Engineering constraint solvers for automatic analysis of probabilistic hybrid automata

Fränzle, Martin; Teige, Tino; Eggers, Andreas

doi:10.1016/j.jlap.2010.07.003

Cited by 37 publications

(30 citation statements)

References 21 publications

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“…In [32], a "symbolic" search through the state space using non-probabilistic methods is performed first, after which a finite-state Markov decision process is constructed and analysed. Instead, [13,10] uses stochastic satisfiability modulo theories to permit the verification of bounded properties.…”

Section: Probabilistic Rectangular Hybrid Automatamentioning

confidence: 99%

Verification and Control of Probabilistic Rectangular Hybrid Automata

Sproston

2015

Lecture Notes in Computer Science

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Abstract. Hybrid systems are characterised by a combination of discrete and continuous components. In many application areas for hybrid systems, such as vehicular control and systems biology, stochastic behaviour is intrinsic. This has led to the development of stochastic extensions of formalisms, such as hybrid automata, for the modelling of hybrid systems, together with their associated verification and controller synthesis algorithms, in order to allow reasoning about quantitative properties such as "the vehicle's speed will reach 50kph within 10 seconds with probability at least 0.99". We consider probabilistic rectangular hybrid automata, which generalise the well-known class of rectangular hybrid automata with the possibility of representing random behavior of the discrete components of the system, permitting the modeling of the likelihood of faults, choices in randomised algorithms and message losses. We highlight the differences between verification and control problems for probabilistic rectangular hybrid automata and the corresponding problems for non-probabilistic rectangular hybrid automata. Furthermore, we will describe the effect of assumptions on the underlying model of time (discrete or continuous) on the considered verification and control problems. Finally, we will also consider how probabilistic rectangular hybrid automata can be used as abstract models for more general classes of stochastic hybrid systems.

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Section: Probabilistic Rectangular Hybrid Automatamentioning

confidence: 99%

Verification and Control of Probabilistic Rectangular Hybrid Automata

Sproston

2015

Lecture Notes in Computer Science

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“…Stochastic hybrid systems [Davis 1984;Ghosh et al 1997;Hu et al 2000;Bujorianu and Lygeros 2006;Cassandras and Lygeros 2006;Meseguer and Sharykin 2006;Koutsoukos and Riley 2008;Fränzle et al 2010;Platzer 2011b] are dynamical systems that combine the dynamics of stochastic processes [Karatzas and Shreve 1991;Øksendal There is more than one way in which stochasticity has been added into hybrid systems models; see, e.g., Figure 3. Stochasticity might be restricted to the discrete dynamics, as in piecewise deterministic Markov decision processes [Davis 1984], restricted to the continuous and switching behavior as in switching diffusion processes [Ghosh et al 1997], or allowed in many parts as in so-called General Stochastic Hybrid Systems; see [Bujorianu and Lygeros 2006;Cassandras and Lygeros 2006] for an overview.…”

Section: Stochastic Hybrid Systemsmentioning

confidence: 99%

“…But dynamical systems are more general and can also describe and analyze chemical processes [Riley et al 2010;Kerkez et al 2010], biological systems [Tiwari 2011], medical models [Grosu et al 2011;Kim et al 2011], and many other behavioral phenomena. Since dynamical systems occur in so many different contexts, different variations of dynamical system models are relevant for applications, including discrete dynamical systems described by difference equations or discrete transitions relations [Galor 2010], continuous dynamical systems described by differential equations [Hirsch et al 2003;Perko 2006], hybrid dynamical systems alias hybrid systems combining discrete and continuous dynamics [Maler et al 1991;Branicky 1995;Branicky et al 1998;Davoren and Nerode 2000;Alur et al 2000;Platzer 2008a;Platzer 2010a;Platzer 2008b;Platzer 2010b;Platzer 2012b], distributed hybrid systems or multi-agent hybrid systems Rounds 2004;Kratz et al 2006;Gilbert et al 2009;Platzer 2010c;Platzer 2012a], and stochastic hybrid systems that take stochastic effects into account [Davis 1984;Ghosh et al 1997;Hu et al 2000;Bujorianu and Lygeros 2006;Cassandras and Lygeros 2006;Meseguer and Sharykin 2006;Koutsoukos and Riley 2008;Fränzle et al 2010;Platzer 2011b].…”

Section: Introductionmentioning

confidence: 99%

Logics of Dynamical Systems

Platzer

2012

2012 27th Annual IEEE Symposium on Logic in Computer Science

122

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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 cyber-physical 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 multi-agent 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 first-order 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 first-order modal logic operators, dynamic logics can express a rich variety of system properties, including safety, controllability, reactivity, liveness, and quantified parametrized properties, even about relations between multiple dynamical systems. In this survey, we focus on some of the representatives of the family of differential dynamic logics, which share the ability to express properties of dynamical systems having continuous dynamics described by various forms of differential equations.We explain the dynamical system models, dynamic logics of dynamical systems, their semantics, their axiomatizations, and proof calculi for proving logical formulas about these dynamical systems. We study differential invariants, i.e., induction principles for differential equations. We survey theoretical results, including soundness and completeness and deductive power. Differential dynamic logics have been implemented in automatic and interactive theorem provers and have been used successfully to verify safety-critical applications in automotive, aviation, railway, robotics, and analogue electrical circuits.

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“…Fränzle et al [FTE10] show first pieces for continuous-time bounded model checking of probabilistic hybrid automata (no stochastic differential equations).…”

Section: Related Workmentioning

confidence: 99%

“…Stochasticity might be restricted to the discrete dynamics, as in piecewise deterministic Markov decision processes, restricted to the continuous and switching behavior as in switching diffusion processes [GAM97], or allowed in many parts as in so-called General Stochastic Hybrid Systems; see [BL06,CL06] for an overview. Several different forms of combinations of probabilities with hybrid systems and continuous systems have been considered, both for model checking [FTE10,KR08,CL06] and for simulation-based validation [MS06,ZPC10].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Differential Dynamic Logic for Stochastic Hybrid Programs

Platzer¹

2011

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Logic is a powerful tool for analyzing and verifying systems, including programs, discrete systems, real-time 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 of our logic is measurable. We present compositional proof rules, including rules for stochastic differential equations, and prove soundness.

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Engineering constraint solvers for automatic analysis of probabilistic hybrid automata

Cited by 37 publications

References 21 publications

Verification and Control of Probabilistic Rectangular Hybrid Automata

Verification and Control of Probabilistic Rectangular Hybrid Automata

Logics of Dynamical Systems

Stochastic Differential Dynamic Logic for Stochastic Hybrid Programs

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