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31
Efficient solving of large nonlinear arithmetic constraint systems with complex boolean structure
 Journal on Satisfiability, Boolean Modeling and Computation
, 2007
"... In order to facilitate automated reasoning about large Boolean combinations of nonlinear arithmetic constraints involving transcendental functions, we provide a tight integration of recent SAT solving techniques with intervalbased arithmetic constraint solving. Our approach deviates substantially f ..."
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Cited by 85 (11 self)
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In order to facilitate automated reasoning about large Boolean combinations of nonlinear arithmetic constraints involving transcendental functions, we provide a tight integration of recent SAT solving techniques with intervalbased arithmetic constraint solving. Our approach deviates substantially from lazy theorem proving approaches in that it directly controls arithmetic constraint propagation from the SAT solver rather than delegating arithmetic decisions to a subordinate solver. Through this tight integration, all the algorithmic enhancements that were instrumental to the enormous performance gains recently achieved in propositional SAT solving carry over smoothly to the rich domain of nonlinear arithmetic constraints. As a consequence, our approach is able to handle large constraint systems with extremely complex Boolean structure, involving Boolean combinations of multiple thousand arithmetic constraints over some thousands of variables.
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.
Real World Verification
"... Abstract. Scalable handling of real arithmetic is a crucial part of the verification of hybrid systems, mathematical algorithms, and mixed analog/digital circuits. Despite substantial advances in verification technology, complexity issues with classical decision procedures are still a major obstacle ..."
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Cited by 17 (3 self)
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Abstract. Scalable handling of real arithmetic is a crucial part of the verification of hybrid systems, mathematical algorithms, and mixed analog/digital circuits. Despite substantial advances in verification technology, complexity issues with classical decision procedures are still a major obstacle for formal verification of realworld applications, e.g., in automotive and avionic industries. To identify strengths and weaknesses, we examine state of the art symbolic techniques and implementations for the universal fragment of realclosed fields: approaches based on quantifier elimination, Gröbner Bases, and semidefinite programming for the Positivstellensatz. Within a uniform context of the verification tool KeYmaera, we compare these approaches qualitatively and quantitatively on verification benchmarks from hybrid systems, textbook algorithms, and on geometric problems. Finally, we introduce a new decision procedure combining Gröbner Bases and semidefinite programming for the real Nullstellensatz that outperforms the individual approaches on an interesting set of problems.
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
Consistency and the Quantified Constraint Satisfaction Problem
, 2007
"... Constraint satisfaction is a very well studied and fundamental artificial intelligence technique. Various forms of knowledge can be represented with constraints, and reasoning techniques from disparate fields can be encapsulated within constraint reasoning algorithms. However, problems involving unc ..."
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Cited by 16 (1 self)
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Constraint satisfaction is a very well studied and fundamental artificial intelligence technique. Various forms of knowledge can be represented with constraints, and reasoning techniques from disparate fields can be encapsulated within constraint reasoning algorithms. However, problems involving uncertainty, or which have an adversarial nature (for example, games), are difficult to express and solve in the classical constraint satisfaction problem. This thesis is concerned with an extension to the classical problem: the Quantified Constraint Satisfaction Problem (QCSP). QCSP has recently attracted interest. In QCSP, quantifiers are allowed, facilitating the expression of uncertainty. I examine whether QCSP is a useful formalism. This divides into two questions: whether QCSP can be solved efficiently; and whether realistic problems can be represented in QCSP. In attempting to answer these questions, the main contributions of this thesis are the following: • the definition of two new notions of consistency; • four new constraint propagation algorithms (with eight variants in total), along with empirical evaluations;
Computational Complexity of Determining Which Statements about Causality Hold
 in Different SpaceTime Models”, Theoretical Computer Science
"... Causality is one of the most fundamental notions of physics. It is therefore important to be able to decide which statements about causality are correct in different models of spacetime. In this paper, we analyze the computational complexity of the corresponding deciding problems. In particular, we ..."
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Cited by 13 (9 self)
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Causality is one of the most fundamental notions of physics. It is therefore important to be able to decide which statements about causality are correct in different models of spacetime. In this paper, we analyze the computational complexity of the corresponding deciding problems. In particular, we show that: • for Minkowski spacetime, the deciding problem is as difficult as the Tarski’s problem of deciding elementary geometry, while • for a natural model of primordial spacetime, the corresponding deciding problem is of the lowest possible complexity. 1
Providing a basin of attraction to a target region by computation of Lyapunovlike functions
 In IEEE Int. Conf. on Computational Cybernetics
, 2006
"... Abstract — In this paper, we present a method for computing a basin of attraction to a target region for nonlinear ordinary differential equations. This basin of attraction is ensured by a Lyapunovlike polynomial function that we compute using an interval based branchandrelax algorithm. This alg ..."
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Cited by 10 (5 self)
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Abstract — In this paper, we present a method for computing a basin of attraction to a target region for nonlinear ordinary differential equations. This basin of attraction is ensured by a Lyapunovlike polynomial function that we compute using an interval based branchandrelax algorithm. This algorithm relaxes the necessary conditions on the coefficients of the Lyapunovlike function to a system of linear interval inequalities that can then be solved exactly, and iteratively reduces the relaxation error by recursively decomposing the state space into hyperrectangles. Tests on an implementation are promising. I.
Solving Existentially Quantified Constraints with One Equality and Arbitrarily Many Inequalities
 In Proc. of CP’03, LNCS 2833
, 2003
"... This paper contains the first algorithm that can solve disjunctions of constraints of the form g1 . . . ..."
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Cited by 6 (0 self)
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This paper contains the first algorithm that can solve disjunctions of constraints of the form g1 . . .
CalCS: SMT Solving for NonLinear Convex Constraints”, FMCAD
, 2010
"... Abstract—Certain formal verification tasks require reasoning about Boolean combinations of nonlinear arithmetic constraints over the real numbers. In this paper, we present a new technique for satisfiability solving of Boolean combinations of nonlinear constraints that are convex. Our approach app ..."
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Cited by 5 (1 self)
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Abstract—Certain formal verification tasks require reasoning about Boolean combinations of nonlinear arithmetic constraints over the real numbers. In this paper, we present a new technique for satisfiability solving of Boolean combinations of nonlinear constraints that are convex. Our approach applies fundamental results from the theory of convex programming to realize a satisfiability modulo theory (SMT) solver. Our solver, CalCS, uses a lazy combination of SAT and a theory solver. A key step in our algorithm is the use of complementary slackness and duality theory to generate succinct infeasibility proofs that support conflictdriven learning. Moreover, whenever nonconvex constraints are produced from Boolean reasoning, we provide a procedure that generates conservative approximations of the original set of constraints by using geometric properties of convex sets and supporting hyperplanes. We validate CalCS on several benchmarks including formulas generated from bounded model checking of hybrid automata and static analysis of floatingpoint software. I.
Guaranteed setpoint computation with application to the control of a sailboat
 VERSION OF THE SUBMISSION TO IEEE TRANSACTIONS ON ROBOTICS, AUGUST 2012 6 International Journal of Control Automation and Systems
, 2010
"... Abstract: The problem of characterizing in a guaranteed way the set of all feasible setpoints of a control problem is known to be difficult. In the present work, the problem to be solved involves nonlinear equality constraints with variables affected by logical quantifiers. This problem is not sol ..."
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Cited by 5 (3 self)
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Abstract: The problem of characterizing in a guaranteed way the set of all feasible setpoints of a control problem is known to be difficult. In the present work, the problem to be solved involves nonlinear equality constraints with variables affected by logical quantifiers. This problem is not solvable by current symbolic methods like quantifier elimination, which is commonly used for solving this class of problems. We propose the utilization of guaranteed setcomputation techniques based on interval analysis, in particular a solver referred to as Quantified Set Inversion (QSI). As an application example, the problem of simultaneously controlling the speed and the orientation of a sailboat is presented. For this purpose, the combination of QSI solver and feedback linearization techniques is employed.