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We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…
A first step towards more reliable software is to execute each statement and each control-flow path in a method once. In this paper, we present a formal method to automatically compute test cases for this purpose based on the idea of a…
Bounded verification has proved useful to detect bugs and to increase confidence in the correctness of a program. In contrast to unbounded verification, reasoning about calls via (bounded) inlining and about loops via (bounded) unrolling…
We study discrete probabilistic programs with potentially unbounded looping behaviors over an infinite state space. We present, to the best of our knowledge, the first decidability result for the problem of determining whether such a…
The paper aims at the development of an apparatus for analysis and construction of near optimal solutions of singularly perturbed (SP) optimal controls problems (that is, problems of optimal control of SP systems) considered on the infinite…
Context. Variability-intensive programs (program families) appear in many application areas and for many reasons today. Different family members, called variants, are derived by switching statically configurable options (features) on and…
Among the biggest challenges in property-based testing (PBT) is the constrained random generation problem: given a predicate on program values, randomly sample from the set of all values satisfying that predicate, and only those values.…
Program synthesis is a class of regression problems where one seeks a solution, in the form of a source-code program, mapping the inputs to their corresponding outputs exactly. Due to its precise and combinatorial nature, program synthesis…
One of the fundamental problems of interest for discrete-time linear systems is whether its input sequence may be recovered given its output sequence, a.k.a. the left inversion problem. Many conditions on the state space geometry, dynamics,…
Tiwari proved that termination of linear programs (loops with linear loop conditions and updates) over the reals is decidable through Jordan forms and eigenvectors computation. Braverman proved that it is also decidable over the integers.…
We present necessary and sufficient conditions for the termination of linear homogeneous programs. We also develop a complete method to check termination for this class of programs. Our complete characterization of termination for such…
In order to verify programs or hybrid systems, one often needs to prove that certain formulas are unsatisfiable. In this paper, we consider conjunctions of polynomial inequalities over the reals. Classical algorithms for deciding these not…
Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of…
Mixed integer nonlinear programs (MINLPs) are arguably among the hardest optimization problems, with a wide range of applications. MINLP solvers that are based on linear relaxations and spatial branching work similar as mixed integer…
Template abstract domains allow to express more interesting properties than classical abstract domains. However, template generation is a challenging problem when one uses template abstract domains for program analysis. In this paper, we…
The paper presents a new algorithmic construction of a finite generating set of rational invariants for the rational action of an algebraic group on the affine space. The construction provides an algebraic counterpart of the moving frame…
Linear complementarity problems are a powerful tool for modeling many practically relevant situations such as market equilibria. They also connect many sub-areas of mathematics like game theory, optimization, and matrix theory. Despite…
The Monniaux Problem in abstract interpretation asks, roughly speaking, whether the following question is decidable: given a program $P$, a safety (\emph{e.g.}, non-reachability) specification $\varphi$, and an abstract domain of invariants…
Loop-invariant synthesis is the basis of program verification. Due to the undecidability of the problem in general, a tool for invariant synthesis necessarily uses heuristics. Despite the common belief that the design of heuristics is vital…
Debugging is difficult. Recent studies show that automatic bug localization techniques have limited usefulness. One of the reasons is that programmers typically have to understand why the program fails before fixing it. In this work, we aim…