To verify the safety of nonlinear dynamical systems based on inductive invariants,key issues include defining the most complete inductive condition and discovering an inductive invariant that satisfies the specified inductive condition. In this paper,to lay a solid foundation for future research into the safety verification of semialgebraic dynamical systems,we first establish a formal framework for evaluating the quality of continuous inductive conditions. In addition,we propose a new complete and computable inductive condition for verifying the safety of semi-algebraic dynamical systems. Compared with the existing complete and computable inductive condition,this new inductive condition can be easily adapted to achieve a set of sufficient inductive conditions with different level of conservativeness and computational complexity,which provides us with a means to trade off between the verification power and complexity. These inductive conditions can be solved by quantifier elimination and SMT solvers.
Hui KongFei HeXiaoyu SongMing GuHongyan TanJiaguang Sun
Differential privacy enables sensitive data to be analyzed in a privacy-preserving manner.In this paper,we focus on the online setting where each analyst is assigned a privacy budget and queries the data interactively.However, existing differentially private data analytics systems such as PINQ process each query independently,which may cause an unnecessary waste of the privacy budget.Motivated by this,we present a satisfiability modulo theories (SMT)-based query tracking approach to reduce the privacy budget usage. In brief,our approach automatically locates past queries that access disjoint parts of the dataset with respect to the current query to save the privacy cost using the SMT solving techniques.To improve efficiency,we further propose an op- timization based on explicitly specified column ranges to facilitate the search process.We have implemented a prototype of our approach with Z3,and conducted several sets of experiments.The results show our approach can save a considerable amount of the privacy budget and each query can be tracked efficiently within milliseconds.
