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A.: MinimalModelGuided Approaches to Solving Polynomial Constraints and Extensions
"... Abstract. In this paper we present new methods for deciding the satisfiability of formulas involving integer polynomial constraints. In previous work we proposed to solve SMT(NIA) problems by reducing them to SMT(LIA): nonlinear monomials are linearized by abstracting them with fresh variables and ..."
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Abstract. In this paper we present new methods for deciding the satisfiability of formulas involving integer polynomial constraints. In previous work we proposed to solve SMT(NIA) problems by reducing them to SMT(LIA): nonlinear monomials are linearized by abstracting them with fresh variables and by performing case splitting on integer variables with finite domain. When variables do not have finite domains, artificial ones can be introduced by imposing a lower and an upper bound, and made iteratively larger until a solution is found (or the procedure times out). For the approach to be practical, unsatisfiable cores are used to guide which domains have to be relaxed (i.e., enlarged) from one iteration to the following one. However, it is not clear then how large they have to be made, which is critical. Here we propose to guide the domain relaxation step by analyzing minimal models produced by the SMT(LIA) solver. Namely, we consider two different cost functions: the number of violated artificial domain bounds, and the distance with respect to the artificial domains. We compare these approaches with other techniques on benchmarks coming from constraintbased program analysis and show the potential of the method. Finally, we describe how one of these minimalmodelguided techniques can be smoothly adapted to deal with the extension MaxSMT of SMT(NIA) and then applied to program termination proving. 1
Formal modelling, analysis and verification of hybrid systems
 In Unifying Theories of Programming and Formal Engineering Methods, volume 8050 of LNCS
, 2013
"... Abstract. Hybrid systems is a mathematical model of embedded systems, and has been widely used in the design of complex embedded systems. In this chapter, we will introduce our systematic approach to formal modelling, analysis and verification of hybrid systems. In our framework, a hybrid system i ..."
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Abstract. Hybrid systems is a mathematical model of embedded systems, and has been widely used in the design of complex embedded systems. In this chapter, we will introduce our systematic approach to formal modelling, analysis and verification of hybrid systems. In our framework, a hybrid system is modelled using Hybird CSP (HCSP), and specified and reasoned about by Hybrid Hoare Logic (HHL), which is an extension of Hoare logic to hybrid systems. For deductive verification of hybrid systems, a complete approach to generating polynomial invariants for polynomial hybrid systems is proposed; meanwhile, a theorem prover for HHL that can provide tool support for the verification has been implemented. We give some case studies from realtime world, for instance, Chinese HighSpeed Train Control System at Level 3 (CTCS3). In addition, based on our invariant generation approach, we consider how to synthesize a switching logic for a considered hybrid system by reduction to constraint solving, to meet a given safety, liveness, optimality requirement, or any of their combinations. We also discuss other issues of hybrid systems, e.g., stability analysis.
Verification of Periodically Controlled Hybrid Systems: Application to An Autonomous Vehicle
 Special Issue of the ACM Transactions on Embedded Computing Systems (TECS
"... This paper introduces Periodically Controlled Hybrid Automata (PCHA) for modular specification of hybrid control systems. In a PCHA, control actions that change the control input to the plant occur roughly periodically, while other actions that update the state of the controller may occur in the int ..."
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This paper introduces Periodically Controlled Hybrid Automata (PCHA) for modular specification of hybrid control systems. In a PCHA, control actions that change the control input to the plant occur roughly periodically, while other actions that update the state of the controller may occur in the interim, changing the setpoint of the system. Such actions could model, for example, sensor updates and information received from higherlevel planning modules that change the setpoint of the controller. Based on periodicity and subtangential conditions, a new sufficient condition for verifying invariant properties of PCHAs is presented. Checking these conditions can be automated using, for example, the constraintbased approach, quantifier elimination, or sum of squares decomposition. The proposed technique is used to verify safety and progress properties of the plannercontroller subsystem of an autonomous ground vehicle. Geometric properties of planner generated paths are derived which guarantee that such paths can be safely followed by the controller.
Automatic Abstraction of NonLinear Systems Using Change of Bases Transformations.
"... We present abstraction techniques that transform a given nonlinear dynamical system into a linear system, such that, invariant properties of the resulting linear abstraction can be used to infer invariants for the original system. The abstraction techniques rely on a change of bases transformation ..."
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We present abstraction techniques that transform a given nonlinear dynamical system into a linear system, such that, invariant properties of the resulting linear abstraction can be used to infer invariants for the original system. The abstraction techniques rely on a change of bases transformation that associates each state variable of the abstract system with a function involving the state variables of the original system. We present conditions under which a given change of basis transformation for a nonlinear system can define an abstraction. Furthermore, we present a technique to discover, given a nonlinear system, if a change of bases transformation involving degreebounded polynomials yielding a linear system abstraction exists. If so, our technique yields the resulting abstract linear system, as well. This approach is further extended to search for a change of bases transformation that abstracts a given nonlinear system into a system of linear differential inclusions. Our techniques enable the use of analysis techniques for linear systems to infer invariants for nonlinear systems. We present preliminary evidence of the practical feasibility of our ideas using a prototype implementation.
Program verification by using DISCOVERER
 the Proc. VSTTE’05 held in Zürich
, 2005
"... Abstract. Recent advances in program verification indicate that various verification problems can be reduced to semialgebraic system (SAS for short) solving. An SAS consists of polynomial equations and polynomial inequalities. Algorithms for quantifier elimination of real closed fields are the ge ..."
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Abstract. Recent advances in program verification indicate that various verification problems can be reduced to semialgebraic system (SAS for short) solving. An SAS consists of polynomial equations and polynomial inequalities. Algorithms for quantifier elimination of real closed fields are the general method for those problems. But the general method usually have low efficiency for specific problems. To overcome the bottleneck of program verification with symbolic approach, one has to combine special techniques with the general method. Based on the work of complete discrimination systems of polynomials [32, 30], we invented new theories and algorithms [31, 29, 33] for SAS solving and partly implemented them as a real symbolic computation tool in Maple named DISCOVERER. In this paper, we first summarize the results that we have done so far both on SASsolving and program verification with DISCOVERER, and then discuss the future work in this direction, including SASsolving itself, termination analysis and invariant generation of programs, and reachability computation of hybrid systems etc. K
M.: Exponentialconditionbased barrier certificate generation for safety verification of hybrid systems
 In: CAV’13
, 2013
"... Abstract. A barrier certificate is an inductive invariant function which can be used for the safety verification of a hybrid system. Safety verification based on barrier certificate has the benefit of avoiding explicit computation of the exact reachable set which is usually intractable for nonlinea ..."
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Abstract. A barrier certificate is an inductive invariant function which can be used for the safety verification of a hybrid system. Safety verification based on barrier certificate has the benefit of avoiding explicit computation of the exact reachable set which is usually intractable for nonlinear hybrid systems. In this paper, we propose a new barrier certificate condition, called Exponential Condition, for the safety verification of semialgebraic hybrid systems. The most important benefit of Exponential Condition is that it has a lower conservativeness than the existing convex condition and meanwhile it possesses the property of convexity. On the one hand, a less conservative barrier certificate forms a tighter overapproximation for the reachable set and hence is able to verify critical safety properties. On the other hand, the property of convexity guarantees its solvability by semidefinite programming method. Some examples are presented to illustrate the effectiveness and practicality of our method.
Characterizing Algebraic Invariants by Differential Radical Invariants ⋆
"... Abstract We prove that any invariant algebraic set of a given polynomial vector field can be algebraically represented by one polynomial and a finite set of its successive Lie derivatives. This socalled differential radical characterization relies on a sound abstraction of the reachable set of solu ..."
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Abstract We prove that any invariant algebraic set of a given polynomial vector field can be algebraically represented by one polynomial and a finite set of its successive Lie derivatives. This socalled differential radical characterization relies on a sound abstraction of the reachable set of solutions by the smallest variety that contains it. The characterization leads to a differential radical invariant proof rule that is sound and complete, which implies that invariance of algebraic equations over realclosed fields is decidable. Furthermore, the problem of generating invariant varieties is shown to be as hard as minimizing the rank of a symbolic matrix, and is therefore NPhard. We investigate symbolic linear algebra tools based on Gaussian elimination to efficiently automate the generation. The approach can, e.g., generate nontrivial algebraic invariant equations capturing the airplane behavior during takeoff or landing in longitudinal motion.
A Differential Operator Approach to Equational Differential Invariants
, 2012
"... Hybrid systems, i.e., dynamical systems combining discrete and continuous dynamics, have a complete axiomatization in differential dynamic logic relative to differential equations. Differential invariants are a natural induction principle for proving properties of the remaining differential equatio ..."
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Hybrid systems, i.e., dynamical systems combining discrete and continuous dynamics, have a complete axiomatization in differential dynamic logic relative to differential equations. Differential invariants are a natural induction principle for proving properties of the remaining differential equations. We study the equational case of differential invariants using a differential operator view. We relate differential invariants to Lie’s seminal work and explain important structural properties resulting from this view. Finally, we study the connection of differential invariants with partial differential equations in the context of the inverse characteristic method for computing differential invariants.
Generating Box Invariants
"... Box invariant sets are boxshaped positively invariant sets. We show that box invariants are computable for a large class of nonlinear and hybrid systems. The technique for computing these invariants is based on nonlinear constraint solving. This paper also shows that the class of multiaffine syst ..."
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Box invariant sets are boxshaped positively invariant sets. We show that box invariants are computable for a large class of nonlinear and hybrid systems. The technique for computing these invariants is based on nonlinear constraint solving. This paper also shows that the class of multiaffine systems, which has been used successfully for modeling and analyzing regulatory and biochemical reaction networks, can be generalized to the class of componentwise monotone and componentwise quasi monotone systems without losing any of its nice properties.