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28
Stochastic Differential Dynamic Logic for Stochastic Hybrid Programs
, 2011
"... should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution or government. A conference version of this report has appeared at CADE [Pla11].Keywords: Dynamic logic, proof calculus, stochastic differential equations, stochastic hybrid Lo ..."
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Cited by 19 (14 self)
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should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution or government. A conference version of this report has appeared at CADE [Pla11].Keywords: Dynamic logic, proof calculus, stochastic differential equations, stochastic hybrid Logic is a powerful tool for analyzing and verifying systems, including programs, discrete systems, realtime systems, hybrid systems, and distributed systems. Some applications also have a stochastic behavior, however, either because of fundamental properties of nature, uncertain environments, or simplifications to overcome complexity. Discrete probabilistic systems have been studied using logic. But logic has been chronically underdeveloped in the context of stochastic hybrid systems, i.e., systems with interacting discrete, continuous, and stochastic dynamics. We aim at overcoming this deficiency and introduce a dynamic logic for stochastic hybrid systems. Our results indicate that logic is a promising tool for understanding stochastic hybrid systems and can help taming some of their complexity. We introduce a compositional model for stochastic hybrid systems. We prove adaptivity, càdlàg, and Markov time properties, and prove that the semantics
Logics of Dynamical Systems
"... We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded ..."
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Cited by 18 (17 self)
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We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded systems and cyberphysical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. Distributed hybrid systems combine distributed systems with hybrid systems, i.e., they are multiagent hybrid systems that interact through remote communication or physical interaction. Stochastic hybrid systems combine stochastic
Hybrid Numerical Solution of the Chemical Master Equation
, 2010
"... We present a numerical approximation technique for the analysis of continuoustime Markov chains that describe networks of biochemical reactions and play an important role in the stochastic modeling of biological systems. Our approach is based on the construction of a stochastic hybrid model in whic ..."
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Cited by 17 (1 self)
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We present a numerical approximation technique for the analysis of continuoustime Markov chains that describe networks of biochemical reactions and play an important role in the stochastic modeling of biological systems. Our approach is based on the construction of a stochastic hybrid model in which certain discrete random variables of the original Markov chain are approximated by continuous deterministic variables. We compute the solution of the stochastic hybrid model using a numerical algorithm that discretizes time and in each step performs a mutual update of the transient probability distribution of the discrete stochastic variables and the values of the continuous deterministic variables. We implemented the algorithm and we demonstrate its usefulness and efficiency on several case studies from systems biology.
R.: Specification and analysis of distributed objectbased stochastic hybrid systems
 In: HSCC
, 2006
"... Abstract. In practice, many stochastic hybrid systems are not autonomous: they are objects that communicate with other objects by exchanging messages through an asynchronous medium such as a network. Issues such as: how to compositionally specify distributed objectbased stochastic hybrid systems ..."
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Cited by 14 (2 self)
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Abstract. In practice, many stochastic hybrid systems are not autonomous: they are objects that communicate with other objects by exchanging messages through an asynchronous medium such as a network. Issues such as: how to compositionally specify distributed objectbased stochastic hybrid systems (OBSHS), how to formally model them, and how to verify their properties seem therefore quite important. This paper addresses these issues by: (i) defining a mathematical model for such systems that can be naturally regarded as a generalized stochastic hybrid system (GSHS) in the sense of [7]; (ii) proposing a formal OBSHS specification language in which system transitions are specified in a modular way by probabilistic rewrite rules; and (iii) showing how these systems can be subjected to statistical model checking analysis to verify their probabilistic temporal logic properties. 1
Stochastic Satisfiability Modulo Theory: A Novel Technique for the Analysis of Probabilistic Hybrid Systems
 In Proceedings of the 11th International Conference on Hybrid Systems: Computation and Control (HSCC’08
, 2008
"... Abstract. The analysis of hybrid systems exhibiting probabilistic behaviour is notoriously difficult. To enable mechanised analysis of such systems, we extend the reasoning power of arithmetic satisfiabilitymodulotheory solving (SMT) by a comprehensive treatment of randomized (a.k.a. stochastic) qu ..."
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Cited by 13 (5 self)
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Abstract. The analysis of hybrid systems exhibiting probabilistic behaviour is notoriously difficult. To enable mechanised analysis of such systems, we extend the reasoning power of arithmetic satisfiabilitymodulotheory solving (SMT) by a comprehensive treatment of randomized (a.k.a. stochastic) quantification over discrete variables within the mixed Booleanarithmetic constraint system. This provides the technological basis for a fully symbolic analysis of probabilistic hybrid automata. Generalizing SMTbased bounded modelchecking of hybrid automata [2, 11], stochastic SMT permits the direct and fully symbolic analysis of probabilistic bounded reachability problems of probabilistic hybrid automata without resorting to approximation by intermediate finitestate abstractions. 1
Almost sure convergence of numerical approximations for Piecewise Deterministic Markov Processes
 J. Comp. Appl. Math
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A unifying formulation of the FokkerPlanckKolmogorov equation for general stochastic hybrid systems
 In Proceedings of the 17th IFAC World Congress
, 2008
"... A general formulation of the FokkerPlanckKolmogorov (FPK) equation for stochastic hybrid systems is presented, within the framework of Generalized Stochastic Hybrid Systems (GSHS). The FPK equation describes the time evolution of the probability law of the hybrid state. Our derivation is based on ..."
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Cited by 6 (0 self)
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A general formulation of the FokkerPlanckKolmogorov (FPK) equation for stochastic hybrid systems is presented, within the framework of Generalized Stochastic Hybrid Systems (GSHS). The FPK equation describes the time evolution of the probability law of the hybrid state. Our derivation is based on the concept of mean jump intensity, which is related to both the usual stochastic intensity (in the case of spontaneous jumps) and the notion of probability current (in the case of forced jumps). This work unifies all previously known instances of the FPK equation for stochastic hybrid systems, and provides GSHS practitioners with a tool to derive the correct evolution equation for the probability law of the state in any given example. Key words: Stochastic hybrid systems, Stochastic system with jumps, Markov processes, FokkerPlanck equation 1.
Probabilistic reachability for stochastic hybrid systems: Theory, computations, and applications
, 2007
"... Copyright c © 2007 by Alessandro Abate Probabilistic Reachability for Stochastic Hybrid Systems: ..."
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Cited by 4 (0 self)
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Copyright c © 2007 by Alessandro Abate Probabilistic Reachability for Stochastic Hybrid Systems:
Towards a Formal Framework for Multidimensional Codesign
"... Abstract. Multidimensional codesign is a recently proposed paradigm for integrating different system dimensions in sensor networks. Examples of such dimensions are logical and physical mobility, continuous and discrete transitions, deterministic and random evolutions and features resulting from thei ..."
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Cited by 2 (2 self)
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Abstract. Multidimensional codesign is a recently proposed paradigm for integrating different system dimensions in sensor networks. Examples of such dimensions are logical and physical mobility, continuous and discrete transitions, deterministic and random evolutions and features resulting from their interaction, like deterministic and stochastic hybrid behaviours. In this paper, we propose a unifying computational model that considers multiple dimensions, inspired by the Hilbertian Formal Methods paradigm. We couple this model with an integration framework based on domain theory. In this framework new dimensions can be incrementally added, and we illustrate this technique by adding logical mobility to the computational model. The new model has a very promising modelling power, offering all formal ingredients of a neural network. We further investigate bisimulation for systems mixing physical and logical mobility. We identify and solve a compatibility problem between bisimulation relations arising from mobility and continuous behaviours.
Dynamic Logics of Dynamical Systems
"... We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important for modeling and understanding many applications, including embedded ..."
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Cited by 1 (1 self)
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We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important for modeling and understanding many applications, including embedded systems and cyberphysical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. Distributed hybrid systems combine distributed systems with hybrid systems, i.e., they are multiagent hybrid systems that interact through remote communication or physical interaction. Stochastic hybrid systems combine stochastic dynamics with hybrid systems. We survey dynamic logics for specifying and verifying properties for each of those classes of dynamical systems. A dynamic logic is a firstorder modal logic with a pair of parametrized modal operators for each dynamical system to express necessary or possible properties of their transition behavior. Due to their full basis of firstorder modal logic operators, dynamic logics can express a rich variety of system properties, including safety, controllability, reactivity, liveness, and quantified parametrized properties, even about