Transforming semantics by abstract interpretation

In 1997, Cousot introduced a hierarchy where semantics are related with each other by abstract interpretation. In this field we consider the standard abstract domain transformers, devoted to refine abstract domains in order to include attribute independent and relational information, respectively the reduced product and power of abstract domains, as domain operations to systematically design and compare semantics of programming languages by abstract interpretation. We first prove that natural semantics can be decomposed in terms of complementary attribute independent observables, leading to an algebraic characterization of the symmetric structure of the hierarchy. Moreover, we characterize some structural property of semantics, such as their compositionality, in terms of simple abstract domain equations. This provides an equational presentation of most well known semantics, which is parametric on the observable and structural property of the semantics, making it possible to systematically derive abstract semantics, e.g. for program analysis, as solutions of abstract domain equations

Mathematical Abstraction, Conceptual Variation and Identity

One of the key features of modern mathematics is the adoption of the abstract method. Our goal in this paper is to propose an explication of that method that is rooted in the history of the subject

A power consensus algorithm for DC microgrids

A novel power consensus algorithm for DC microgrids is proposed and analyzed. DC microgrids are networks composed of DC sources, loads, and interconnecting lines. They are represented by differential-algebraic equations connected over an undirected weighted graph that models the electrical circuit. A second graph represents the communication network over which the source nodes exchange information about the instantaneous powers, which is used to adjust the injected current accordingly. This give rise to a nonlinear consensus-like system of differential-algebraic equations that is analyzed via Lyapunov functions inspired by the physics of the system. We establish convergence to the set of equilibria consisting of weighted consensus power vectors as well as preservation of the weighted geometric mean of the source voltages. The results apply to networks with constant impedance, constant current and constant power loads.Comment: Abridged version submitted to the 20th IFAC World Congress, Toulouse, Franc