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384
Logics for Hybrid Systems
 Proceedings of the IEEE
, 2000
"... This paper offers a synthetic overview of, and original contributions to, the use of logics and formal methods in the analysis of hybrid systems ..."
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Cited by 138 (13 self)
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This paper offers a synthetic overview of, and original contributions to, the use of logics and formal methods in the analysis of hybrid systems
A New Class of Decidable Hybrid Systems
 In Hybrid Systems : Computation and Control
, 1999
"... One of the most important analysis problems of hybrid systems is the reachability problem. State of the art computational tools perform reachability computation for timed automata, multirate automata, and rectangular automata. In this paper, we extend the decidability frontier for classes of lin ..."
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Cited by 112 (8 self)
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One of the most important analysis problems of hybrid systems is the reachability problem. State of the art computational tools perform reachability computation for timed automata, multirate automata, and rectangular automata. In this paper, we extend the decidability frontier for classes of linear hybrid systems, which are introduced as hybrid systems with linear vector fields in each discrete location. This result is achieved by showing that any such hybrid system admits a finite bisimulation, and by providing an algorithm that computes it using decision methods from mathematical logic.
Differential Dynamic Logic for Hybrid Systems
, 2007
"... Hybrid systems are models for complex physical systems and are defined as dynamical systems with interacting discrete transitions and continuous evolutions along differential equations. With the goal of developing a theoretical and practical foundation for deductive verification of hybrid systems, ..."
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Cited by 76 (44 self)
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Hybrid systems are models for complex physical systems and are defined as dynamical systems with interacting discrete transitions and continuous evolutions along differential equations. With the goal of developing a theoretical and practical foundation for deductive verification of hybrid systems, we introduce a dynamic logic for hybrid programs, which is a program notation for hybrid systems. As a verification technique that is suitable for automation, we introduce a free variable proof calculus with a novel combination of realvalued free variables and Skolemisation for lifting quantifier elimination for real arithmetic to dynamic logic. The calculus is compositional, i.e., it reduces properties of hybrid programs to properties of their parts. Our main result proves that this calculus axiomatises the transition behaviour of hybrid systems completely relative to differential equations. In a case study with cooperating traffic agents of the European Train Control System, we further show that our calculus is wellsuited for verifying realistic hybrid systems with parametric system dynamics.
Definable compactness and definable subgroups of ominimal groups
 J. LONDON MATH. SOC
, 1999
"... We introduce the notion of definable compactness and within the context of ominimal structures prove several topological properties of definably compact spaces. In particular a definable set in an ominimal structure is definably compact (with respect to the subspace topology) if and only if it is c ..."
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Cited by 55 (2 self)
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We introduce the notion of definable compactness and within the context of ominimal structures prove several topological properties of definably compact spaces. In particular a definable set in an ominimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. We then apply definable compactness to the study of groups and rings in ominimal structures. The main result we prove, Theorem 1.2, is that any infinite definable group in an ominimal structure that is not definably compact contains a definable torsionfree subgroup of dimension one. Using this theorem we give a complete characterization of all rings without zero divisors that are definable in ominimal structures. The paper concludes with several examples illustrating some limitations on extending Theorem 1.2.
Definably compact abelian groups
 Journal of Mathematical Logic
, 2004
"... Let M be an o–minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n– dimensional group definable in M. We show the following: the o–minimal fundamental group of G is isomorphic to Z n; for each k> 0, the k–torsion subgroup of G is isomorphic to (Z/kZ ..."
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Cited by 51 (24 self)
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Let M be an o–minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n– dimensional group definable in M. We show the following: the o–minimal fundamental group of G is isomorphic to Z n; for each k> 0, the k–torsion subgroup of G is isomorphic to (Z/kZ) n, and the o–minimal cohomology algebra over Q of G is isomorphic to the exterior algebra over Q with n generators of degree one. 1
Weakly ominimal structures and real closed fields
 Trans. Amer. Math. Soc
"... Abstract. A linearly ordered structure is weakly ominimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly ominimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results ..."
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Cited by 40 (6 self)
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Abstract. A linearly ordered structure is weakly ominimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly ominimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly ominimal structures. Foremost among these, we show that every weakly ominimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly ominimal structures, patterned, as much as possible, after that for ominimal structures. 1.
Reach Set Computations Using Real Quantifier Elimination
, 2000
"... Reach set computations are of fundamental importance in control theory. We consider the reach set problem for openloop systems described by parametric inhomogeneous linear dierential systems and use real quanti er elimination methods to get exact and approximate solutions. For simple elementar ..."
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Cited by 36 (1 self)
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Reach set computations are of fundamental importance in control theory. We consider the reach set problem for openloop systems described by parametric inhomogeneous linear dierential systems and use real quanti er elimination methods to get exact and approximate solutions. For simple elementary functions we give an exact calculation of the cases where exact semialgebraic transcendental implicitization is possible. For the negative cases we provide approximate alternating using discrete point checking or safe estimations of reach sets and control parameter sets. The method employs a reduction of forward and backward reach set and control parameter set problem to the transcendental implicitization problem for the components of special solutions of simpler nonparametric systems. Numerous examples are computed using the redlog and qepcad packages.