Cylindrical algebraic decomposition is one of the most important tools for computing with semi-algebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set F ⊂ R[y1, . . . , yn] we apply comprehensive triangular decomposition in order to obtain an F -invariant cylindrical decomposition of the n-dimensional complex space, from which we extract an F -invariant cylindrical algebraic decomposition of the n-dimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.
A barrier certificate can separate the state space of a considered hybrid system (HS) into safe and unsafe parts according to the safety property to be verified. Therefore this notion has been widely used in the verification of HSs. A stronger condition on barrier certificates means that less expressive barrier certificates can be synthesized. On the other hand, synthesizing more expressive barrier certificates often means high complexity. In [9], Kong et al considered how to relax the condition of barrier certificates while still keeping their convexity so that one can synthesize more expressive barrier certificates efficiently using semi-definite programming (SDP). In this paper, we first discuss how to relax the condition of barrier certificates in a general way, while still keeping their convexity. Particularly, one can then utilize different weaker conditions flexibly to synthesize different kinds of barrier certificates with more expressiveness efficiently using SDP. These barriers give more opportunities to verify the considered system. We also show how to combine two functions together to form a combined barrier certificate in order to prove a safety property under consideration, whereas neither of them can be used as a barrier certificate separately, even according to any relaxed condition. Another contribution of this paper is that we discuss how to discover certificates from the general relaxed condition by SDP. In particular, we focus on how to avoid the unsoundness because of numeric error caused by SDP with symbolic checking.
Interpolation-based techniques have been widely and successfully applied in the verification of hardware and software, e.g., in bounded-model checking, CEGAR, SMT, etc., whose hardest part is how to synthesize interpolants. Various work for discovering interpolants for propositional logic, quantifier-free fragments of first-order theories and their combinations have been proposed. However, little work focuses on discovering polynomial interpolants in the literature. In this paper, we provide an approach for constructing non-linear interpolants based on semidefinite programming, and show how to apply such results to the verification of programs by examples.Note that δ 1 can be represented as 923.42(0.90 + 0.7y − 0.1y + 0.43x ) 2 + 252.84(0.42 − 0.28y + 0.21y − 0.84x ) 2 + 461.69(−0.1 − 0.83y + 0.44y + 0.34x ) 2 + 478(−0.06 + 0.48y + 0.87y +0.03x ) 2 +578.94(x) 2 . Similarly, δ 2 and δ 3 can be represented as sums of squares also.Moreover, using the approach in [25], we can prove θ is an inductive invariant of the loop, therefore, error() will never be executed.
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