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Bisimulation for general stochastic hybrid systems
 In HSCC 2005
, 2005
"... Abstract. In this paper we define a bisimulation concept for some very general models for stochastic hybrid systems (general stochastic hybrid systems). The definition of bisimulation builds on the ideas of Edalat and of Larsen and Skou and of Joyal, Nielsen and Winskel. The main result is that this ..."
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Cited by 21 (11 self)
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Abstract. In this paper we define a bisimulation concept for some very general models for stochastic hybrid systems (general stochastic hybrid systems). The definition of bisimulation builds on the ideas of Edalat and of Larsen and Skou and of Joyal, Nielsen and Winskel. The main result is that this bisimulation for GSHS is indeed an equivalence relation. The secondary result is that this bisimulation relation for the stochastic hybrid system models used in this paper implies the same kind of bisimulation for their continuous parts and respectively for their jumping structures.
S.: A characterization of meaningful schedulers for continuoustime Markov decision processes. In: Formal Modeling and Analysis of Timed Systems
 LNCS
, 2006
"... Abstract. Continuoustime Markov decision process are an important variant of labelled transition systems having nondeterminism through labels and stochasticity through exponential firetime distributions. Nondeterministic choices are resolved using the notion of a scheduler. In this paper we chara ..."
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Cited by 17 (1 self)
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Abstract. Continuoustime Markov decision process are an important variant of labelled transition systems having nondeterminism through labels and stochasticity through exponential firetime distributions. Nondeterministic choices are resolved using the notion of a scheduler. In this paper we characterize the class of measurable schedulers, which is the most general one, and show how a measurable scheduler induces a unique probability measure on the sigmaalgebra of infinite paths. We then give evidence that for particular reachability properties it is sufficient to consider a subset of measurable schedulers. Having analyzed schedulers and their induced probability measures we finally show that each probability measure on the sigmaalgebra of infinite paths is indeed induced by a measurable scheduler which proves that this class is complete. 1
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 9 (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...
A computable approach to measure and integration theory
"... We introduce a computable framework for Lebesgue’s measure and integration theory in the spirit of domain theory. For an effectively given second countable locally compact Hausdorff space and an effectively given finite Borel measure on the space, we define a recursive measurable set, which extends ..."
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Cited by 7 (2 self)
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We introduce a computable framework for Lebesgue’s measure and integration theory in the spirit of domain theory. For an effectively given second countable locally compact Hausdorff space and an effectively given finite Borel measure on the space, we define a recursive measurable set, which extends the corresponding notion due to ˜Sanin for the Lebesgue measure on the real line. We also introduce the stronger notion of a computable measurable set, where a measurable set is approximated from inside and outside by sequences of closed and open subsets respectively. The set of recursive measurable subsets and that of computable measurable subsets are both closed under complementation, finite unions and finite intersections. We then introduce intervalvalued measurable functions and develop the notion of recursive and computable measurable functions using intervalvalued simple functions. This leads us to the interval versions of the main results in classical measure theory. The Lebesgue integral is shown to be a continuous operator on the domain of intervalvalued measurable functions and the intervalvalued Lebesgue integral provides a computable framework for integration: The Lebesgue integral of a bounded recursive or computable measurable function with respect to an effectively given finite Borel measure on an effectively given locally compact second countable Hausdorff space can be computed up to any required accuracy. Key Words: Domain theory, data type, recursive and computable measurable set, intervalvalued measurable function, intervalvalued Lebesgue integral. Double Dedication: This paper is dedicated to the historical memory of Jamshid Kashani (d. 1429), the Iranian mathematician who was the first to use the recursive fixed point method in analysis with which he computed sin1 ◦ correct to 9 sexagesimal places; he is also wellknown for computing π to 16 decimal places [3, pages 7 and 151]. This work is also dedicated to my colleague and friend Giuseppe Longo on the occasion of his sixtieth birthday to commend him for his wide range of interdisciplinary research interests and for his internationalist outlook on human culture. 1.
Abstractions of Stochastic Hybrid Systems
, 2008
"... In this paper we define a stochastic bisimulation concept for a very general class of stochastic hybrid systems, which subsumes most classes of stochastic hybrid systems. The definition of this bisimulation builds on the concept of zigzag morphism defined for strong Markov processes. The main result ..."
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Cited by 4 (4 self)
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In this paper we define a stochastic bisimulation concept for a very general class of stochastic hybrid systems, which subsumes most classes of stochastic hybrid systems. The definition of this bisimulation builds on the concept of zigzag morphism defined for strong Markov processes. The main result is that this stochastic bisimulation is indeed an equivalence relation. The secondary result is that this bisimulation relation for the stochastic hybrid system models used in this paper implies the same kind of bisimulation for their continuous parts and respectively for their jumping structures.
Probabilistic Bisimulation: Naturally on Distributions
"... Abstract. In contrast to the usual understanding of probabilistic systems as stochastic processes, recently these systems have also been regarded as transformers of probabilities. In this paper, we give a natural definition of strong bisimulation for probabilistic systems corresponding to this view ..."
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Cited by 1 (0 self)
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Abstract. In contrast to the usual understanding of probabilistic systems as stochastic processes, recently these systems have also been regarded as transformers of probabilities. In this paper, we give a natural definition of strong bisimulation for probabilistic systems corresponding to this view that treats probability distributions as firstclass citizens. Our definition applies in the same way to discrete systems as well as to systems with uncountable state and action spaces. Several examples demonstrate that our definition refines the understanding of behavioural equivalences of probabilistic systems. In particular, it solves a longstanding open problem concerning the representation of memoryless continuous time by memoryfull continuous time. Finally, we give algorithms for computing this bisimulation not only for finite but also for classes of uncountably infinite systems. 1
ON CATEGORIES OF COHESIVE, ACTIVE SETS AND OTHER DYNAMIC SYSTEMS BY
"... This work is intended to contribute to a program of developing general concepts and methods for applying category theory to the modeling and solution o f scientic problems. It was motivated by my experiences as an engineering student studying continuum and kinetic models of
uid
ows. The language o ..."
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This work is intended to contribute to a program of developing general concepts and methods for applying category theory to the modeling and solution o f scientic problems. It was motivated by my experiences as an engineering student studying continuum and kinetic models of
uid
ows. The language of category theory is employed because it facilitates precise comparisons between diverse types of structures. Using this language to investigate relationships between
uid
ow models requires categorical specications of constitutive relations and of idioms occurring across classications of dynamic systems. It is shown that a categorytheoretic denition of chaotic system applies not only to the Smale horseshoe, a standard chaotic system, but also to Conway's \Game of Life " automaton. Symbolic dynamics of the \Dining Philosophers " relational system is computed. A category composed of stochastic matrices is dened and some of its elementary properties are developed. A categorical variant of symbolic dynamics is applied to a nite stochastic process. Using pointwise Kan extension formulas, conditions ensuring existence of certain representations among categories of dynamic systems are proved. iii For my parents iv Acknowledgments Thanks, Mom and Dad. You are the greatest. I am nally done! Thanks to my wife Susan for being supportive, generous, and trusting while I've worked on this and while we lived far apart. Thanks to my grandparents Byron Oliver, Wesley Wojtowicz, and Josephine Oliver. You are greatly missed. Aloha and thank you Janet Wojtowicz for the many things you still do for our family. Thanks Michele and Shane for the wellwishes.
www.elsevier.com/locate/ic
, 2004
"... Bisimulation and cocongruence for probabilistic systems ..."
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