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MetiTarski: An Automatic Theorem Prover for RealValued Special Functions
"... Abstract Many theorems involving special functions such as ln, exp and sin can be proved automatically by MetiTarski: a resolution theorem prover modified to call a decision procedure for the theory of real closed fields. Special functions are approximated by upper and lower bounds, which are typica ..."
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Cited by 44 (7 self)
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Abstract Many theorems involving special functions such as ln, exp and sin can be proved automatically by MetiTarski: a resolution theorem prover modified to call a decision procedure for the theory of real closed fields. Special functions are approximated by upper and lower bounds, which are typically rational functions derived from Taylor or continued fraction expansions. The decision procedure simplifies clauses by deleting literals that are inconsistent with other algebraic facts. MetiTarski simplifies arithmetic expressions by conversion to a recursive representation, followed by flattening of nested quotients. Applications include verifying hybrid and control systems.
Recent progress in continuous and hybrid reachability analysis
 In Proc. IEEE International Symposium on ComputerAided Control Systems Design. IEEE Computer
, 2006
"... Abstract — Setbased reachability analysis computes all possible states a system may attain, and in this sense provides knowledge about the system with a completeness, or coverage, that a finite number of simulation runs can not deliver. Due to its inherent complexity, the application of reachabilit ..."
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Cited by 30 (1 self)
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Abstract — Setbased reachability analysis computes all possible states a system may attain, and in this sense provides knowledge about the system with a completeness, or coverage, that a finite number of simulation runs can not deliver. Due to its inherent complexity, the application of reachability analysis has been limited so far to simple systems, both in the continuous and the hybrid domain. In this paper we present recent advances that, in combination, significantly improve this applicability, and allow us to find better balance between computational cost and accuracy. The presentation covers, in a unified manner, a variety of methods handling increasingly complex types of continuous dynamics (constant derivative, linear, nonlinear). The improvements include new geometrical objects for representing sets, new approximation schemes, and more flexible combinations of graphsearch algorithm and partition refinement. We report briefly some preliminary experiments that have enabled the analysis of systems previously beyond reach. I.
Hybrid Systems: From Verification to Falsification by Combining Motion Planning and Discrete Search
 FORMAL METHODS IN SYSTEM DESIGN
, 2007
"... We propose HyDICE, Hybrid DIscrete Continuous Exploration, a multilayered approach for hybridsystem falsification that combines motion planning with discrete search and discovers safety violations by computing witness trajectories to unsafe states. The discrete search uses discrete transitions and ..."
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Cited by 24 (8 self)
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We propose HyDICE, Hybrid DIscrete Continuous Exploration, a multilayered approach for hybridsystem falsification that combines motion planning with discrete search and discovers safety violations by computing witness trajectories to unsafe states. The discrete search uses discrete transitions and a statespace decomposition to guide the motion planner during the search for witness trajectories. Experiments on a nonlinear hybrid robotic system with over one million modes and experiments with an aircraft conflictresolution protocol with highdimensional continuous state spaces demonstrate the effectiveness of HyDICE. Comparisons to related work show computational speedups of up to two orders of magnitude.
Symbolic model checking of hybrid systems using template polyhedra
 In TACAS’08  Tools and Algorithms for
, 2008
"... Abstract. We propose techniques for the verification of hybrid systems using template polyhedra, i.e., polyhedra whose inequalities have fixed expressions but with varying constant terms. Given a hybrid system description and a set of template linear expressions as inputs, our technique constructs o ..."
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Cited by 21 (7 self)
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Abstract. We propose techniques for the verification of hybrid systems using template polyhedra, i.e., polyhedra whose inequalities have fixed expressions but with varying constant terms. Given a hybrid system description and a set of template linear expressions as inputs, our technique constructs overapproximations of the reachable states using template polyhedra. Therefore, operations used in symbolic model checking such as intersection, union and postcondition across discrete transitions over template polyhedra can be computed efficiently using template polyhedra without requiring expensive vertex enumeration. Additionally, the verification of hybrid systems requires techniques to handle the continuous dynamics inside discrete modes. We propose a new flowpipe construction algorithm using template polyhedra. Our technique uses higherorder Taylor series expansion to approximate the time trajectories. The terms occurring in the Taylor series expansion are bounded using repeated optimization queries. The location invariant is used to enclose the remainder term of the Taylor series, and thus truncate the expansion. Finally, we have implemented our technique as a part of the tool TimePass for the analysis of affine hybrid automata. 1
Guaranteed termination in the verification of LTL properties of nonlinear robust discrete time hybrid systems
 IN PELED, D., TSAY, Y.K., EDS.: ATVA. VOLUME 3707 OF LNCS
, 2005
"... We present a novel approach to the automatic verification of LTL requirements of nonlinear discretetime hybrid systems. The verification tool uses an intervalbased constraint solver for nonlinear robust constraints to compute incrementally refined abstractions. Although the problem is undecidabl ..."
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Cited by 19 (6 self)
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We present a novel approach to the automatic verification of LTL requirements of nonlinear discretetime hybrid systems. The verification tool uses an intervalbased constraint solver for nonlinear robust constraints to compute incrementally refined abstractions. Although the problem is undecidable, we prove termination of abstraction refinement based verification of such properties for the class of robust nonlinear hybrid systems, thus significantly extending previous semidecidability results. We argue, that safety critical control applications are robust hybrid systems. We give first results on the application of this approach to a variant of an aircraft collision avoidance protocol.
Model checking of hybrid systems: From reachability towards stability
 Hybrid Systems: Computation and Control, volume 3927 of LNCS
, 2006
"... Abstract. We call a hybrid system stable if every trajectory inevitably ends up in a given region. Our notion of stability deviates from classical definitions in control theory. In this paper, we present a model checking algorithm for stability in the new sense. The idea of the algorithm is to reduc ..."
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Cited by 18 (3 self)
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Abstract. We call a hybrid system stable if every trajectory inevitably ends up in a given region. Our notion of stability deviates from classical definitions in control theory. In this paper, we present a model checking algorithm for stability in the new sense. The idea of the algorithm is to reduce the stability proof for the whole system to a set of (smaller) proofs for several onemode systems. 1
Z.: Constraints for continuous reachability in the verification of hybrid systems
 AISC 2006. LNCS (LNAI
"... Abstract. The method for verification of hybrid systems by constraint propagation based abstraction refinement that we introduced in an earlier paper is based on an overapproximation of continuous reachability information of ordinary differential equations using constraints that do not contain diff ..."
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Cited by 17 (3 self)
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Abstract. The method for verification of hybrid systems by constraint propagation based abstraction refinement that we introduced in an earlier paper is based on an overapproximation of continuous reachability information of ordinary differential equations using constraints that do not contain differentiation symbols. The method uses an interval constraint propagation based solver to solve these constraints. This has the advantage that—without complicated algorithmic changes—the method can be improved by just changing these constraints. In this paper, we discuss various possibilities of such changes, we prove some properties about the amount of overapproximations introduced by the new constraints, and provide some timings that document the resulting improvement. 1
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 15 (15 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
The Structure of Differential Invariants and Differential Cut Elimination
, 2011
"... not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution or government. Keywords: Proof theory, differential equations, differential cut elimination, logics of programs, The biggest challenge in hybrid systems verification is the handling o ..."
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Cited by 14 (12 self)
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not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution or government. Keywords: Proof theory, differential equations, differential cut elimination, logics of programs, The biggest challenge in hybrid systems verification is the handling of differential equations. Because computable closedform solutions only exist for very simple differential equations, proof certificates have been proposed for more scalable verification. Search procedures for these proof certificates are still rather adhoc, though, because the problem structure is only understood poorly. We investigate differential invariants, which can be checked for invariance along a differential equation just by using their differential structure and without having to solve the differential equation. We study the structural properties of differential invariants. To analyze tradeoffs for proof search complexity, we identify more than a dozen relations between several classes of differential invariants and compare their deductive power. As our main results, we analyze the deductive power of differential cuts and the deductive power of differential invariants with auxiliary differential variables. We refute the differential cut elimination hypothesis and show that differential cuts are fundamental proof principles that strictly increase the deductive power. We also prove that
Relational Abstractions For Continuous and Hybrid Systems
"... Abstract. There has been much recent progress on invariant generation techniques for continuous systems whose dynamics are described by Ordinary Differential Equations (ODE). In this paper, we present a simple abstraction scheme for hybrid systems that abstracts continuous dynamics by relating any s ..."
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Cited by 11 (5 self)
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Abstract. There has been much recent progress on invariant generation techniques for continuous systems whose dynamics are described by Ordinary Differential Equations (ODE). In this paper, we present a simple abstraction scheme for hybrid systems that abstracts continuous dynamics by relating any state of the system to a state that can potentially be reached at some future time instant. Such relations are then interpreted as discrete transitions that model the continuous evolution of states over time. We adapt templatebased invariant generation techniques for continuous dynamics to derive relational abstractions for continuous systems with linear as well as nonlinear dynamics. Once a relational abstraction hasbeen derived,theresultingsystemis apurelydiscrete, infinitestatesystem. Therefore, techniquessuchas kinductioncan be directly applied to this abstraction to prove properties, and bounded modelchecking techniques applied to find potential falsifications. We present the basic underpinnings of our approach and demonstrate its use on many benchmark systems to derive simple and usable abstractions. 1