous but the time steps are discrete. We allow such processes to interact with the environment by syn...
Formalizing and reasoning about quality
, 2012
"... Abstract. Traditional formal methods are based on a Boolean satisfaction notion: a reactive system satisfies, or not, a given specification. We generalize formal methods to also address the quality of systems. As an adequate specification formalism we introduce the linear temporal logic LTL[F]. The ..."
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Cited by 9 (6 self)
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Abstract. Traditional formal methods are based on a Boolean satisfaction notion: a reactive system satisfies, or not, a given specification. We generalize formal methods to also address the quality of systems. As an adequate specification formalism we introduce the linear temporal logic LTL[F]. The satisfaction value of an LTL[F] formula is a number between 0 and 1, describing the quality of the satisfaction. The logic generalizes traditional LTL by augmenting it with a (parameterized) set F of arbitrary functions over the interval [0, 1]. For example, F may contain the maximum or minimum between the satisfaction values of subformulas, their product, and their average. The classical decision problems in formal methods, such as satisfiability, model checking, and synthesis, are generalized to search and optimization problems in the quantitative setting. For example, model checking asks for the quality in which a specification is satisfied, and synthesis returns a system satisfying the specification with the highest quality. Reasoning about quality gives rise to other natural questions, like the distance between specifications. We formalize these basic questions and study them for LTL[F]. By extending the automatatheoretic approach for LTL to a setting that takes quality into an account, we are able to solve the above problems and show that reasoning about LTL[F] has roughly the same complexity as reasoning about traditional LTL. 1
Conditional Expectation and the Approximation of Labelled Markov Processes
 IN: CONCUR 2003  CONCURRENCY THEORY, LECTURE NOTES IN COMPUTER SCIENCE 2761 (2003
, 2003
"... We develop a new notion of approximation of labelled Markov processes based on the use of conditional expectations. The key idea is to approximate a system by a coarsegraining of the state space and using averages of the transition probabilities. This is unlike any of the previous notions based ..."
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Cited by 9 (4 self)
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We develop a new notion of approximation of labelled Markov processes based on the use of conditional expectations. The key idea is to approximate a system by a coarsegraining of the state space and using averages of the transition probabilities. This is unlike any of the previous notions based on the use of simulation. The resulting approximations are customizable, more accurate and stay within the world of LMPs. The use of averages and expectations may well also make the approximations more robust. We introduce a novel condition  called "granularity"  which leads to unique conditional expectations and which turns out to be a key concept despite its simplicity.
OntheFly Exact Computation of Bisimilarity Distances
 In TACAS, volume 7795 of Lecture Notes in Computer Science
"... Abstract. This paper proposes an algorithm for exact computation of bisimilarity distances between discretetime Markov chains introduced by Desharnais et. al. Our work is inspired by the theoretical results presented by Chen et. al. at FoSSaCS’12, proving that these distances can be computed in po ..."
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Cited by 9 (6 self)
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Abstract. This paper proposes an algorithm for exact computation of bisimilarity distances between discretetime Markov chains introduced by Desharnais et. al. Our work is inspired by the theoretical results presented by Chen et. al. at FoSSaCS’12, proving that these distances can be computed in polynomial time using the ellipsoid method. Despite its theoretical importance, the ellipsoid method is known to be inefficient in practice. To overcome this problem, we propose an efficient onthefly algorithm which, unlike other existing solutions, computes exactly the distances between given states and avoids the exhaustive state space exploration. It is parametric in a given set of states for which we want to compute the distances. Our technique successively refines overapproximations of the target distances using a greedy strategy which ensures that the state space is further explored only when the current approximations are improved. Tests performed on a consistent set of randomly generated Markov chains shows that our algorithm improves, in average, the efficiency of the corresponding iterative algorithms with orders of magnitude. 1