Automata-Based Symbolic Representations of Polyhedra

This work describes a 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. We also develop an algorithm for computing Boolean combinations of polyhedra represented by IRVA.