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259
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 98 (7 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 84 (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.
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 33 (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.
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 33 (1 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 28 (12 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
Mixed RealInteger Linear Quantifier Elimination
, 1999
"... Consider the elementary theory T of the real numbers in the language L having 0, 1 as constants, addition and subtraction and integer part as operations, and equality, order and congruences modulo natural number constants as relations. We show that T admits an effective quantifier elimination proced ..."
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Cited by 26 (1 self)
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Consider the elementary theory T of the real numbers in the language L having 0, 1 as constants, addition and subtraction and integer part as operations, and equality, order and congruences modulo natural number constants as relations. We show that T admits an effective quantifier elimination procedure and is decidable. Moreover this procedure provides sample answers for existentially quantified variables. The procedure comprises as special cases linear elimination for the reals and for Presburger arithmetic. We provide closely matching upper and lower bounds for the complexity of the quantifier elimination and decision problem for T . Applications include a characterization of T definable subsets of the real line, and the modeling of parametric mixed integer linear optimization, of continuous phenomena with periodicity, and the simulation and analysis of hybrid control systems. We also consider the elementary theory of reals in variations of this language in view of quantifier elimination...