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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.
Solving NonLinear Arithmetic
"... We present a new algorithm for deciding satisfiability of nonlinear arithmetic constraints. The algorithm performs a ConflictDriven Clause Learning (CDCL)style search for a feasible assignment, while using projection operators adapted from cylindrical algebraic decomposition to guide the search aw ..."
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Cited by 29 (3 self)
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We present a new algorithm for deciding satisfiability of nonlinear arithmetic constraints. The algorithm performs a ConflictDriven Clause Learning (CDCL)style search for a feasible assignment, while using projection operators adapted from cylindrical algebraic decomposition to guide the search away from the conflicting states.
Flow*: An Analyzer for NonLinear Hybrid Systems
"... Abstract. The tool FLOW * performs Taylor modelbased flowpipe construction for nonlinear (polynomial) hybrid systems. FLOW * combines wellknown Taylor model arithmetic techniques for guaranteed approximations of the continuous dynamics in each mode with a combination of approaches for handling mo ..."
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Cited by 23 (2 self)
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Abstract. The tool FLOW * performs Taylor modelbased flowpipe construction for nonlinear (polynomial) hybrid systems. FLOW * combines wellknown Taylor model arithmetic techniques for guaranteed approximations of the continuous dynamics in each mode with a combination of approaches for handling mode invariants and discrete transitions. FLOW * supports a wide variety of optimizations including adaptive step sizes, adaptive selection of approximation orders and the heuristic selection of template directions for aggregating flowpipes. This paper describes FLOW * and demonstrates its performance on a series of nonlinear continuous and hybrid system benchmarks. Our comparisons show that FLOW * is competitive with other tools. 1 Overview of FLOW* In this paper, we present the FLOW * tool to generate flowpipes for nonlinear hybrid systems using Taylor Models (TMs). TMs were originally proposed by Berz and Makino [1] to represent functions by means of higherorder Taylor polynomial expansions, bloated by an interval to represent the approximation error. TMs support
Automatic verification of control system implementations
 In proceedings of EMSOFT
, 2010
"... Software implementations of controllers for physical subsystems form the core of many modern safetycritical systems such as aircraft flight control and automotive engine control. A fundamental property of such implementations is stability, the guarantee that the physical plant converges to a desire ..."
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Cited by 19 (10 self)
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Software implementations of controllers for physical subsystems form the core of many modern safetycritical systems such as aircraft flight control and automotive engine control. A fundamental property of such implementations is stability, the guarantee that the physical plant converges to a desired behavior under the actions of the controller. We present a methodology and a tool to perform automated static analysis of embedded controller code for stability of the controlled physical system. The design of controllers for physical systems provides not only the controllers but also mathematical proofs of their stability under idealized mathematical models. Unfortunately, since these models do not capture most of the implementation details, it is not always clear if the stability
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
Solving Nonlinear Polynomial Arithmetic via SAT Modulo Linear Arithmetic
"... Polynomial constraintsolving plays a prominent role in several areas of engineering and software verification. In particular, polynomial constraint solving has a long and successful history in the development of tools for proving termination of programs. Wellknown and very efficient techniques, l ..."
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Cited by 13 (5 self)
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Polynomial constraintsolving plays a prominent role in several areas of engineering and software verification. In particular, polynomial constraint solving has a long and successful history in the development of tools for proving termination of programs. Wellknown and very efficient techniques, like SAT algorithms and tools, have been recently proposed and used for implementing polynomial constraint solving algorithms through appropriate encodings. However, powerful techniques like the ones provided by the SMT (SAT modulo theories) approach for linear arithmetic constraints (over the rationals) are underexplored to date. In this paper we show that the use of these techniques for developing polynomial constraint solvers outperforms the best existing solvers and provides a new and powerful approach for implementing better and more general solvers for termination provers.
Integrating icp and lra solvers for deciding nonlinear real arithmetic problems
"... Abstract—We propose a novel integration of interval constraint propagation (ICP) with SMT solvers for linear real arithmetic (LRA) to decide nonlinear real arithmetic problems. We use ICP to search for interval solutions of the nonlinear constraints, and use the LRA solver to either validate the sol ..."
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Cited by 12 (3 self)
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Abstract—We propose a novel integration of interval constraint propagation (ICP) with SMT solvers for linear real arithmetic (LRA) to decide nonlinear real arithmetic problems. We use ICP to search for interval solutions of the nonlinear constraints, and use the LRA solver to either validate the solutions or provide constraints to incrementally refine the search space for ICP. This serves the goal of separating the linear and nonlinear solving stages, and we show that the proposed methods preserve the correctness guarantees of ICP. Experimental results show that such separation is useful for enhancing efficiency. I.
SAT modulo ODE: A direct SAT approach to hybrid systems
, 2008
"... In order to facilitate automated reasoning about large Boolean combinations of nonlinear arithmetic constraints involving ordinary differential equations (ODEs), we provide a seamless integration of safe numeric overapproximation of initialvalue problems into a SATmodulotheory (SMT) approach to ..."
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Cited by 10 (3 self)
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In order to facilitate automated reasoning about large Boolean combinations of nonlinear arithmetic constraints involving ordinary differential equations (ODEs), we provide a seamless integration of safe numeric overapproximation of initialvalue problems into a SATmodulotheory (SMT) approach to intervalbased arithmetic constraint solving. Intervalbased safe numeric approximation of ODEs is used as an interval contractor being able to narrow candidate sets in phase space in both temporal directions: postimages of ODEs (i.e., sets of states reachable from a set of initial values) are narrowed based on partial information about the initial values and, vice versa, preimages are narrowed based on partial knowledge about postsets. In contrast to the related CLP(F) approach of Hickey and Wittenberg [10], we do (a) support coordinate transformations mitigating the wrapping effect encountered upon iterating intervalbased overapproximations of reachable state sets and (b) embed the approach into an SMT framework, thus accelerating the solving process through the algorithmic enhacements of recent SAT solving technology.
SAT Modulo Linear Arithmetic for Solving Polynomial Constraints
"... Polynomial constraint solving plays a prominent role in several areas of hardware and software analysis and verification, e.g., termination proving, program invariant generation and hybrid system verification, to name a few. In this paper we propose a new method for solving nonlinear constraints ba ..."
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Cited by 10 (4 self)
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Polynomial constraint solving plays a prominent role in several areas of hardware and software analysis and verification, e.g., termination proving, program invariant generation and hybrid system verification, to name a few. In this paper we propose a new method for solving nonlinear constraints based on encoding the problem into an SMT problem considering only linear arithmetic. Unlike other existing methods, our method focuses on proving satisfiability of the constraints rather than on proving unsatisfiability, which is more relevant in several applications as we illustrate with several examples. Nevertheless, we also present new techniques based on the analysis of unsatisfiable cores that allow one to efficiently prove unsatisfiability too for a broad class of problems. The power of our approach is demonstrated by means of extensive experiments comparing our prototype with stateoftheart tools on benchmarks taken both from the academic and the industrial world.
Improving SAT Modulo ODE for hybrid systems analysis by combining different enclosure methods
 Software Engineering and Formal Methods, volume 7041 of Lecture Notes in Computer Science
, 2011
"... Abstract. Aiming at automatic verification and analysis techniques for hybrid systems, we present a novel combination of enclosure methods for ordinary differential equations (ODEs) with the iSAT solver for large Boolean combinations of arithmetic constraints. Improving on our previous work, the co ..."
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Cited by 9 (1 self)
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Abstract. Aiming at automatic verification and analysis techniques for hybrid systems, we present a novel combination of enclosure methods for ordinary differential equations (ODEs) with the iSAT solver for large Boolean combinations of arithmetic constraints. Improving on our previous work, the contribution of this paper lies in combining iSAT with VNODELP, as a stateoftheart enclosure method for ODEs, and with bracketing systems which exploit monotonicity properties to find enclosures for problems that VNODELP alone cannot enclose tightly. We apply our method to the analysis of a nonlinear hybrid system by solving predicative encodings of an inductive stability argument and evaluate the impact of different methods and their combination. 1