In this paper, we consider an extended concept of invariant for polynomial dynamical systems (PDSs) with domain and initial condition, and establish a sound and complete criterion for checking semi-algebraic invariants (SAIs) for such PDSs. The main idea is encoding relevant dynamical properties as conditions on the high order Lie derivatives of polynomials occurring in the SAI. A direct consequence of this criterion is a relatively complete method of SAI generation based on template assumption and semi-algebraic constraint solving. Relative completeness means if there is an SAI in the form of a predefined template, then our method can indeed find one.

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.

No abstract

We formalize the theory of quantum Hoare logic (QHL) [TOPLAS 33(6),19], an extension of Hoare logic for reasoning about quantum programs. In particular, we formalize the syntax and semantics of quantum programs in Isabelle/HOL, write down the rules of quantum Hoare logic, and verify the soundness and completeness of the deduction system for partial correctness of quantum programs. As preliminary work, we formalize some necessary mathematical background in linear algebra, and define tensor products of vectors and matrices on quantum variables. As an application, we verify the correctness of Grover's search algorithm. To our best knowledge, this is the first time a Hoare logic for quantum programs is formalized in an interactive theorem prover, and used to verify the correctness of a nontrivial quantum algorithm.

No abstract

Abstract. Quantitative loop invariants are an essential element in the verification of probabilistic programs. Recently, multivariate Lagrange interpolation has been applied to synthesizing polynomial invariants. In this paper, we propose an alternative approach. First, we fix a polynomial template as a candidate of a loop invariant. Using Stengle's Positivstellensatz and a transformation to a sum-of-squares problem, we find sufficient conditions on the coefficients. Then, we solve a semidefinite programming feasibility problem to synthesize the loop invariants. If the semidefinite program is unfeasible, we backtrack after increasing the degree of the template. Our approach is semi-complete in the sense that it will always lead us to a feasible solution if one exists and numerical errors are small. Experimental results show the efficiency of our approach.

No abstract