Implicit Real Vector Automata

This thesis introduces a new data structure, the Implicit Real Vector Automaton (IRVA), suited for representing symbolically polyhedra, i.e., regions of n-dimensional space defined by finite Boolean combinations of linear inequalities. IRVA can represent exactly arbitrary convex and non-convex polyhedra, including features such as open and closed boundaries, unconnected parts, and non-manifold components. In addition, they provide efficient procedures for deciding whether a point belongs to a given polyhedron, and determining the polyhedron component (vertex, edge, facet, ...) that contains a point. An advantage of IRVA is that they can easily be minimized into a canonical form, which leads to a simple and efficient test for equality between represented polyhedra. Elementary IRVA representing primitive polyhedra, such as linear
(in)equations and vector spaces are easily constructed and algorithms have been developed for computing Boolean combinations as well as projections of polyhedra represented by IRVA. These algorithms are illustrated by complete examples of executions as a support for the comprehension of their mechanisms. Another contribution is a first prototype implementation of an IRVA library, containing functions for building and manipulating arbitrary n-dimensional polyhedra. We reinforce the presentation of the implementation by discussing some design choices. Such choices include the use of exact arithmetic. Finally, experimental results are presented and discussed. These experiments pave the way to future adaptations and improvements.