In this paper, we show that it is possible to abstract program fragments using real variables using formulas in the theory of real closed fields. This abstraction is compositional and modular. We first propose domain (in a wide class including intervals and octagons), we then show how to obtain an optimal abstraction of program fragments with respect to that domain. This abstraction allows computing optimal fixed points inside that abstract domain, without the need for a widening operator. 1

Abstract — In this paper, we present a method for computing a basin of attraction to a target region for non-linear ordinary differential equations. This basin of attraction is ensured by a Lyapunov-like polynomial function that we compute using an interval based branch-and-relax algorithm. This algorithm relaxes the necessary conditions on the coefficients of the Lyapunov-like 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 hyper-rectangles. Tests on an implementation are promising. I.

Linearity analysis determines which variables depend on which other variables and whether the dependence is linear or nonlinear. One of the many applications of this analysis is determining whether a loop involves only linear loop-carried dependences and therefore the adjoint of the loop may be reversed and fused with the computation of the original function. This paper specifies the data-flow equations that compute linearity analysis. In addition, the paper describes using linearity analysis with array dependence analysis to determine whether a loop-carried dependence is linear or nonlinear.

Abstract. We present a verification environment for imperative programs (using Hoare logic) and for functional programs (using fixpoint theory) in the frame of the Theorema system (www.theorema.org). In particular, we discuss some methods for finding the invariants of loops and of specifications of auxiliary tail recursive functions. These methods use algorithms from (polynomial) algebra and combinatorics, namely Groebner bases, variable elimination and symbolic summation (the Gosper algorithm, the technique of generating functions). The techniques are demonstrated on several examples which have been treated automatically by our implementation.

We discuss interesting properties of a general technique for inferring polynomial invariants for a subfamily of imperative loops, called the P-solvable loops, with assignments only. The approach combines algorithmic combinatorics, polynomial algebra and computational logic, and it is implemented in a new software package called Aligator. We present a collection of examples illustrating the power of the framework.

Abstract. We describe practical experiments of program verification in the frame of the Theorema system (www.theorema.org). This includes both functional programs (using fixpoint theory), as well as imperative programs (using Hoare logic). By comparing different approaches we are trying to find general schemes which are useful for practical work. The Theorema system offers facilities for working with higher-order predicate logic formulae (including various general and domain-oriented provers) and also for defining and testing algorithms both in functional and in imperative styles. We generate verification conditions as natural-style predicate logic formulae, which can be then proven by Theorema, by issuing natural-style proofs which are human–readable.

We present methods for checking the partial correctness of, respectively to optimize, imperative programs, using polynomial algebra methods, namely resultant computation and quantifier elimination (QE) by cylindrical algebraic decomposition (CAD). The results are very promising but also show that there is room for improvement of algebraic algorithms. 1