Implicit Real Vector Automata

Abstract

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.

Symbolic methods and automata

“…Finite-state representations are not a panacea; however, since the size of NDD and RVA recognising linear constraints can grow linearly with the magnitude of their coe cients, the representations can become unnecessarily large. A possible solution to this problem, consisting of representing some internal structures of automata by algebraic means, is being investigated, see [11] and [17]. Another problem is that the presence of dual encodings is also a source of ine ciency.…”

Automata-Based Symbolic Representations of Polyhedra

“…It has been shown that the SCC of A are related to the polyhedral components of Π [4,3,7]. This can intuitively be explained as follows.…”

“…Second, in the particular case where one has dim(VS (s )) = dim(VS (s)) + 1, it is essential to ensure that the state s does not represent a boundary of the polyhedral component associated to s. This is done by checking that, among the words w such that s w → s , at least two of them have leading symbols −(i) and +(i) with an identical face number and opposite polarities. This tricky particular case was overlooked in [3].…”

“…The main drawback of RVA is their size that can grow linearly with the coefficients of linear constraints, which makes those symbolic representations unmanageable in some applications. This drawback is alleviated by Implicit Real-Vector Automata (IRVA), which intuitively operate on similar principles as RVA, but replace some of their unnecessarily large internal structures by more concise algebraic objects [3]. Interestingly enough, it has been shown that the RVA structures replaced in IRVA closely match the internal components of SNC representations of polyhedra, and that their reachability properties represent the incidence relation between them.…”